COMPARISON OF DESIGN OPTIMALITY CRITERIA OF REDUCED MODELS
FOR RESPONSE SURFACE DESIGNS IN A SPHERICAL DESIGN REGION
by
Boonorm Chomtee
A dissertation submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Statistics
MONTANA STATE UNIVERSITY
Bozeman, Montana
April 2003
c
COPYRIGHT
by
Boonorm Chomtee
2003
All Rights Reserved
ii
APPROVAL
of a dissertation submitted by
Boonorm Chomtee
This dissertation has been read by each member of the dissertation committee and
has been found to be satisfactory regarding content, English usage, format, citations,
bibliographic style, and consistency, and is ready for submission to the College of
Graduate Studies.
Dr. John J. Borkowski
(Signature)
Date
Approved for the Department of Statistics
Dr. Kenneth L. Bowers
(Signature)
Date
Approved for the College of Graduate Studies
Dr. Bruce R. McLeod
(Signature)
Date
iii
STATEMENT OF PERMISSION TO USE
In presenting this dissertation in partial fulfillment of the requirements for a
doctoral degree at Montana State University, I agree that the Library shall make it
available to borrowers under rules of the Library. I further agree that copying of
this dissertation is allowable only for scholarly purposes, consistent with “fair use” as
prescribed in the U. S. Copyright Law. Requests for extensive copying or reproduction
of this dissertation should be referred to Bell & Howell Information and Learning,
300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted “the
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Signature
Date
iv
ACKNOWLEDGEMENTS
First of all, I wish to thank my advisor, Dr. John J. Borkowski, for his invaluable
guidance and assistance during the preparation of this dissertation. I sincerely thank
for his understanding and patience.
In addition, I would like to thank Professor Robert J. Boik and Dr. Steve Cherry
who are my reading committee for their time, useful comments and suggestions as
well as the other members of my committee Dr. William F. Quimby and Dr. Lisa
Stanley.
Finally, a special thank to my parents for their love and support. I thank my
husband, my sisters, brothers and friends for their love and encouragement.
v
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
1. RESPONSE SURFACE METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Response Surface Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Level Fractional Factorial Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design Optimality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
5
10
14
2. RESPONSE SURFACE DESIGNS IN A SPHERICAL
DESIGN REGION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Central Composite Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Box-Behnken Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotatability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Small Composite Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plackett-Burman Composite Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hybrid Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Uniform Shell Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The X0X Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The (X0X)−1 Matrix for Symmetric Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The (X0X)−1 matrix for a CCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The (X0X)−1 matrix for a BBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The (X0X)−1 matrix for a hybrid 311B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The |X0X| for Symmetric Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spherical Prediction Variance Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
24
27
28
30
31
34
35
54
55
58
59
61
62
3. OPTIMALITY CRITERIA FOR A SPHERICAL RESPONSE SURFACE DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
Optimality Criteria for the Full Second Order Model . . . . . . . . . . . . . . . . . . . . . . . .
Design Criteria Comparison Ranking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
VIFs and the Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reduced Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimality Criteria for Reduced Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Robustness of the Response Surface Designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
68
73
80
85
88
vi
4. ROBUSTNESS OF SPHERICAL RESPONSE SURFACE DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
The Robustness of 3-Factor Response Surface Designs . . . . . . . . . . . . . . . . . . . . . . .
The Central Composite Designs (CCDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Box-Behnken Designs (BBDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Small Composite Designs (SCDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Uniform Shell Designs (UNFSDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Hybrid 310 Designs (310s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Hybrid 311A Designs (311As) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Hybrid 311B Designs (311Bs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Robustness of 4-Factor Response Surface Designs . . . . . . . . . . . . . . . . . . . . . . .
The Central Composite Designs (CCDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Box-Behnken Designs (BBDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Small Composite Designs (SCDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Plackett-Burman Composite Designs (PBCDs) . . . . . . . . . . . . . . . . . . . . .
The Uniform Shell Designs (UNFSDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Hybrid 416A Designs (416As) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Hybrid 416B Designs (416Bs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Hybrid 416C Designs (416Cs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
General Results for the Reduced Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D-Efficiencies Greater Than 100% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of Design Optimality Criteria of Reduced Models . . . . . . . . . . . . . .
103
104
127
139
143
147
152
156
161
161
196
200
206
212
216
220
224
229
233
234
5. WEIGHTED DESIGN OPTIMALITY CRITERIA FOR
SPHERICAL RESPONSE SURFACE DESIGNS. . . . . . . . . . . . . . . . . . . . . . . . . 262
Inheritance Principles for Reduced Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weighted Design Optimality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weighted Design Optimality Criteria Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . .
262
267
269
271
277
6. CONCLUSION AND FUTURE RESEARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
APPENDIX A – Tables of D, A, G, and IV Criteria
Values for 3 and 4 Factor Response Surface Designs . . . . . . . . . . . . . . . . . . . . .
APPENDIX B – D, A, G, and IV Criteria Plots for Small
Composite, Uniform Shell, and Hybrid Designs for 3 Factors . . . . . . . . . . . .
APPENDIX C – D, A, G, snd IV Criteria Plots for
Small Composite, Plackett-Burman Composite, Uniform Shell, and Hybrid Designs for 4 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX D – Programming Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
302
339
360
403
vii
LIST OF TABLES
Table
Page
1. 26−2 Design with Generators ABCE and BCDF . . . . . . . . . . . . . . . . . . . . . . . .
12
2. A 15-Point Central Composite Design
(CCD) for
√
Three Factors (K = 3) and α = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3. Box-Behnken Designs (BBDs) for 3 and 4 Factors . . . . . . . . . . . . . . . . . . . . . . .
26
4. Small Composite Designs (SCDs) for 3 and 4 Factors . . . . . . . . . . . . . . . . . . .
29
5. Plackett-Burman Composite Design (PBCD) for 4 Factors. . . . . . . . . . . . . .
31
6. Hybrid Designs for 3 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
7. Hybrid Designs for 4 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
8. Uniform Shell Designs (UNFSDs) for 3 and 4 Factors . . . . . . . . . . . . . . . . . . .
34
9. The Optimality Criteria for K = 3 Design Variables . . . . . . . . . . . . . . . . . . . . .
66
10. The Optimality Criteria for K = 4 Design Variables . . . . . . . . . . . . . . . . . . . . .
67
11. Design Optimality Criteria Comparison Ranking for
K = 3, n0 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
12. Design Optimality Criteria Comparison Ranking for
K = 3, n0 = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
13. Design Optimality Criteria Comparison Ranking for
K = 4, n0 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
14. Design Optimality Criteria Comparison Ranking for
K = 4, n0 = 2, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
15. VIFs for the 3-Factor Response Surface Designs . . . . . . . . . . . . . . . . . . . . . . . . .
75
16. VIFs for the 4-Factor Response Surface Designs . . . . . . . . . . . . . . . . . . . . . . . . .
76
viii
17. Mean VIFs, Criteria Values, and Ranks for 3-Factor,
11-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
18. Mean VIFs, Criteria Values, and Ranks for 3-Factor,
13-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
19. Mean VIFs, Criteria Values, and Ranks for 3-Factor,
15-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
20. Mean VIFs, Criteria Values, and Ranks for 4-Factor,
17-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
21. Mean VIFs, Criteria Values, and Ranks for 4-Factor,
19-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
22. Mean VIFs, Criteria Values, and Ranks for 4-Factor,
21-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
23. Mean VIFs, Criteria Values, and Ranks for 4-Factor,
23-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
24. Mean VIFs, Criteria Values, and Ranks for 4-Factor,
25-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
25. Mean VIFs, Criteria Values, and Ranks for 4-Factor,
27-Point Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
26. Reduced Models (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
27. Reduced Models (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
28. Q-Paths for K = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
29. Q-Paths for K = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
30. C-Paths for K = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
31. C-Paths for K = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
32. The Optimality Criteria Across the Reduced Models
for the CCD (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
ix
33. The Optimality Criteria Across the Reduced Models
for the BBD (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
34. The Optimality Criteria Across the Reduced Models
for the SCD (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
35. The Optimality Criteria Across the Reduced Models
for the UNFSDs (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
36. The Optimality Criteria Across the Reduced Models
for the 310s (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
37. The Optimality Criteria Across the Reduced Models
for the 311As (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
38. The Optimality Criteria Across the Reduced Models
for the 311Bs (K = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
39. The Optimality Criteria Across the Reduced Models
for the CCD (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
40. The Optimality Criteria Across the Reduced Models
for the BBD (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
41. The Optimality Criteria Across the Reduced Models
for the SCD (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
42. The Optimality Criteria Across the Reduced Models
for the PBCD (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
43. The Optimality Criteria Across the Reduced Models
for the UNFSD (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
44. The Optimality Criteria Across the Reduced Models
for the 416A (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
45. The Optimality Criteria Across the Reduced Models
for the 416B (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
46. The Optimality Criteria Across the Reduced Models
for the 416C (K = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
x
47. The Number of Models the D, A, and G-Criteria
Values are Greater Than (for dv = 3), or Smaller
Than (for dv = 1, 2) the Full Second-Order Model
Criteria Values when K = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
48. The Number of Models the D, A, and G-Criteria
Values are Greater Than (for dv = 4), or Smaller
Than (for dv = 1, 2, and 3) the Full Second-Order
Model Criteria Values when K = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
49. Comparisons of D, A, G, and IV Criteria for K = 3,
N = 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
50. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
3, N = 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
51. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 3, N = 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
52. Design Criteria Comparison Ranking for K = 3, N = 11 . . . . . . . . . . . . . . . . 238
53. Comparisons of D, A, G, and IV Criteria for K = 3,
N = 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
54. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
3, N = 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
55. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 3, N = 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
56. Design Criteria Comparison Ranking for K = 3, N = 13 . . . . . . . . . . . . . . . . 241
57. Comparisons of D, A, G, and IV Criteria for K = 3,
N ≥ 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
58. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
3, N ≥ 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
xi
59. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 3, N ≥ 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
60. Design Criteria Comparison Ranking for K = 3, N = 15 . . . . . . . . . . . . . . . . 246
61. Comparisons of D, A, G, and IV Criteria for K = 4,
N = 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
62. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
4, N = 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
63. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 4, N = 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
64. Design Criteria Comparison Ranking for K = 4, N = 17 . . . . . . . . . . . . . . . . 250
65. Comparisons of D, A, G, and IV Criteria for K = 4,
N = 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
66. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
4, N = 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
67. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 4, N = 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
68. Design Criteria Comparison Ranking for K = 4, N = 19 . . . . . . . . . . . . . . . . 254
69. Comparisons of D, A, G, and IV Criteria for K = 4,
N = 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
70. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
4, N = 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
71. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 4, N = 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
xii
72. Comparisons of D, A, G, and IV Criteria for K = 4,
N = 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
73. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
4, N = 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
74. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 4, N = 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
75. Design Criteria Comparison Ranking for K = 4, N =
21 and 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
76. Comparisons of D, A, G, and IV Criteria for K = 4,
N = 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
77. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One Squared Term) for K =
4, N = 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
78. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two Squared Terms) for
K = 4, N = 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
79. Design Criteria Comparison Ranking for K = 4, N = 25 . . . . . . . . . . . . . . . . 259
80. Comparisons of D, A, G, and IV Criteria for K = 4,
N = 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
81. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least One squared Term) for K =
4, N = 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
82. Comparisons of D, A, G, and IV Criteria (Across
Models with at Least Two squared Terms) for
K = 4, N = 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
83. Design Criteria Comparison Ranking for K = 4, N = 27 . . . . . . . . . . . . . . . . 261
84. Optimality Criteria of a 15-Point CCD for WH Models, K = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
xiii
85. Optimality Criteria of a 15-Point CCD for SH Models,
K = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
86. The WH and SH Model Probabilities for a 3 Factor
15-Point CCD with pl = .9, p1 = .4, p2 = .5, and
pq = .7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
87. Weighted Optimality Criteria for the 3-Factor 15Point CCD Across WH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
88. Coefficients of WH Dw Models for 3-Factor Designs . . . . . . . . . . . . . . . . . . . . . 279
89. Coefficients of WH Aw Models for 3-Factor Designs . . . . . . . . . . . . . . . . . . . . . 280
90. Coefficients of WH Gw Models for 3-Factor Designs . . . . . . . . . . . . . . . . . . . . . 281
91. Coefficients of WH IVw Models for 3-Factor Designs. . . . . . . . . . . . . . . . . . . . . 282
92. Coefficients of SH Ds Models for 3-Factor Designs . . . . . . . . . . . . . . . . . . . . . . . 283
93. Coefficients of SH As Models for 3-Factor Designs . . . . . . . . . . . . . . . . . . . . . . . 284
94. Coefficients of SH Gs Models for 3-Factor Designs . . . . . . . . . . . . . . . . . . . . . . . 285
95. Coefficients of SH IVs Models for 3-Factor Designs . . . . . . . . . . . . . . . . . . . . . . 286
96. Coefficients of WH Dw Models for 4-Factor Designs . . . . . . . . . . . . . . . . . . . . . 287
97. Coefficients of WH Aw Models for 4-Factor Designs . . . . . . . . . . . . . . . . . . . . . 288
98. Coefficients of WH IVw Models for 4-Factor Designs. . . . . . . . . . . . . . . . . . . . . 289
99. Coefficients of SH Ds Models for 4-Factor Designs . . . . . . . . . . . . . . . . . . . . . . . 290
100.Coefficients of SH As Models for 4-Factor Designs . . . . . . . . . . . . . . . . . . . . . . . 291
101.Coefficients of SH IVs Models for 4-Factor Designs . . . . . . . . . . . . . . . . . . . . . . 292
102.Full Model Optimality Criteria for Small Response
Surface Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
xiv
103.Weighted Optimality Criteria for Small Response Surface Designs Across WH Models with pl = .9,
p1 = .4, p2 = .5, and pq = .7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
104.Weighted Optimality Criteria for Small Response Surface Designs Across SH Models with pl = .8,
p2 = .5, and pq = .5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
xv
LIST OF FIGURES
Figure
Page
1. Spherical Coordinates for K = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2. Example of the D-Efficiency Plot (Plotting Symbol
= Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3. Example of the D-Efficiency Plot (Plotting Symbol
= C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4. D-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5. The Change in D-Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6. D-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7. The Change in D-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . 114
8. A-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9. The Change in A-Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10. A-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11. The Change in A-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . 118
12. G-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
13. The Change in G-Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 120
xvi
14. G-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
15. The Change in G-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . 122
16. IV -Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
17. The Change in IV -Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 124
18. IV -Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
19. The Change in IV -Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor CCDs . . . . . . . . . . . . . . . . . . . 126
20. D-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
21. The Change in D-Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . . . . . . . . 131
22. D-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
23. The Change in D-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . 132
24. A-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
25. The Change in A-Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . . . . . . . . 133
26. A-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
27. The Change in A-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . 134
xvii
28. G-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
29. The Change in G-Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . . . . . . . . 135
30. G-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
31. The Change in G-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . 136
32. IV -Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
33. The Change in IV -Efficiency Plots by Reduction of
Squared Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . . . . . . . . 137
34. IV -Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
35. The Change in IV -Efficiency Plots by Reduction of
Cross-Product Terms in Model for 3 Factor BBDs . . . . . . . . . . . . . . . . . . . 138
36. D-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
37. D-Efficiency Plots for 4 Factor CCDs for dv = 1, 2,
and 3 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
38. The Change in D-Efficiency Plots by Reduction of
Squared Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 170
39. A-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
40. A-Efficiency Plots for 4 Factor CCDs for dv = 1, 2,
and 3 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
41. The Change in A-Efficiency Plots by Reduction of
Squared Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 173
xviii
42. G-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol = Q-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
43. G-Efficiency Plots for 4 Factor CCDs for dv = 1, 2,
and 3 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
44. The Change in G-Efficiency Plots by Reduction of
Squared Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 176
45. IV -Efficiency Plots for 4 Factor CCDs for dv = 4
(Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
46. IV -Efficiency Plots for 4 Factor CCDs for dv = 1, 2,
and 3 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
47. The Change in IV -Efficiency Plots by Reduction of
Squared Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . 179
48. D-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
49. D-Efficiency Plots for 4 Factor CCDs for dv = 4 → 3
and 4 → 3 → 2 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . 181
50. D-Efficiency Plots for 4 Factor CCDs for dv = 3,
4 → 3 and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path) . . . . . . . . . . . . 182
51. The Change in D-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . 183
52. A-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
53. A-Efficiency Plots for 4 Factor CCDs for dv = 4 → 3
and 4 → 3 → 2 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . 185
54. A-Efficiency Plots for 4 Factor CCDs for dv = 3,
4 → 3 and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path) . . . . . . . . . . . . 186
55. The Change in A-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . 187
xix
56. G-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
57. G-Efficiency Plots for 4 Factor CCDs for dv = 4 → 3
and 4 → 3 → 2 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . 189
58. G-Efficiency Plots for 4 Factor CCDs for dv = 3,
4 → 3 and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path) . . . . . . . . . . . . 190
59. The Change in G-Efficiency Plots by Reduction of
Cross-Product Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . 191
60. IV -Efficiency Plots for 4 Factor CCDs for dv = 4
(Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
61. IV -Efficiency Plots for 4 Factor CCDs for dv = 4 → 3
and 4 → 3 → 2 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . 193
62. IV -Efficiency Plots for 4 Factor CCDs for dv = 3,
4 → 3 and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path) . . . . . . . . . . . . 194
63. The Change in IV -Efficiency Plots by Reduction of
Cross-Product Terms in Model for 4 Factor CCDs . . . . . . . . . . . . . . . . . . . 195
64. The D, A, G, and IV -Criteria Comparison Plots for
13-Point 3-Factor Designs (Plotting Symbol = q) . . . . . . . . . . . . . . . . . . . . 242
65. The A and IV -Criteria Comparison Plots for 15-Point
3-Factor Designs (Plotting Symbol = q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
66. The D-Criterion Comparison Plots for 17-Point 4Factor Designs (Plotting Symbol = q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
67. The D-Criterion Comparison Plot for 19-Point 4Factor Designs (Plotting Symbol = q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
68. The IV -Criterion Comparison Plot for 25-Point 4Factor Designs (Plotting Symbol = q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
69. The D, A, G, and IV -Criteria Plots for 3 Factor
SCDs (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
xx
70. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 3 Factor
SCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
71. The D, A, G, and IV -Criteria Plots for 3 Factor
SCDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
72. The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
3 Factor SCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
73. The D, A, G, and IV -Criteria Plots for 3 Factor
UNFSDs (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
74. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 3 Factor
UNFSDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
75. The D, A, G, and IV -Criteria Plots for 3 Factor
UNFSDs (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
76. The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
3 Factor UNFSDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
77. The D, A, G, and IV -Criteria Plots for 3 Factor 310
Designs (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
78. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 3 Factor
310 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
79. The D, A, G, and IV -Criteria Plots for 3 Factor 310
Designs (Plotting Symbol = C-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
80. The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
3 Factor 310 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
81. The D, A, G, and IV -Criteria Plots for 3 Factor 311A
Designs (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
xxi
82. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 3 Factor
311A Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
83. The D, A, G, and IV -Criteria Plots for 3 Factor 311A
Designs (Plotting Symbol = C-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
84. The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
3 Factor 311A Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
85. The D, A, G, and IV -Criteria Plots for 3 Factor 311B
Designs (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
86. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 3 Factor
311B Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
87. The D, A, G, and IV -Criteria Plots for 3 Factor 311B
Designs (Plotting Symbol = C-Path). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
88. The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
3 Factor 311B Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
89. The D, A, G, and IV -Criteria Plots for 4 Factor
SCDs for dv = 4 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . . . 361
90. The D, A, G, and IV -Criteria Plots for 4 Factor
SCDs for dv = 1, 2, and 3 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . 362
91. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 4 Factor
SCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
92. The D, A, G, and IV -Criteria Plots for 4 Factor
SCDs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . 364
93. The D, A, G, and IV -Criteria Plots for 4 Factor
SCDs for dv = 4 → 3 and 4 → 3 → 2 (Plotting
Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
xxii
94. The D, A, G, and IV -Criteria Plots for 4 Factor
SCDs for dv = 3, 4 → 3, and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
95. The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
4 Factor SCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
96. The D, A, G, and IV -Criteria Plots for 4 Factor
PBCDs for dv = 4 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . 368
97. The D, A, G, and IV -Criteria Plots for 4 Factor
PBCDs for dv = 1, 2, and 3 (Plotting Symbol =
Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
98. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 4 Factor
PBCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
99. The D, A, G, and IV -Criteria Plots for 4 Factor
PBCDs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . 371
100.The D, A, G, and IV -Criteria Plots for 4 Factor
PBCDs for dv = 4 → 3 and 4 → 3 → 2 (Plotting
Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
101.The D, A, G, and IV -Criteria Plots for 4 Factor
PBCDs for dv = 3, 4 → 3, and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
102.The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
4 Factor PBCDs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
103.The D, A, G, and IV -Criteria Plots for 4 Factor
UNFSDs for dv = 4 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . 375
104.The D, A, G, and IV -Criteria Plots for 4 Factor
UNFSDs for dv = 1, 2, and 3 (Plotting Symbol =
Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
xxiii
105.The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 4 Factor
UNFSDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
106.The D, A, G, and IV -Criteria Plots for 4 Factor
UNFSDs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . 378
107.The D, A, G, and IV -Criteria Plots for 4 Factor
UNFSDs for dv = 4 → 3 and 4 → 3 → 2 (Plotting
Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
108.The D, A, G, and IV -Criteria Plots for 4 Factor
UNFSDs for dv = 3, 4 → 3, and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
109.The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
4 Factor UNFSDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
110.The D, A, G, and IV -Criteria Plots for 4 Factor 416A
Designs for dv = 4 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . 382
111.The D, A, G, and IV -Criteria Plots for 4 Factor 416A
Designs for dv = 1, 2, and 3 (Plotting Symbol =
Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
112.The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 4 Factor
416A Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
113.The D, A, G, and IV -Criteria Plots for 4 Factor 416A
Designs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . 385
114.The D, A, G, and IV -Criteria Plots for 4 Factor 416A
Designs for dv = 4 → 3 and 4 → 3 → 2 (Plotting
Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
115.The D, A, G, and IV -Criteria Plots for 4 Factor 416A
Designs for dv = 3, 4 → 3, and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
xxiv
116.The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
4 Factor 416A Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
117.The D, A, G, and IV -Criteria Plots for 4 Factor 416B
Designs for dv = 4 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . 389
118.The D, A, G, and IV -Criteria Plots for 4 Factor 416B
Designs for dv = 1, 2, and 3 (Plotting Symbol =
Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
119.The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 4 Factor
416B Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
120.The D, A, G, and IV -Criteria Plots for 4 Factor 416B
Designs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . 392
121.The D, A, G, and IV -Criteria Plots for 4 Factor 416B
Designs for dv = 4 → 3 and 4 → 3 → 2 (Plotting
Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
122.The D, A, G, and IV -Criteria Plots for 4 Factor 416B
Designs for dv = 3, 4 → 3, and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
123.The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
4 Factor 416B Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
124.The D, A, G, and IV -Criteria Plots for 4 Factor 416C
Designs for dv = 4 (Plotting Symbol = Q-Path) . . . . . . . . . . . . . . . . . . . . . 396
125.The D, A, G, and IV -Criteria Plots for 4 Factor 416C
Designs for dv = 1, 2, and 3 (Plotting Symbol =
Q-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
126.The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared Terms in Models for 4 Factor
416C Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
127.The D, A, G, and IV -Criteria Plots for 4 Factor 416C
Designs for dv = 4 (Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . 399
xxv
128.The D, A, G, and IV -Criteria Plots for 4 Factor 416C
Designs for dv = 4 → 3 and 4 → 3 → 2 (Plotting
Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
129.The D, A, G, and IV -Criteria Plots for 4 Factor 416C
Designs for dv = 3, 4 → 3, and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
130.The Change in D, A, G, and IV -Criteria Plots by
Reduction of Cross-Product Terms in Models for
4 Factor 416C Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
xxvi
ABSTRACT
In this dissertation, the major objective is to compare 3 and 4 factor response
surface designs in a spherical design region by studying design optimality criteria (D,
A, G, and IV -criteria) over sets of reduced models. Hence, theoretical and computational details of evaluating optimality criteria for reduced models for response
surface designs in a spherical design region have been described. Specifically, robustness results of the spherical response surface designs and the comparison of design
optimality criteria of the response surface designs across the full second-order model
and sets of reduced models for 3 and 4 design variables based on the four optimality
criteria (D, A, G, and IV -criteria) are presented.
Also, new types of D, A, G, and IV optimality criteria for response surface designs
in a spherical design region are developed by using prior probability assignment to
model effects (for some specified values of pl , pq , p1 , and p2 ). The four new D, A, G,
and IV optimality criteria will be referred to as weighted design optimality criteria.
The weighted design optimality criteria of the response surface designs across the
weak heredity and strong heredity reduced models for 3 and 4 design variables are
evaluated.
1
CHAPTER 1
RESPONSE SURFACE METHODOLOGY
Introduction
In their book, Response Surface Methodology, Myers and Montgomery [45] defined
response surface methodology (RSM) as a collection of statistical and mathematical
techniques useful for developing, improving, and optimizing processes. The response
surface procedures involve experimental strategy, mathematical methods, and statistical inference and when combined, enable the experimenter to make an efficient
empirical exploration of the system or process of interest. Response surface methodology also has important applications in the design, development, and formulation of
new products, as well as in the improvement of existing product designs.
In 1976, Myers [43] stated in his book, Response Surface Methodology, that the
primary objective of RSM is to aid the statistician and other users of statistics in
applying response surface procedures to appropriate problems in many technical fields.
Historically, according to A.I. Khuri and J.A. Cornell [32]:
The roots of response surface methodology (RSM) can be traced back to the
works of J. Wishart, C.P. Winsor, E.A. Mitscherlich, F. Yates, and others
in the early 1930s or even earlier. However, it was not until 1951 that RSM
was formally developed by G.E.P. Box and K.B. Wilson and other colleagues
at Imperial Chemical Industries in England. Their objective was to explore
relationships such as those between the yield of a chemical process and a
set of input variables presumed to influence the yield. Since the pioneering
work of Box and his co-workers, RSM has been successfully used and applied
2
in many diverse fields such as chemical engineering, industrial development
and process improvement, agricultural and biological research, even computer
simulation, to name just a few.
Myers [43] adds:
An important textbook, written by a team of chemists and statisticians and
edited by O.L. Davies, contains a chapter entitled “The Determinations of
Optimum Conditions” which deals with the exploration of response surfaces.
Many other papers have been published on this topic; among them are articles which contribute to the theory, and accounts which show the successful
application of known RSM techniques in such areas as chemistry, engineering, biology, agronomy, textiles, the food industry, education, psychology, and
others . . . . At the outset, early workers in the response surface area actually
introduced little in the way of new statistical or mathematical techniques in
the response surface analysis. Rather, the set of methods represented an ingenious common sense approach to problem solving, coupled with the use of
reasonably well-known statistical and mathematical methods.
RSM is useful in the solution of many types of industrial problems. Generally,
these problems are classified into three categories (Myers and Montgomery [45]):
1. Mapping a Response Surface Over a Particular Region of Interest. If the true
unknown response function has been approximated over a region around the current
operating conditions with a suitable fitted response surface, then the experimenter
can predict in advance the changes in response that will result from any readjustments
to process variables.
2. Optimization of the Response. In the industrial world, a very important problem is to determine the conditions that optimize the process. A second-order model
could be used to approximate the response in a narrow region and from examination
of this approximating response surface the optimum levels or condition for process
variables could be chosen.
3
3. Selection of Operating Conditions to Achieve Specifications or Customer Requirements. In most response surface problems there are several responses that must
be simultaneously considered. For example, in a chemical process, suppose that yield,
cost and concentration are responses. We would like to maintain a high yield, while
simultaneously keeping the cost low and satisfying customer-imposed specifications
for concentration.
Suppose the problem faced by an experimenter is the exploration and optimization
of a response surface, where, the response variable of interest is y and there is a set
of independent variables x1 , x2 , . . . , xK . Myers and Montgomery [45] stated that in
some systems the nature of the relationship between y and the x’s might be known
“exactly” based on the underlying engineering, chemical, or physical principles. For
this case, we could write a model of the form y = g(x1 , x2 , . . . , xK ) + , where the
term in the model represents the “error” in the system. This type of relationship is
called a mechanical model.
However, we consider the more common situation where the experimenter is
concerned with a product, process, or system involving a response variable of interest y that depends on the controllable independent variables x1 , x2 , . . . , xK but
the underlying mechanism is not fully understood. In this situation, the experimenter must approximate the unknown function g with an appropriate empirical
model y = f (x1 , x2 , . . . , xK ) + . This empirical model is called a response surface
model.
4
Like the mechanical model g, the form of the true response function f is unknown
and is a term that represents other sources of variability not accounted for in f such
as measurement error on the response, other sources of variation that are inherent in
the process or system ( background noise, or common cause variation in the language
of statistical process control), the effect of other variables, and so on. We will treat
as an error, often assuming it to follow a normal distribution with mean zero and
variance σ 2 . If E() = 0, then
E(y) = E[f (ξ1 , ξ2 , . . . , ξK )] + E() = η
where η = f (ξ1 , ξ2 , . . . , ξK ).
(1.1)
The variables ξ1 , ξ2 , . . . , ξK in Equation 1.1 are usually called the natural variables,
because they are expressed in the natural units of measurement, such as time (hr),
pounds per square inch (psi), etc. In RSM, it is convenient to transform the natural
variables to coded variables x1 , x2 , . . . , xK , where these coded variables are dimensionless each with mean zero and the same range or standard deviation (Myers and
Montgomery [45]). Without loss of generality, the true response function 1.1 can be
written in terms of the coded variables as
η = f (x1 , x2 , . . . , xK ).
(1.2)
5
Response Surface Designs
Although the true response functional form is, in general, unknown, we assume
that it can be approximated. Among the various model fitting techniques, the least
squares criterion is used most often to fit the empirical model. A polynomial function
is frequently employed as an approximating model over the experimental region. Usually, the first-order model or main effects model is likely to be appropriate in some
relatively small region of the independent variable space in a location where there
is little curvature. If the approximating function is the first-order model, it can be
written as:
y = β0 +
K
X
β i xi + i=1
where the β’s are parameter coefficients, y is the measured response and is an error
term which accounts for random error and the deviation between the true model and
the first-order polynomial. This model has K + 1 parameters. If there is interaction
between these variables, the approximating model, referred to as the interaction model
can be written as:
y = β0 +
K
X
β i xi +
i=1
This model has 1 + K +
K
2
=1+
K
X
βij xi xj + .
i<j
K+1
2
parameters.
If the true response surface has curvature, the first-order and interaction models
are inadequate and may suffer from significant lack-of-fit because the curvature is
6
ignored. In this situation, a second-order model will likely be required as an approximating model. The full second-order model has the form:
y = β0 +
K
X
β i xi +
i=1
This model has 1 + K + K +
K
2
K
X
βii x2i
i=1
=
K+2
2
+
K
X
βij xi xj + .
(1.3)
1≤i<j
parameters.
It is noted that there is a close connection between RSM and linear regression
analysis. For example, in linear regression analysis, suppose the model in Equation
1.3 is considered where the β’s are a set of unknown parameters. To estimate the
values of β, we must collect data on the system we are studying. Regression analysis
is a branch of statistical model building that uses these data to estimate the β’s.
Because, in general, polynomial models are linear functions of the unknown β’s, we
refer to the technique as linear regression analysis. Also, we will see that careful
planning is very important in the data collection phase in a response surface study.
The experimental designs used in this phase are called response surface designs (Myers
and Montgomery [45]).
Myers and Montgomery [45] also stated that the second-order model is widely
used in RSM for several reasons. Among these are the following:
1. The second-order model is very flexible. It can take on a variety of functional
forms, so it will often work well as an approximating model to the true response
surface model.
2. It is easy to estimate the parameters (β’s) in the second-order model. The
method of least squares can be used for this purpose.
7
3. There is considerable practical experience indicating that second-order models
work well in solving real response surface problems.
In addition, Atkinson and Donev [1] mentioned in the book, Optimum Experimental Designs that from previous experience, it was expected that a second-order
polynomial model would adequately describe the response and it is appropriate in
the region of a maximum (minimum) of the response. Once the empirical model is
fit, answering questions concerning the model adequacy, influential independent variables, a suitable operation region for the independent variables, or the optimization
of a process can be addressed directly using RSM.
Myers and Montgomery [45] mentioned that most applications of response surface
methodology are classified into three phases as follows:
Phase zero: Generate a list of factors or independent variables that are potentially important in modeling the response in a response surface study. This leads to
an experimental design to investigate these factors with the goal of eliminating the
unimportant factors. This type of experiment is called a screening experiment. Its
objective is to reduce the list of candidate variables or factors so that subsequent
experiments will be more efficient and require fewer runs or tests. Two common
screening designs are the Plackett-Burman designs [37, 47] and two-level fractional
factorial designs which will be discussed in the next section.
Phase one: When the important independent variables are identified, the experimenter’s objective is to determine if the current levels of the independent variables
8
yield a value of the response that is near the optimum. If the current levels of the independent variables are not consistent with optimum performance, the experimenter
must determine a set of adjustments to the process variables that will move the process toward the optimum. If the optimization goal is to maximize (minimize) the
response, this phase makes considerable use of the first-order model and an optimization technique called the method of steepest ascent (descent), a method whereby the
experimenter proceeds sequentially along the path of maximum increase (decrease) in
response. For more detailed information of the method of steepest ascent (descent),
see Box and Draper [15], Myers [43], and Myers and Montgomery [45].
Phase two: When the process is near the optimum, the experimenter usually
wants a model that will accurately approximate the true response function within a
relatively small region around the optimum. Since the true response surface usually
exhibits curvature near the optimum, a second-order model (or perhaps some higherorder polynomial) will be used. Once an appropriate approximating model has been
obtained, this model may be analyzed using canonical analysis and ridge analysis
to determine the optimum conditions for the process which minimize or maximize a
response. For more information of canonical analysis and ridge analysis, see Myers
[43], Myers and Montgomery [45], and Khuri and Cornell [32].
Myers [43] mentioned the impact of RSM first became apparent in the chemical
industry, where nearly all process-oriented problems involve an optimization phase.
This phase uses experimentation to find levels of the independent variables that give
9
rise to a desirable process yield. Therefore, this optimization activity usually involves
determining the type of experimental plan to be used. Once it is decided what are
the proper ranges for experimental variables, the task remains to decide what combinations of variable levels should be used in the experiment taking into account that
each observation involves a certain amount of cost and effort. It is in the experimental design area of the methodology that the pioneering researchers made their
contribution.
Since the early RSM publications presenting examples on optimization, the general techniques have been refined considerably through work which has led to better
experimental plans. Imaginative criteria for choosing experimental runs have been
established and the resulting plans have been used effectively by research workers. In
some cases, these experimental response surface designs are simply augmentations of
the well-known factorial designs. In other cases, the designs presented are reasonably
simple geometric configurations of design points or experimental runs in the space of
the variables which are pertinent to the system.
Myers and Montgomery [45] also described the minimum properties of response
surface designs for fitting a second-order polynomial model as follows:
1. At least three levels of each design variable.
2. At least 1 + 2K + K(K − 1)/2 =
K+2
2
distinct design points, so the
parameters of a second-order polynomial model can be estimated.
K+2
2
10
The sequential experimental process for these three phases is usually performed
within some region of the independent variable space called the operability region,
i.e., the region over which experiments could be conducted. For more information
on response surface methodology, see Box and Draper [15], Khuri and Cornell [32],
Myers [43], and Myers and Montgomery [45].
Two-Level Fractional Factorial Designs
Because the two-level factorial designs and fractional-factorial designs play a major role in the variable screening process as part of phase zero of RSM and as a
component for many second-order response surface designs, we review the two-level
fractional factorial designs.
The 2K factorial design is a K-factor factorial design with each of the K factors
having two levels. Hence, a statistical model for a 2K design could include K main
effects,
K
2
two-factor interactions,
K
3
three-factor interactions, . . ., and one K-
factor interaction. That is, for a full 2K factorial design, the “complete” model
contains 2K − 1 effects. See Myers and Montgomery [45] for a discussion of 2K
factorial designs including their construction, analysis, and orthogonal blocking.
For 2K factorial designs, the number of runs required for a complete replicate of
the design rapidly increases as the number of factors K increases. Furthermore, it
often happens that the experimenter does not have enough available resources to run
the full factorial design. Thus, if the experimenter can reasonably assume that certain
11
high-order interactions are negligible, then information on the main effects and loworder interactions may be obtained by running a fraction of size 2K−P , (P ≥1) of the
full factorial design. This design is called a 2K−P fractional factorial of the 2K design
or a 1/2P fraction of the 2K design.
When selecting a 1/2P fraction, we want to select design points that allow estimation of the effects of interest. Generation of such a design uses P high order
interactions. The P interactions used to generate a 1/2P fraction are called the
generators of the fractional factorial design. Design generation begins with the full
factorial design for K − P factors. If the factor levels are coded as 0 and 1, then the
P generators determine the levels of the remaining P factors by equating them with
certain interactions. The levels can be written as the sum (mod 2) of the columns
corresponding to factors comprising that interaction. Or, equivalently, if the factor
levels are coded as +1 and −1, the P generators can be written as the product of
the columns corresponding to factors comprising that interaction as shown in Table
1. For example, suppose we want to generate a 26−2 design with factors A, B, C, D,
E, F . Let factors A, B, C, D correspond to the columns defining a 24 design. The
remaining two factors E and F are defined by equating or confounding them with
the ABC and BCD three-factor interactions, respectively. Thus, we write E = ABC
and F = BCD. Then, the defining relation is given by
I = ABCE = BCDF = ADEF
where I is the identity column (or in matrix form, I is a column of ones).
12
Table 1. 26−2 Design with Generators ABCE and BCDF .
A
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
B
−
−
+
+
−
−
+
+
−
−
+
+
−
−
+
+
C
−
−
−
−
+
+
+
+
−
−
−
−
+
+
+
+
D
−
−
−
−
−
−
−
−
+
+
+
+
+
+
+
+
E = ABC
−
+
+
−
+
−
−
+
−
+
+
−
+
−
−
+
F = BCD
−
−
+
+
+
+
−
−
+
+
−
−
−
−
+
+
Note: the column products, ABCE, BCDF , and ADEF = I, where I is a column
of ones. In this case, ABCE and BCDF are the generators of the defining relation
and ADEF is their generalized interaction. A generalized interaction is determined
by taking the product of columns of generators. For this 26−2 of design,
ABCE · BCDF = AB 2 C 2 DEF = ADEF.
In general for any 2K−P design, there are P generators and 2P − P − 1 generalized
interactions formed by all two-way, three-way, . . . , P -way products of the set of
generators. An element in the defining relation will be called a word. Therefore,
there are 2P − 1 words in the defining relation. The factors in a word will be called
letters, and the number of letters in a word will be called the length of the word.
13
Since only 1/2P of the full factorial design is run, each of the 2K effects (including
the intercept) is aliased with 2P − 1 other effects. That is, estimation of aliased effects
are calculated identically and cannot be separated from each other. For example, if
I = ADEF , when main effects A, D, E, F are estimated, in reality the A + DEF ,
D + AEF , E + ADF and F + ADE combined effects are actually estimated. In
other words, A and DEF , D and AEF , E and ADF , and F and ADE are aliases.
Moreover, the alias structure of a 2K−P design is associated with the resolution of the
design.
A design is of Resolution R if no p-factor effect is aliased with another effect
containing less than R − p factors. In terms of the defining relation, the Resolution
R of a 2K−P fractional factorial design is the length of the shortest word in the
defining relation. For example, the 27−2 design with defining relation I = ABCDE =
CDEF G = ABF G is of resolution IV because the shortest word is of length four.
Designs of resolution III, IV, and V are particularly important. The definitions of
these designs are as follows:
1. Resolution III designs are designs in which (i) no main effect is aliased with any
other main effect and (ii) some or all main effects are aliased with two-factor
interactions and two-factor interactions may be aliased with each other.
2. Resolution IV designs are designs in which (i) no main effect is aliased with
any other main effect or two-factor interaction and (ii) some or all two-factor
interactions are aliased with each other.
14
3. Resolution V designs are designs in which (i) no main effect or two-factor interaction is aliased with any other main effect or two-factor interaction and (ii)
some or all two-factor interactions are aliased with three-factor interactions.
Therefore, a fractional factorial design is often employed when a restriction is placed
on the amount of available resources. For example, Resolution III and IV designs are
often used as screening designs. Moreover, the highest possible resolution of a fractional factorial design is desired because the higher the resolution, the less restrictive
the assumptions that are required regarding which interactions are negligible. As resolution increases, the number of aliased effects decreases, so that an improved analysis
of the data is obtained. For more information on two-level fractional factorial designs,
see Montgomery [42], Myers and Montgomery [45], and Box and Draper [15]. For a
catalog of two-level fractional factorial designs, see Dey [21] and National Bureau of
Standards Applied Mathematics Series 48 [53].
Design Optimality Criteria
When an experimenter has to decide which experimental design should be run,
beyond considering the physical, time, money and the design size constraints, design
optimality criteria are often used to evaluate a proposed experimental design. Box
and Draper [15] noted that orthogonality was an important design principle when
R.A. Fisher and F. Yates developed the first full and fractional factorial designs.
For a response surface design used to fit a second-order model, at least three levels
15
of each design variable are required. Hence, to require orthogonality for a response
surface design is also to require the response surface design to have a large number of
runs when the second-order model is to be fit. For example, although 3K designs are
orthogonal designs, they also require 3K design points. Because of the impracticality
of such large design sizes, alternative criteria to orthogonality, called design optimality
criteria, were developed to evaluate and compare response surface designs.
Design optimality criteria are primarily concerned with “optimal properties” of
the X0X matrix for the design matrix X. By studying the optimality criteria, the
adequacy of a proposed experimental design can be assessed prior to running it. In
addition, if several alternative designs are proposed, their optimality properties can
be compared to aid in the choice of design.
Because the most common empirical statistical model used to approximate the
true model over the experimental region is a polynomial model, the use of the X0X
matrix in design evaluation stresses the importance of the assumption that the empirical model is adequate, or, equivalently, the optimality criteria are highly model
dependent. Also, the experimenter needs to be aware that although a design may be
best among several designs by one optimality criterion, it may perform poorly when
evaluated by a different optimality criterion. Hence, the choice of design will also
depend upon the choice of the evaluation criterion. Box and Draper [15] discuss the
|X0X| criterion and matters related to it.
16
Many of the design optimality criteria for evaluation and comparison response
surface designs, as well as the algorithms for computer-generated designs, are based
on the foundational work of Kiefer [33, 34] and Kiefer and Wolfowitz [35]. Prior to
their research, it was routinely assumed that each point in an experimental design is
assigned an equal weight. However, Keifer and his colleagues generalized this established concept to allow for alternate weighting schemes for the set of design points.
Their research introduced the mathematically insightful concepts which allowed a
design to be considered a probability measure on the design space.
Because design optimality criteria are characterized by letters of the alphabet,
they are often called alphabetic optimality criteria (Box and Draper [15]). Four commonly used alphabetic optimality criteria are the D, A, G, and IV criteria:
1. D-Optimality is based on |X0X| which is inversely proportional to the square
of the volume of the confidence region on the regression coefficients. It is an indicator of how well the set of coefficients are estimated. Hence, a smaller |X0X|,
or, equivalently, a larger |(X0X)−1 | implies poorer estimation of the regression
coefficients in the model. Also, the elements of (X0X)−1 are proportional to the
variances and covariances of the regression coefficients, scaled by N/σ 2 . Thus,
control of the |X0X| by design results in control of the variances and covariances
of the regression coefficients. Hence, where X is the design matrix, the goal of
D-optimality is to
maximize |X0X|, or equivalently, minimize |(X0X)−1 |.
17
2. A-Optimality is based on the individual variances of regression coefficients
and the goal of A-optimality is to,
minimize trace (X0X)−1 ,
where X is the design matrix, and trace is the sum of the scaled variances of
the regression coefficients.
3. G-Optimality is based on V (x) = N f 0 (x)(X0X)−1 f (x), the scaled prediction
variance. G-optimality is a minimax criterion. That is, the goal is to minimize
the maximum prediction variance in the design region. Hence, the goal of Goptimality is to,
minimize max N f 0 (x)(X0X)−1 f (x) ,
x∈X
where X is the design matrix, x is any point in the design region X , f (x) =
[f1 (x), . . . , fp (x)]0 is a vector of p real-valued functions based on the p parameter
model terms, and N is the design size.
4. IV -Optimality is also based on V (x) = N f 0 (x)(X0X)−1 f (x). The goal of IV optimality is to minimize the average prediction variance in the design region.
Hence, the goal of IV -optimality is to
minimize average N f 0 (x)(X0X)−1 f (x) over x ∈ X ,
where X is the design matrix, x is any point in the design region X , f (x) =
[f1 (x), . . . , fp (x)]0 is a vector of p real-valued functions based on the p parameter
model terms, and N is the design size.
18
When considering an experimental design for implementation, several of its properties can be determined by computing measures of design efficiency. When calculating design efficiencies, the optimal values must first be found. Ideally, a design’s
alphabetic criterion value is close to the optimal value for the theoretically optimal
design. For D, A, and G efficiencies, larger values imply a better design, while for
IV criterion, a smaller value implies a better design (Borkowski and Valeroso [12]).
In the review paper, Response Surface Methodology: 1966-1988, Myers, Khuri,
and Carter [44], point out that both Kiefer and Box agree that design selection should
be guided by more than one criterion because a design may be best among several
designs by one optimality criterion, it may be poorer when evaluated by a different
optimality criterion.
Many studies have been performed to compare experimental designs by using
these D, A, G, IV efficiencies. In his paper Which Response Surface Design is Best,
Lucas [39] compared several types of quadratic response surface designs in hypersphere
and hypercube regions. This paper includes comparisons of composite designs, BoxBehnken designs, Uniform Shell designs, Hoke designs, Pesotchinsky designs, and BoxDraper designs using D and G efficiencies. In A Comparison of Design Optimality
Criteria of Reduced Models for Response Surface Designs in the Hypercube, Borkowski
and Valeroso [12], provided a comparison of the Central Composite, Small Composite,
Notz, Hoke, Box-Draper and several computer-generated designs using D, G, A, and
IV efficiency measures. For more information on D-optimality, see St. John and
19
Draper [52], Mitchell [41]; on G-optimality, see Borkowski [3]; on D and G-optimality,
see Lucas [38, 39], Myers, Khuri, and Carter [44], Kiefer and Wolfowitz [35]; on D,
A, and G-optimality, see Kiefer [33, 34]; on D, A, G, and IV -optimality, see Myers
and Montgomery [45], Box and Draper [15], Atkinson and Donev [1], Borkowski [6],
Borkowski and Valeroso [9, 10, 12]; on IV -optimality, see Borkowski [7].
Another application of optimality criteria for evaluating response surface design
was presented by Borkowski and Valeroso [12]. In this paper, they quantified the robustness of designs in the hypercube against model misspecification by calculating the
D, A, G, and IV criteria for “reasonable” reduced models for the second-order model
in Equation 1.3 that are formed by removing terms based on hierarchy. Specifically:
1. If a model contains an x2i term, then it must contain the corresponding xi term.
2. If a model contains an xi xj term, then it must contain the corresponding xi
and/or xj term.
This set of reduced models is consistent with the definition of weak heredity given in
Chipman [18] and Chipman and Hamada [19].
The objective of the dissertation is to expand the work of Borkowski and Valeroso
[12] by comparing the design optimality criteria of reduced models for response surface
design in a spherical design region. This dissertation adopts the set of reduced models
that are formed by removing terms based on hierarchy for the case K = 3 and K = 4
design variables.
20
The D, A, G, and IV design optimality measures used in this dissertation and
calculated over reduced models of the second-order model can be written as:
D − efficiency
=
A − efficiency
=
G − efficiency
=
IV − criterion
=
|X0 X|1/p
N
p
100
trace [N (X0X)−1 ]
p
100
2
Nσ
bmax
100
2
N σave
(1.4)
(1.5)
(1.6)
(1.7)
where X is the design matrix, p is the number of model parameters, N is the design
2
size, σ
bmax
is the maximum of f 0 (x)(X0 X)−1 f (x) approximated over the set of candi-
2
is the average of f 0 (x)(X0 X)−1 f (x) over the design space. These
date points, and σave
D and A-efficiency measures represent the percent of the number of runs required by
a hypothetical orthogonal design to achieve the same |X0 X| and trace [N (X0X)−1 ].
G-efficiency and the IV -criterion are based on the scaled prediction variance function. Also, the evaluation of the IV -criterion, like the G-efficiency, is over a continuous design region. For example, in a cuboidal or spherical design region, the
IV -criterion involves integration over the cuboidal or spherical space, that is, IV criterion = ω −1
R
X
V (x) dx, where ω =
R
X
dx is the volume of the cuboidal or
spherical design region X . In the dissertation, these design optimality measures were
calculated using Matlab software (Mathworks [40]) for response surface designs in the
spherical region for the set of reduced models. Many practitioners use the designgenerating capability of the SAS OPTEX procedure (SAS Institute [49]). The D,
21
A, G, and IV design efficiency measures contained in output of the SAS OPTEX
procedure are as follows:
SAS D − efficiency
=
SAS A − efficiency
=
SAS G − efficiency
=
SAS IV − criterion
=
|X0 X|1/p
)
ND
p/ND
)
100 (
trace [(X0X)−1 ]
v
u
u
p/ND
)
100 (t
max x0 (X0X)−1 x
x∈C
p
avex∈C x0 (X0X)−1 x
100 (
where p is the number of parameters in the linear model, ND is the number of design
points, and C is the set of candidate points (i.e., a user-supplied set of potential
design points). The SAS D and SAS A-efficiencies are identical to those used in this
dissertation given in ( 1.4) and ( 1.5). The SAS G-efficiency is based on the square
root of the maximum scaled prediction variance while the G-efficiency given in ( 1.6)
is not based on the square root. Equation 1.6 is based on the actual maximum scaled
prediction variance. The IV -criterion or I-optimality in SAS is the square root of
the average scaled variance for prediction (APV) over the candidate points. That is,
the SAS OPTEX procedure calculates G and IV -criteria by approximation based on
the finite user-supplied candidate points. However, this approximation of APV can
be very poor (Borkowski [7]).
In this dissertation, Chapter 2 contains a review of response surface designs with
emphasis on designs in a spherical design region. The structure of X0X and (X0X)−1
for symmetric response surface designs, and closed-form expressions for |X0X| for the
22
symmetric response surface designs are developed. Optimality criterion values are
calculated and the results are presented in Chapter 3. Research results summarizing
the design criteria comparisons for the full second-order model for 3 and 4 factor
spherical response surface designs are also presented in Chapter 3. The research is
then extended to a study of reduced models. Specifically, the robustness of these
designs and a comparison of design optimality criterion across reduced models for 3
and 4 design variables in the spherical region based on D, A, G, and IV criteria are
presented in Chapter 4. Weighted design optimality criteria are newly-developed criteria for assessing design optimality properties across sets of reduced models. Chapter
5 contains the results of the research for weighted design optimality criteria for response surface designs having 3 and 4 design variables using the principles of weak
and strong heredity (Chipman [18] and Chipman and Hamada [19]) in the spherical
design region.
23
CHAPTER 2
RESPONSE SURFACE DESIGNS IN A SPHERICAL DESIGN REGION
Central Composite Designs
The class of composite designs was first introduced by Box and Wilson [16] in
1951. A central composite design (CCD) consists of:
1. nf = f rf points from rf replicates of an f = 2K−P full (P = 0) or fractional
(P > 0) factorial design of at least Resolution V , where K is the number of design
variables. Each point is of the form (x1 , . . . , xK ) = (±1, ±1, . . . , ±1). This portion is
called the factorial portion of the design. The factorial portion allows estimation of
all linear (βi ) and product (βij ) term coefficients in the model.
2. ns = 2Krs points from rs replicates of the 2K star or axial points at a distance α
from the center. Each star point is of the form (x1 , . . . , xK ) = (0, . . . , 0, ±α, 0, . . . , 0).
This portion is called the star portion or the axial portion of the design. The star
points allow estimation of squared term coefficients (βii ) in the model.
3. n0 center points at (x1 , . . . , xK ) = (0, 0, . . . , 0). The center points provide an
internal estimate of pure error used to test for lack of fit and also contribute toward
estimation of the squared terms.
Thus, the total number of CCD points is N = f rf + 2Krs + n0 . The values of
the star distance (α) generally varies from 1.0 to
√
K. When the star point distance
24
α = 1, the CCD is called a face-centered cube design and when the star point distance
α=
√
α =
√
K, the CCD is called a spherical CCD. This dissertation is concerned with the
K case. See Table 2 for a CCD example. For more information on central
composite designs, see Box and Wilson [16], Hartley [31], Lucas [38, 39], Draper [24],
Myers and Montgomery [45], Borkowski [3, 4, 5], and Borkowski and Valeroso [9].
Box-Behnken Designs
In 1960, Box and Behnken [13] introduced designs that are now known as the
Box-Behnken designs (BBDs). Many of these designs are formed by combining twolevel factorial designs with a balanced incomplete block design (BIBD). Associated
with BIBDs, and hence, many BBDs are the following parameters:
K = the number of design variables.
b = the number of blocks in the BIBD.
t = the number of design variables per block.
r = the number of blocks in which a design variable appears.
λ = the number of times that each pair of design variables
appears in the same block. It must hold that λ =
r(t−1)
.
K−1
To generate a BBD, the t design variables appearing in each block in the BIBD
are replaced with the t columns defining a 2t factorial design with levels ±1. The
remaining K − t columns are set at mid-level 0 and n0 center points are included in
the design. The total number of design points is N = f Kr/t + n0 = f b + n0 where
25
Table 2. √
A 15-Point Central Composite Design (CCD) for Three Factors (K = 3)
and α = 3.
Points
nf = 8
y
ns = 6
i
n0 = 1
x1
1
1
1
1
−1
−1
−1
−1
±α
0
0
0
x2
1
1
−1
−1
1
1
−1
−1
0
±α
0
0
x3
1
−1
1
−1
1
−1
1
−1
0
0
±α
0
m
w
i
y
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
by
i
y
b
b
b
b
b
b
b
b
b
1
b
b
b
b
b
b
b
b
by
i
b
b
b
b
b
b
b b
b y
b
b
b
b
b
b
b
α
b
b
b
b
b
b
bi
b
y
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
by
-
y
bb
bb
i
y
i
26
f = 2t . For larger number of design variables K (e.g., when K = 6), a fractional
factorial is suggested. See Table 3 for BBD design matrices for 3 and 4 factors having
one center point. The levels are coded so that they lie on a sphere of radius α =
√
K.
For more information on BBDs, see Box and Behnken [13], Lucas [39], Borkowski
[4, 5], Borkowski and Valeroso [9], and Myers and Montgomery [45].
Table 3. Box-Behnken Designs (BBDs) for 3 and 4 Factors.
A 25 Point BBD, α =
A 13 Point BBD, α =
x1
−1.2247
−1.2247
1.2247
1.2247
−1.2247
−1.2247
1.2247
1.2247
0
0
0
0
0
x2
−1.2247
1.2247
−1.2247
1.2247
0
0
0
0
−1.2247
−1.2247
1.2247
1.2247
0
√
3
x3
0
0
0
0
−1.2247
1.2247
−1.2247
1.2247
−1.2247
1.2247
−1.2247
1.2247
0
x1
−1.4142
−1.4142
1.4142
1.4142
−1.4142
−1.4142
1.4142
1.4142
−1.4142
−1.4142
1.4142
1.4142
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
−1.4142
1.4142
−1.4142
1.4142
0
0
0
0
0
0
0
0
−1.4142
−1.4142
1.4142
1.4142
−1.4142
−1.4142
1.4142
1.4142
0
0
0
0
0
x3
0
0
0
0
−1.4142
1.4142
−1.4142
1.4142
0
0
0
0
−1.4142
1.4142
−1.4142
1.4142
0
0
0
0
−1.4142
−1.4142
1.4142
1.4142
0
√
4
x4
0
0
0
0
0
0
0
0
−1.4142
1.4142
−1.4142
1.4142
0
0
0
0
−1.4142
1.4142
−1.4142
1.4142
−1.4142
1.4142
−1.4142
1.4142
0
27
Rotatability
Montgomery [42] stated that it is important for second-order model to provide good predictions throughout the region of interest. To define “good” is to
require that the model have a reasonably consistent and stable variance, V [ŷ (x)]
= σ 2 N f 0 (x)(X0X)−1 f (x), or scaled prediction variance, N V [ŷ (x)] /σ 2 , of the predicted response at points of interest x, where f (x) is a vector corresponding to the
model terms. The scaled prediction variance is often used in design comparison studies because the division by σ 2 makes the quantity scale-free and the multiplication by
the design size (N ) allows this quantity to reflect variance on a per observation basis.
That is, when two designs are compared, scaling by N penalizes the larger design.
Thus, emphasis is placed on design size efficiency.
“Rotatability” is a property to be considered for a spherical design region but not
for a cuboidal design region. A design is rotatable if V [ŷ (x)] is the same at all points
x which are the same distance from the design center. That is, the prediction variance
is constant on spheres. This constant variance property is desirable when accurate
predictions are needed and the experimenter initially does not know where the optimum response occurs in the design space. However, rotatability or near-rotatability
often is easy to achieve without sacrificing other important design properties. It can
be shown that any CCD is rotatable when α =
p
4
f /rs , where f is the number of
factorial points and rs is the number of replications for star or axial points, and the
28
4 factor BBD is also rotatable (Borkowski [5], Khuri and Cornell [32], and Myers and
Montgomery [45]).
Small Composite Designs
In 1959, small composite designs (SCDs) were suggested by Hartley [31]. The SCD
has the same basic construction as the CCD. That is, an SCD consists of a factorial
portion, an axial or star portion, and center points. However, unlike the CCD, the
SCD employs a two-level fractional factorial design of Resolution III, provided that
two-factor interactions are not aliased with other two-factor interactions. As a result,
the total run size is reduced from that of the CCD. See Table 4 for 3 and 4 factor
SCD design matrices having one center point in a spherical region. For K = 3,
the factorial portion contains the points from the fractional factorial design having
generator I = ABC. For K = 4, the factorial portion contains the points from
the fractional factorial design having generator I = ABD. For more information
on SCDs, see Hartley [31], Draper [24], Draper and Lin [25], Box and Draper [15],
Giovannitti-Jensen and Myers [29], Myers et al. [46], and Borkowski and Valeroso
[12].
29
Table 4. Small Composite Designs (SCDs) for 3 and 4 Factors.
√
A 10 Point SCD, α = 3
x1
x2
x3
1
1
1
1
−1
−1
−1
1
−1
−1
−1
1
1.732
0
0
−1.732
0
0
0
1.732
0
0 −1.732
0
0
0
1.732
0
0 −1.732
A 16 Point SCD, α =
x1
x2
x3
−1
−1
−1
1
−1
−1
−1
1
−1
−1
−1
1
1
1
−1
1
−1
1
−1
1
1
1
1
1
2
0
0
−2
0
0
0
2
0
0
−2
0
0
0
2
0
0
−2
0
0
0
0
0
0
√
4
x4
1
−1
−1
1
1
−1
−1
1
0
0
0
0
0
0
2
−2
30
Plackett-Burman Composite Designs
In 1946, Plackett and Burman [47] introduced designs that are now known as
the Plackett-Burman designs. They produced a series of two-level fractional factorial
designs for examining up to K = N − 1 design variables in N runs, where N is
a multiple of 4, N is not a power of 2, and N ≤ 100. In general, Plackett-Burman
designs are usful in screening situations in which we examine many factors. Moreover,
Draper [24] and Draper and Lin [25] showed that the Plackett-Burman designs can
be used as the basis for Plackett-Burman Composite Designs (PBCDs). PBCDs are
formed as follows:
1. For the factorial portion use K columns of a Plackett-Burman design. If duplicate runs exist in the Plackett-Burman design, we may remove one of the duplicates
to reduce the sample size.
2. Add 2K star or axial points of radius α.
3. If α =
√
K, then center points are suggested to avoid singularity or near
singularity.
The Plackett-Burman designs are similar to CCDs except that Plackett-Burman
designs are used instead of a factorial or fractional factorial design for the factorial
portion. In this research, a 12-run Plackett-Burman design is used for a K = 4 factor
PBCD, and the design matrix of a one center point PBCD in a spherical region is
shown in Table 5. For more information on PBCDs, see Plackett and Burman [47],
31
Draper [24], Draper and Lin [25], Lin and Draper [37], Myers and Montgomery [45],
and Khuri and Cornell [32].
Table 5. Plackett-Burman Composite Design (PBCD) for 4 Factors.
x1
1
1
−1
1
1
1
−1
−1
−1
1
−1
−1
x2
−1
1
1
−1
1
1
1
−1
−1
−1
1
−1
√
A 20 Point PBCD, α = 4
x3
x4
x1
x2
1
−1
2
0
−1
1
−2
0
1
−1
0
2
1
1
0
−2
−1
1
0
0
1
−1
0
0
1
1
0
0
1
1
0
0
−1
1
−1
−1
−1
−1
−1
−1
x3
0
0
0
0
2
−2
0
0
x4
0
0
0
0
0
0
2
−2
Hybrid Designs
In 1976, hybrid response surface designs were developed by Roquemore [48]. Hybrid designs were created to achieve the same degree of orthogonality as the CCD,
to be near-minimum-point in size, to be near-rotatable, and to possess some ease
in coding. They are created using a CCD for K − 1 factors, and the levels of the
K th factor are chosen to create certain symmetries within the design. The result
is a class of designs that are economical and either rotatable or near-rotatable for
K = 3, 4, 6, and 7. For K = 3, there are three hybrid designs which are denoted 310,
311A, and 311B. For K = 4, there also are three hybrid designs which are denoted
32
416A, 416B, and 416C. The design names indicate the number of variables (3 or 4),
number of points (10, 11, or 16), and a letter designation for different designs of the
same size. The design matrices of hybrid designs in a spherical region are shown
in Table 6 and Table 7. For more information on hybrid designs, see Roquemore
[48], Lucas [39], Giovannitti-Jensen and Myers [29], Myers et al. [46], Myers and
Montgomery [45], and Khuri and Cornell [32].
Table 6. Hybrid Designs for 3 Factors.
A 10 Point H310, α =
x1
0
0
−1.1162
1.1162
−1.1162
1.1162
1.3100
−1.3100
0
0
x2
0
0
−1.1162
−1.1162
1.1162
1.1162
0
0
1.3100
−1.3100
√
x3
1.4406
−.1518
0.7128
0.7128
0.7128
0.7128
−1.0351
−1.0351
−1.0351
−1.0351
A 11 Point H311B, α =
x1
0
0
−0.5308
1.4894
0.5308
−1.4894
0.5308
1.4894
−0.5308
−1.4894
0
x2
0
0
1.4894
0.5308
−1.4894
−0.5308
1.4894
−0.5308
−1.4894
0.5308
0
3
√
3
x3
1.7321
−1.7321
0.7071
0.7071
0.7071
0.7071
−0.7071
−0.7071
−0.7071
−0.7071
0
A 11 Point H311A, α =
x1
0
0
−1.0954
1.0954
−1.0954
1.0954
1.5492
−1.5492
0
0
0
x2
0
0
−1.0954
−1.0954
1.0954
1.0954
0
0
1.5492
−1.5492
0
√
3
x3
1.5492
−1.5492
0.7746
0.7746
0.7746
0.7746
−0.7746
−0.7746
−0.7746
−0.7746
0
33
Table 7. Hybrid Designs for 4 Factors.
A 16 Point H416A, α =
x1
0
0
−1.0449
1.0449
−1.0449
1.0449
−1.0449
1.0449
−1.0449
1.0449
1.7609
−1.7609
0
0
0
0
x2
0
0
−1.0449
−1.0449
1.0449
1.0449
−1.0449
−1.0449
1.0449
1.0449
0
0
1.7609
−1.7609
0
0
x3
0
0
−1.0449
−1.0449
−1.0449
−1.0449
1.0449
1.0449
1.0449
1.0449
0
0
0
0
1.7609
−1.7609
√
4
x4
1.8645
−1.5616
0.6733
0.6733
0.6733
0.6733
0.6733
0.6733
0.6733
0.6733
−0.9482
−0.9482
−0.9482
−0.9482
−0.9482
−0.9482
A 16 Point H416C, α =
x1
0
0
−1.0973
1.0973
−1.0973
1.0973
−1.0973
1.0973
−1.0973
1.0973
1.6127
−1.6127
0
0
0
0
x2
0
0
−1.0973
−1.0973
1.0973
1.0973
−1.0973
−1.0973
1.0973
1.0973
0
0
1.6127
−1.6127
0
0
x3
0
0
−1.0973
−1.0973
−1.0973
−1.0973
1.0973
1.0973
1.0973
1.0973
0
0
0
0
1.6127
−1.6127
√
4
x4
1.9372
0
0.6227
0.6227
0.6227
0.6227
0.6227
0.6227
0.6227
0.6227
−1.1532
−1.1532
−1.1532
−1.1532
−1.1532
−1.1532
A 16 Point H416B, α =
x1
0
0
−1.0838
1.0838
−1.0838
1.0838
−1.0838
1.0838
−1.0838
1.0838
1.6448
−1.6448
0
0
0
0
x2
0
0
−1.0838
−1.0838
1.0838
1.0838
−1.0838
−1.0838
1.0838
1.0838
0
0
1.6448
−1.6448
0
0
x3
0
0
−1.0838
−1.0838
−1.0838
−1.0838
1.0838
1.0838
1.0838
1.0838
0
0
0
0
1.6448
−1.6448
√
4
x4
1.8768
−0.2918
0.6551
0.6551
0.6551
0.6551
0.6551
0.6551
0.6551
0.6551
−1.1377
−1.1377
−1.1377
−1.1377
−1.1377
−1.1377
34
Uniform Shell Designs
In 1970, uniform shell designs (UNFSDs) were developed by Doehlert [22] and
Doehlert and Klee [23]. A UNFSD is so-named because it consists of points uniformly
spaced on concentric spherical shells. As a result, these designs require many levels
of each variable. See Table 8 for the UNFSD matrices in a spherical region. For more
information on UNFSDs, see Doehlert [22], Doehlert and Klee [23], Lucas [39], and
Khuri and Cornell [32].
Table 8. Uniform Shell Designs (UNFSDs) for 3 and 4 Factors.
A 20 Point UNFSD, α =
A 12 Point UNFSD, α =
x1
1.7321
0.8660
−0.8660
−1.7321
−0.8660
0.8660
0.8660
−0.8660
0
−0.8660
0.8660
0
x2
0
1.5000
1.5000
0
−1.5000
−1.5000
0.5000
0.5000
−1.0000
−0.5000
−0.5000
1.0000
√
3
x3
0
0
0
0
0
0
1.4142
1.4142
1.4142
−1.4142
−1.4142
−1.4142
x1
2
1
−1
1
−1
0
1
−1
0
0
−2
−1
1
−1
1
0
−1
1
0
0
x2
0
1.7321
1.7321
0.5774
0.5774
−1.1547
0.5774
0.5774
−1.1547
0
0
−1.7321
−1.7321
−0.5774
−0.5774
1.1547
−0.5774
−0.5774
1.1547
0
x3
0
0
0
1.6330
1.6330
1.6330
0.4082
0.4082
0.4082
−1.2247
0
0
0
−1.6330
−1.6330
−1.6330
−0.4082
−0.4082
−0.4082
1.2247
√
4
x4
0
0
0
0
0
0
1.5811
1.5811
1.5811
1.5811
0
0
0
0
0
0
−1.5811
−1.5811
−1.5811
−1.5811
35
The X0X Matrix
When choosing a design to run from among a set of proposed experimental designs, the researcher may consider the four common design optimality criteria (D, A,
G, and IV criteria). These design optimality criteria are based on optimal properties
of the X0X matrix for the expanded design matrix X. The expanded design matrix
X is formed by adding columns that correspond to the model terms (e.g., the intercept, interaction, and squared terms) to the matrix of levels for the design variables.
The assumption that the empirical model is adequate is essential when evaluating
designs by properties of X0X, or, equivalently, the X0X optimality criteria are highly
model dependent. Polynomials often are used as empirical models for approximating
the true model over the experimental design region as it is mentioned in Chapter 1.
The commonly-used second-order model on K design variables is considered in this
research:
y = β0 +
K
X
i=1
β i xi +
K
X
i=1
βii x2i
+
K
X
βij xi xj + .
(2.1)
1≤i<j
Let X be the expanded design matrix of spherical response surface designs with
associated quadratic response surface model on K design variables x1 , x2 , . . . , xK .
The expanded design matrix X and the associated X0X matrix are now presented for
all of the 3 and 4 factor designs examined in this dissertation. This will be followed
by a discussion of symmetric designs and the X0X matrix.
36
The structure of the expanded design matrix X for a 3 factor CCD where α =
√
3,
rs = 1, and N = f + 2Krs + n0 is as follows:
Points x0
f =8
2Krs
=6
n0
x1
1
1
1
1
1
1
1
1
1 −1
1 −1
1 −1
1 −1
1
α
1 −α
1
0
1
0
1
0
1
0
1
0
x2
x3
x1 x2
x1 x3
1
1
1 −1
−1
1
−1 −1
1
1
1 −1
−1
1
−1 −1
0
0
0
0
α
0
−α
0
0
α
0 −α
0
0
1
1
−1
−1
−1
−1
1
1
0
0
0
0
0
0
0
1
−1
1
−1
−1
1
−1
1
0
0
0
0
0
0
0
x2 x3
x21
x22
x23
1
1
1
1
−1
1
1
1
−1
1
1
1
1
1
1
1
1
1
1
1
−1
1
1
1
−1
1
1
1
1
1
1
1
2
0 α
0
0
0 α2
0
0
0
0 α2
0
2
0
0 α
0
0
0
0 α2
0
0
0 α2
0
0
0
0
The X0X matrix for a 3 factor CCD is determined directly by matrix multiplication
and the resulting block matrix form is as follows:

N
0
0
0

2
0
f
+
2r
α
0
0
s


2
0
0
f + 2rs α
0


0
0
0
f + 2rs α2


0
0
0
0

X0X = 
0
0
0
0



0
0
0
0

2
 f + 2rs α
0
0
0

 f + 2rs α2
0
0
0
2
f + 2rs α
0
0
0
0
0
0
0
f
0
0
0
0
0
0
0
0
0
0
f
0
0
0
0

0 f + 2rs α2 f + 2rs α2 f + 2rs α2

0
0
0
0


0
0
0
0


0
0
0
0


0
0
0
0


0
0
0
0



f
0
0
0

4

0 f + 2rs α
f
f


0
f
f + 2rs α4
f
4
0
f
f
f + 2rs α
37
For a 4 factor CCD, use the structure of the expanded design matrix X where
α=
√
4, rs = 1, and N = f + 2Krs + n0 given below:
Points x0
f = 16
2Krs
=8
n0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1
x2
x3
−1
−1
−1
−1
−1
−1
−1
−1
1
1
1
1
1
1
1
1
−α
α
0
0
0
0
0
0
0
−1
−1
−1
−1
1
1
1
1
−1
−1
−1
−1
1
1
1
1
0
0
−α
α
0
0
0
0
0
−1
−1
1
1
−1
−1
1
1
−1
−1
1
1
−1
−1
1
1
0
0
0
0
−α
α
0
0
0
x4 x1 x2 x1 x3 x1 x4 x2 x3 x2 x4 x3 x4 x21 x22 x23 x24
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
0
0
0
0
0
0
−α
α
0
1
1
1
1
−1
−1
−1
−1
−1
−1
−1
−1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
−1
−1
1
1
−1
−1
−1
−1
1
1
−1
−1
1
1
0
0
0
0
0
0
0
0
0
1
−1
1
−1
1
−1
1
−1
−1
1
−1
1
−1
1
−1
1
0
0
0
0
0
0
0
0
0
1
1
−1
−1
−1
−1
1
1
1
1
−1
−1
−1
−1
1
1
0
0
0
0
0
0
0
0
0
1
−1
1
−1
−1
1
−1
1
1
−1
1
−1
−1
1
−1
1
0
0
0
0
0
0
0
0
0
1 1 1 1 1
−1 1 1 1 1
−1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
−1 1 1 1 1
−1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
−1 1 1 1 1
−1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
−1 1 1 1 1
−1 1 1 1 1
1 1 1 1 1
0 α2 0 0 0
0 α2 0 0 0
0 0 α2 0 0
0 0 α2 0 0
0 0 0 α2 0
0 0 0 α2 0
0 0 0 0 α2
0 0 0 0 α2
0 0 0 0 0
38
The corresponding X0X matrix is













X0X = 







f

f

f
f
N
0
0
0
0
0
f + 2rs α2
0
0
0
0
0
f + 2rs α2
0
0
0
0
f + 2rs α2
0
0
0
0
0
0
f + 2rs α2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+ 2rs α2
0
0
0
0
+ 2rs α2
0
0
0
0
+ 2rs α2
0
0
0
0
+ 2rs α2
0
0
0
0
0 0 0 0 0 0 f + 2rs α2 f + 2rs α2 f + 2rs α2 f + 2rs α2 
000000
0
0
0
0


000000
0
0
0
0


000000
0
0
0
0


000000
0
0
0
0


f 00000
0
0
0
0


0f 0000
0
0
0
0


00f 000
0
0
0
0


000f 00
0
0
0
0


0000f 0
0
0
0
0


00000f
0
0
0
0


0 0 0 0 0 0 f + 2rs α4
f
f
f


000000
f
f + 2rs α4
f
f


4
000000
f
f
f + 2rs α
f
4
000000
f
f
f
f + 2rs α
For Box-Behnken designs, the structures of the expanded design matrix X for
3 and 4 design variables are scaled so that extreme design points are at a distance
α=
√
K. The expanded design matrix X of a 3 factor BBD is
x0
x1
x2
x3
x1 x2
x1 x3
x2 x3
x21
x22
x23
1
1
1
1
1
1
1
1
1
1
1
1
1
−1.2247
−1.2247
1.2247
1.2247
−1.2247
−1.2247
1.2247
1.2247
0
0
0
0
0
−1.2247
1.2247
−1.2247
1.2247
0
0
0
0
−1.2247
−1.2247
1.2247
1.2247
0
0
0
0
0
−1.2247
1.2247
−1.2247
1.2247
−1.2247
1.2247
−1.2247
1.2247
0
1.50
−1.50
−1.50
1.50
0
0
0
0
0
0
0
0
0
0
0
0
0
1.50
−1.50
−1.50
1.50
0
0
0
0
0
0
0
0
0
0
0
0
0
1.50
−1.50
−1.50
1.50
0
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
0
0
0
0
0
1.50
1.50
1.50
1.50
0
0
0
0
1.50
1.50
1.50
1.50
0
0
0
0
0
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
0
39
The total number of design points is N = f (2r − λ) + n0 , where f = 2t and t is the
number of design variables per block, r is the number of blocks in which a design
variable appears, and λ is the number of times that each pair of design variables
appears in the same block (λ =
BBD where α =








X0X = 







N
0
0
0
0
0
0
f rα2
f rα2
f rα2
p
r(t−1)
).
K−1
The block matrix form of X0X for a 3 factor
K/t is the following:
0
f rα2
0
0
0
0
0
0
0
0
0
0
f rα2
0
0
0
0
0
0
0
0
0
0
f rα2
0
0
0
0
0
0
0
0
0
0
f λα4
0
0
0
0
0
0
0
0
0
0
f λα4
0
0
0
0
0
0
0
0
0
0
f λα4
0
0
0
f rα2
0
0
0
0
0
0
f?
f λα4
f λα4
f rα2
0
0
0
0
0
0
f λα4
f?
f λα4
f rα2
0
0
0
0
0
0
f λα4
f λα4
f?
















where f ? is (f (r − λ) + f λ) α4 .
For a 4 factor BBD, the structure of the expanded design matrix X, where the
design is also scaled, is the following:
40
x0
x1
x2
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1.4142
-1.4142
1.4142
1.4142
-1.4142
-1.4142
1.4142
1.4142
-1.4142
-1.4142
1.4142
1.4142
0
0
0
0
0
0
0
0
0
0
0
0
0
-1.4142
1.4142
-1.4142
1.4142
0
0
0
0
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
-1.4142
-1.4142
1.4142
1.4142
0
0
0
0
0
0
0
0
0
-1.4142
1.4142
-1.4142
1.4142
0
0
0
0
-1.4142
1.4142
-1.4142
1.4142
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
0
x4 x1 x2 x1 x3 x1 x4 x2 x3 x2 x4 x3 x4 x21 x22 x23 x24
0
0
0
0
0
0
0
0
-1.4142
1.4142
-1.4142
1.4142
0
0
0
0
-1.4142
1.4142
-1.4142
1.4142
-1.4142
1.4142
-1.4142
1.4142
0
2
-2
-2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
-2
-2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
-2
-2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
-2
-2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
-2
-2
2
0
0
0
0
0
The block matrix form of X0X for a 4 factor BBD where α =

p
N
0
0
0
0
0
0
0
0
0
0
 0 f rα2 0
0
0
0
0
0
0
0
0

2
 0
0
f
rα
0
0
0
0
0
0
0
0

2
 0
0
0
f
rα
0
0
0
0
0
0
0

2
 0
0
0
0
f
rα
0
0
0
0
0
0

4
 0
0
0
0
0
f
λα
0
0
0
0
0

4
 0
0
0
0
0
0
f
λα
0
0
0
0

4
X0X = 
0
0
0
0
0
0
0
f
λα
0
0
0

4
 0
0
0
0
0
0
0
0
f
λα
0
0

4
 0
0
0
0
0
0
0
0
0
f
λα
0

4
 0
0
0
0
0
0
0
0
0
0
f
λα

 f rα2 0
0
0
0
0
0
0
0
0
0

 f rα2 0
0
0
0
0
0
0
0
0
0

 f rα2 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
f rα2 0
where f ? is (f (r − λ) + f λ) α4 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
-2
-2
2
0
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
0
0
0
0
0
0
0
0
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
2
2
2
2
0
0
0
0
2
2
2
2
0
0
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
2
2
2
2
0
0
0
0
2
2
2
2
2
2
2
2
0
K/t is:
f rα2
0
0
0
0
0
0
0
0
0
0
f?
f λα4
f λα4
f λα4
f rα2
0
0
0
0
0
0
0
0
0
0
f λα4
f?
f λα4
f λα4
f rα2
0
0
0
0
0
0
0
0
0
0
f λα4
f λα4
f?
f λα4

f rα2
0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

f λα4 

f λα4 

f λα4 
f?
41
In general, the block matrix form of the X0X matrix for a CCD and for a BBD
based on a BIBD can be written in a form analogous to the form given by Borkowski
and Valeroso [9]. That is,

N
0

φ1
XX =
β JK
where φ1 is a (K +
K
2
φ 01
Diag (di )
φ2

β J 0K

φ 02
0
δ IK + γ JK J K
) × 1 zero matrix, φ2 is a K × (K +
K
2
) zero matrix, JK is a
K × 1 unit column vector, IK is a K × K identity matrix, and Diag(di ) is a diagonal
matrix such that
di =
β, for 1 ≤ i ≤ K
γ, for K + 1 ≤ i ≤ K +
K
2
and define β, δ, and γ as follows:
Design
CCD
BBD
β
f + 2rs α2
f rα2
δ
2rs α4
f (r − λ) α4
γ
f
f λ α4
For K = 4, β = 24, δ = 32, and γ = 16 for both the CCDs having rs = 1 and the
BBDs. Thus, when K = 4, for any n0 , X0X for the CCD having rs = 1 and X0X for
the BBD are identical. Therefore, their D, A, G, and IV criteria values will also be
identical.
42
For a 3 factor small composite design (SCD), use the following structure of the
expanded design matrix X where N = f + 2Krs + n0 = 11, α =
Points
2Krs
=6
n0
3, and rs = 1:
x2
x3
x1 x2
x1 x3
x2 x3
x21
x22
x23
1
1
1
1
1
-1
1
-1
1
1
-1
-1
1 1.732
0
1 -1.732
0
1
0 1.732
1
0 -1.732
1
0
0
1
0
0
1
0
0
1
-1
-1
1
0
0
0
0
1.732
-1.732
0
1
-1
-1
1
0
0
0
0
0
0
0
1
-1
1
-1
0
0
0
0
0
0
0
1
1
-1
-1
0
0
0
0
0
0
0
1
1
1
1
3
3
0
0
0
0
0
1
1
1
1
0
0
3
3
0
0
0
1
1
1
1
0
0
0
0
3
3
0
x0
f =4
√
x1
The X0X matrix for this 3 factor SCD is the following:








0
XX = 







11
0
0
0
0
0
0
10
10
10
0
10
0
0
0
0
4
0
0
0
0
0
10
0
0
4
0
0
0
0
0
0
0
10
4
0
0
0
0
0
0
0
0
4
4
0
0
0
0
0
0
0
4
0
0
4
0
0
0
0
0
4
0
0
0
0
4
0
0
0
10
0
0
0
0
0
0
22
4
4
10
0
0
0
0
0
0
4
22
4
10
0
0
0
0
0
0
4
4
22
















43
For a 4 factor SCD, use the following structure of the expanded design matrix X,
where N = f + 2Krs + n0 = 17, α =
√
4, and rs = 1:
Points x0 x1 x2 x3 x4 x1 x2 x1 x3 x1 x4 x2 x3 x2 x4 x3 x4 x21 x22 x23 x24
f =8
2Krs
=8
n0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
1
-1
-1
1
1
-1
1
2
-2
0
0
0
0
0
0
0
-1
-1
1
-1
1
-1
1
1
0
0
2
-2
0
0
0
0
0
-1
-1
-1
1
-1
1
1
1
0
0
0
0
2
-2
0
0
0
1
-1
-1
1
1
-1
-1
1
0
0
0
0
0
0
2
-2
0
1
-1
-1
1
1
-1
-1
1
0
0
0
0
0
0
0
0
0
1
-1
1
-1
-1
1
-1
1
0
0
0
0
0
0
0
0
0
-1
-1
1
-1
1
-1
1
1
0
0
0
0
0
0
0
0
0
1
1
-1
-1
-1
-1
1
1
0
0
0
0
0
0
0
0
0
-1
1
-1
-1
1
1
-1
1
0
0
0
0
0
0
0
0
0
-1
1
1
1
-1
-1
-1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
4
4
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
4
4
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
4
4
0
0
0
16
0
0
0
0
0
0
0
0
0
0
8
8
40
8
16
0
0
0
0
0
0
0
0
0
0
8
8
8
40
1
1
1
1
1
1
1
1
0
0
0
0
0
0
4
4
0
The X0X matrix for this 4 factor SCD is the following:













0
XX = 












17
0
0
0
0
0
0
0
0
0
0
16
16
16
16
0
16
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
16
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
16
8
0
0
0
0
0
0
0
0
0
0
0
0
0
8
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
16
0
0
0
0
0
0
0
0
0
0
40
8
8
8
16
0
0
0
0
0
0
0
0
0
0
8
40
8
8


























44
For a Plackett-Burman composite design (PBCD), the structure of the expanded
design matrix X for 4 design variables where α =
Points
f = 12
ns
=6
n0
√
4 is:
x0 x1 x2 x3 x4 x1 x2 x1 x3 x1 x4 x2 x3 x2 x4 x3 x4 x21 x22 x23 x24
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
1
1
1
-1
-1
-1
1
-1
-1
2
-2
0
0
0
0
0
0
0
-1
1
1
-1
1
1
1
-1
-1
-1
1
-1
0
0
2
-2
0
0
0
0
0
1
-1
1
1
-1
1
1
1
-1
-1
-1
-1
0
0
0
0
2
-2
0
0
0
-1
1
-1
1
1
-1
1
1
1
-1
-1
-1
0
0
0
0
0
0
2
-2
0
-1
1
-1
-1
1
1
-1
1
1
-1
-1
1
0
0
0
0
0
0
0
0
0
1
-1
-1
1
-1
1
-1
-1
1
-1
1
1
0
0
0
0
0
0
0
0
0
-1
1
1
1
1
-1
-1
-1
-1
-1
1
1
0
0
0
0
0
0
0
0
0
-1
-1
1
-1
-1
1
1
-1
1
1
-1
1
0
0
0
0
0
0
0
0
0
1
1
-1
-1
1
-1
1
-1
-1
1
-1
1
0
0
0
0
0
0
0
0
0
-1
-1
-1
1
-1
-1
1
1
-1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
4
4
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
4
4
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
4
4
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
4
4
0
The X0X matrix for this 4 factor PBCD is the following:













X0X = 












21
0
0
0
0
0
0
0
0
0
0
20
20
20
20
0
20
0
0
0
0
0
0
−4
4
−4
0
0
0
0
0
0
20
0
0
0
−4
4
0
0
−4
0
0
0
0
0
0
0
20
0
−4
0
−4
0
−4
0
0
0
0
0
0
0
0
0
20
4
−4
0
−4
0
0
0
0
0
0
0
0
0
−4
4
12
0
0
0
0
−4
0
0
0
0
0
0
−4
0
−4
0
12
0
0
−4
0
0
0
0
0
0
0
4
−4
0
0
0
12
−4
0
0
0
0
0
0
0
−4
0
0
−4
0
0
−4
12
0
0
0
0
0
0
0
4
0
−4
0
0
−4
0
0
12
0
0
0
0
0
0
−4
−4
0
0
−4
0
0
0
0
12
0
0
0
0
20
0
0
0
0
0
0
0
0
0
0
44
12
12
12
20
0
0
0
0
0
0
0
0
0
0
12
44
12
12
20
0
0
0
0
0
0
0
0
0
0
12
12
44
12
20
0
0
0
0
0
0
0
0
0
0
12
12
12
44


























45
For 3 factor of hybrid designs, the structure of the expanded design matrix X for
a hybrid 310 design is the following:
x2 x3
x21
x22
x23
1
0
0 1.4406
0
0
0
1
0
0 -0.1518
0
0
0
1 -1.1162 -1.1162 0.7128 1.2459 -0.7957 -0.7957
1 1.1162 -1.1162 0.7128 -1.2459 0.7957 -0.7957
1 -1.1162 1.1162 0.7128 -1.2459 -0.7957 0.7957
1 1.1162 1.1162 0.7128 1.2459 0.7957 0.7957
1 1.3100
0 -1.0351
0 -1.3559
0
1 -1.3100
0 -1.0351
0 1.3559
0
1
0 1.3100 -1.0351
0
0 -1.3559
1
0 -1.3100 -1.0351
0
0 1.3559
0
0
1.2459
1.2459
1.2459
1.2459
1.7161
1.7161
0
0
0
0
1.2459
1.2459
1.2459
1.2459
0
0
1.7161
1.7161
2.0753
0.0230
0.5081
0.5081
0.5081
0.5081
1.0714
1.0714
1.0714
1.0714
x0
x1
x2
x3
x1 x2
x1 x3
The X0X matrix for a hybrid 310 design is:








0
XX = 







0
0
0
0
0
0
8.42 8.42 8.42
10
0 8.42 0
0
0
0
0
0
0
0
0
0 8.42 0
0
0
0
0
0
0
0
0
0 8.42 0
0
0
0
0
0
0
0
0
0 6.21 0
0
0
0
0
0
0
0
0
0 6.21 0
0
0
0
0
0
0
0
0
0 6.21
0
0
0
8.42 0
0
0
0
0
0 12.10 6.21 6.21
8.42 0
0
0
0
0
0
6.21 12.10 6.21
8.42 0
0
0
0
0
0
6.21 6.21 9.93
















46
The structure of the expanded design matrix X for a hybrid 311A design is
x2 x3
x21
x22
x23
1
0
0 1.5492
0
0
0
1
0
0 -1.5492
0
0
0
1 -1.0954 -1.0954 0.7746 1.2000 -0.8485 -0.8485
1 1.0954 -1.0954 0.7746 -1.2000 0.8485 -0.8485
1 -1.0954 1.0954 0.7746 -1.2000 -0.8485 0.8485
1 1.0954 1.0954 0.7746 1.2000 0.8485 0.8485
1 1.5492
0 -0.7746
0 -1.2000
0
1 -1.5492
0 -0.7746
0 1.2000
0
1
0 1.5492 -0.7746
0
0 -1.2000
1
0 -1.5492 -0.7746
0
0 1.2000
1
0
0
0
0
0
0
0
0
1.2000
1.2000
1.2000
1.2000
2.4000
2.4000
0
0
0
0
0
1.2000
1.2000
1.2000
1.2000
0
0
2.4000
2.4000
0
2.4000
2.4000
0.6000
0.6000
0.6000
0.6000
0.6000
0.6000
0.6000
0.6000
0
x0
x1
x2
x3
x1 x2
x1 x3
and the X0X matrix for a hybrid 311A design is the following:








0
XX = 







11
0
0
0
0
0
0
9.60
9.60
9.60
0
9.60
0
0
0
0
0
0
0
0
0
0
9.60
0
0
0
0
0
0
0
0
0
0
9.60
0
0
0
0
0
0
0
0
0
0
5.76
0
0
0
0
0
0
0
0
0
0
5.76
0
0
0
0
0
0
0
0
0
0
5.76
0
0
0
9.60
0
0
0
0
0
0
17.28
5.76
5.76
9.60
0
0
0
0
0
0
5.76
17.28
5.76
9.60
0
0
0
0
0
0
5.76
5.76
14.40
















47
The structure of the expanded design matrix X for a hybrid 311B design is
Point x0
x1
x2
x3
x1 x2
x1 x3
x2 x3
x21
x22
x23
1
1
1
1
1
1
1
1
1
1
1
0
0
-0.5308
1.4894
0.5308
-1.4894
0.5308
1.4894
-0.5308
-1.4894
0
0
0
1.4894
0.5308
-1.4894
-0.5308
1.4894
-0.5308
-1.4894
0.5308
0
1.7321
-1.7321
0.7071
0.7071
0.7071
0.7071
-0.7071
-0.7071
-0.7071
-0.7071
0
0
0
-0.7906
0.7906
-0.7906
0.7906
0.7906
-0.7906
0.7906
-0.7906
0
0
0
-0.3754
1.0532
0.3754
-1.0532
-0.3754
-1.0532
0.3754
1.0532
0
0
0
1.0532
0.3754
-1.0532
-0.3754
-1.0532
0.3754
1.0532
-0.3754
0
0
0
0.2818
2.2182
0.2818
2.2182
0.2818
2.2182
0.2818
2.2182
0
0
0
2.2182
0.2818
2.2182
0.2818
2.2182
0.2818
2.2182
0.2818
0
3.0000
3.0000
0.5000
0.5000
0.5000
0.5000
0.5000
0.5000
0.5000
0.5000
0
ns
=2
f =8
n0
The X0X matrix for a hybrid 311B design is the following:








0
XX = 







11
0
0
0
0
0
0
10
10
10
0
10
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
5
0
0
0
10
0
0
0
0
0
0
20
5
5
10
0
0
0
0
0
0
5
20
5
10
0
0
0
0
0
0
5
5
20
















48
For a 4 factor hybrid 416A design with one center point, the structure of the
expanded design matrix X is:
x0
x1
x2
x3
x4
x1 x2
x1 x3
x1 x4
x2 x3
x2 x4
x3 x4
x21
x22
x23
x24
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
-1.0449
1.0449
-1.0449
1.0449
-1.0449
1.0449
-1.0449
1.0449
1.7609
-1.7609
0
0
0
0
0
0
0
-1.0449
-1.0449
1.0449
1.0449
-1.0449
-1.0449
1.0449
1.0449
0
0
1.7609
-1.7609
0
0
0
0
0
-1.0449
-1.0449
-1.0449
-1.0449
1.0449
1.0449
1.0449
1.0449
0
0
0
0
1.7609
-1.7609
0
1.8645
-1.5616
0.6733
0.6733
0.6733
0.6733
0.6733
0.6733
0.6733
0.6733
-0.9482
-0.9482
-0.9482
-0.9482
-0.9482
-0.9482
0
0
0
1.0918
-1.0918
-1.0918
1.0918
1.0918
-1.0918
-1.0918
1.0918
0
0
0
0
0
0
0
0
0
1.0918
-1.0918
1.0918
-1.0918
-1.0918
1.0918
-1.0918
1.0918
0
0
0
0
0
0
0
0
0
-0.7035
0.7035
-0.7035
0.7035
-0.7035
0.7035
-0.7035
0.7035
-1.6698
1.6698
0
0
0
0
0
0
0
1.0918
1.0918
-1.0918
-1.0918
-1.0918
-1.0918
1.0918
1.0918
0
0
0
0
0
0
0
0
0
-0.7035
-0.7035
0.7035
0.7035
-0.7035
-0.7035
0.7035
0.7035
0
0
-1.6698
1.6698
0
0
0
0
0
-0.7035
-0.7035
-0.7035
-0.7035
0.7035
0.7035
0.7035
0.7035
0
0
0
0
-1.6698
1.6698
0
0
0
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
3.1009
3.1009
0
0
0
0
0
0
0
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
0
0
3.1009
3.1009
0
0
0
0
0
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
1.0918
0
0
0
0
3.1009
3.1009
0
3.4763
2.4385
0.4534
0.4534
0.4534
0.4534
0.4534
0.4534
0.4534
0.4534
0.8991
0.8991
0.8991
0.8991
0.8991
0.8991
0
The X0X matrix for a hybrid 416A design having one center point is:

17
0
0
0
0
0 0 0 0 0 0 14.94 14.94 14.94 14.94
 0 14.94 0
0
0
0 0 0 0 0 0
0
0
0
0 



 0
0
14.94
0
0
0
0
0
0
0
0
0
0
0
0


 0
0
0 14.94 0
0 0 0 0 0 0
0
0
0
0 


 0
0
0
0 14.94 0 0 0 0 0 0
0
0
0
0 



 0
0
0
0
0
9.54
0
0
0
0
0
0
0
0
0




0
0
0
0
0 9.54 0 0 0 0
0
0
0
0 
 0


X0X =  0
0
0
0
0
0 0 9.54 0 0 0
0
0
0
0 


 0
0
0
0
0
0 0 0 9.54 0 0
0
0
0
0 


 0
0
0
0
0
0 0 0 0 9.54 0
0
0
0
0 


 0
0
0
0
0
0 0 0 0 0 9.54 0
0
0
0 


 14.94 0
0
0
0
0 0 0 0 0 0 28.77 9.54 9.54 9.54 


 14.94 0
0
0
0
0 0 0 0 0 0 9.54 28.77 9.54 9.54 


 14.94 0
0
0
0
0 0 0 0 0 0 9.54 9.54 28.77 9.54 
14.94 0
0
0
0
0 0 0 0 0 0 9.54 9.54 9.54 24.53

49
The structure of the expanded design matrix X for a hybrid 416B design with
one center point is:
x0
x1
x2
x3
x4
x1 x2
x1 x3
x1 x4
x2 x3
x2 x4
x3 x4
x21
x22
x23
x24
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
-1.0838
1.0838
-1.0838
1.0838
-1.0838
1.0838
-1.0838
1.0838
1.6448
-1.6448
0
0
0
0
0
0
0
-1.0838
-1.0838
1.0838
1.0838
-1.0838
-1.0838
1.0838
1.0838
0
0
1.6448
-1.6448
0
0
0
0
0
-1.0838
-1.0838
-1.0838
-1.0838
1.0838
1.0838
1.0838
1.0838
0
0
0
0
1.6448
-1.6448
0
1.8768
-0.2918
0.6551
0.6551
0.6551
0.6551
0.6551
0.6551
0.6551
0.6551
-1.1377
-1.1377
-1.1377
-1.1377
-1.1377
-1.1377
0
0
0
1.1746
-1.1746
-1.1746
1.1746
1.1746
-1.1746
-1.1746
1.1746
0
0
0
0
0
0
0
0
0
1.1746
-1.1746
1.1746
-1.1746
-1.1746
1.1746
-1.1746
1.1746
0
0
0
0
0
0
0
0
0
-0.7100
0.7100
-0.7100
0.7100
-0.7100
0.7100
-0.7100
0.7100
-1.8714
1.8714
0
0
0
0
0
0
0
1.1746
1.1746
-1.1746
-1.1746
-1.1746
-1.1746
1.1746
1.1746
0
0
0
0
0
0
0
0
0
-0.7100
-0.7100
0.7100
0.7100
-0.7100
-0.7100
0.7100
0.7100
0
0
-1.8714
1.8714
0
0
0
0
0
-0.7100
-0.7100
-0.7100
-0.7100
0.7100
0.7100
0.7100
0.7100
0
0
0
0
-1.8714
1.8714
0
0
0
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
2.7055
2.7055
0
0
0
0
0
0
0
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
0
0
2.7055
2.7055
0
0
0
0
0
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
1.1746
0
0
0
0
2.7055
2.7055
0
3.5223
0.0851
0.4292
0.4292
0.4292
0.4292
0.4292
0.4292
0.4292
0.4292
1.2945
1.2945
1.2945
1.2945
1.2945
1.2945
0
The X0X matrix for a hybrid 416B design having one center point is:













0
XX = 











17
0
0
0
0
0
0
0
0
0
0
14.81 14.81 14.81 14.81
0
14.81
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14.81
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14.81
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14.81
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.04
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.04
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.04
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.04
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.04
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.04
0
0
0
0
14.81
0
0
0
0
0
0
0
0
0
0
25.68 11.04 11.04 11.04
14.81
0
0
0
0
0
0
0
0
0
0
11.04 25.68 11.04 11.04
14.81
0
0
0
0
0
0
0
0
0
0
11.04 11.04 25.68 11.04
0
0
0
0
0
0
0
0
0
0
11.04 11.04 11.04 23.94
14.81

























50
The structure of the expanded design matrix X for a hybrid 416C design is:
x0
x1
x2
x3
x4
x1 x2
x1 x3
x1 x4
x2 x3
x2 x4
x3 x4
x21
x22
x23
x24
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
-1.0973
1.0973
-1.0973
1.0973
-1.0973
1.0973
-1.0973
1.0973
1.6127
-1.6127
0
0
0
0
0
0
-1.0973
-1.0973
1.0973
1.0973
-1.0973
-1.0973
1.0973
1.0973
0
0
1.6127
-1.6127
0
0
0
0
-1.0973
-1.0973
-1.0973
-1.0973
1.0973
1.0973
1.0973
1.0973
0
0
0
0
1.6127
-1.6127
1.9372
0
0.6227
0.6227
0.6227
0.6227
0.6227
0.6227
0.6227
0.6227
-1.1532
-1.1532
-1.1532
-1.1532
-1.1532
-1.1532
0
0
1.2041
-1.2041
-1.2041
1.2041
1.2041
-1.2041
-1.2041
1.2041
0
0
0
0
0
0
0
0
1.2041
-1.2041
1.2041
-1.2041
-1.2041
1.2041
-1.2041
1.2041
0
0
0
0
0
0
0
0
-0.6833
0.6833
-0.6833
0.6833
-0.6833
0.6833
-0.6833
0.6833
-1.8597
1.8597
0
0
0
0
0
0
1.2041
1.2041
-1.2041
-1.2041
-1.2041
-1.2041
1.2041
1.2041
0
0
0
0
0
0
0
0
-0.6833
-0.6833
0.6833
0.6833
-0.6833
-0.6833
0.6833
0.6833
0
0
-1.8597
1.8597
0
0
0
0
-0.6833
-0.6833
-0.6833
-0.6833
0.6833
0.6833
0.6833
0.6833
0
0
0
0
-1.8597
1.8597
0
0
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
2.6008
2.6008
0
0
0
0
0
0
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
0
0
2.6008
2.6008
0
0
0
0
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
1.2041
0
0
0
0
2.6008
2.6008
3.7527
0
0.3878
0.3878
0.3878
0.3878
0.3878
0.3878
0.3878
0.3878
1.3298
1.3298
1.3298
1.3298
1.3298
1.3298
The X0X matrix for a hybrid 416C design is:













X0X = 











16
0
0
0
0
0
0
0
0
0
0
14.83 14.83 14.83 14.83
0
14.83
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14.83
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14.83
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14.83
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.60
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.60
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10.65
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11.60
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10.65
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10.65
0
0
0
0
14.83
0
0
0
0
0
0
0
0
0
0
25.13 11.60 11.60 10.65
14.83
0
0
0
0
0
0
0
0
0
0
11.60 25.13 11.60 10.65
0
0
0
0
0
0
0
0
0
0
11.60 11.60 25.13 10.65
14.83
14.83
0
0
0
0
0
0
0
0
0
0
10.65 10.65 10.65 25.90

























51
For a 3 factor uniform shell design (UNFSD), the structure of the expanded design
matrix X is:
x0
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1.7321
0.8660
-0.8660
-1.7321
-0.8660
0.8660
0.8660
-0.8660
0
-0.8660
0.8660
0
0
x2
x3
x1 x2
x1 x3
x2 x3
x21
x22
x23
0
0
0
0
0 3.0000
0
0
1.5000
0 1.2990
0
0 0.7500 2.2500
0
1.5000
0 -1.2990
0
0 0.7500 2.2500
0
0
0
0
0
0 3.0000
0
0
-1.5000
0 1.2990
0
0 0.7500 2.2500
0
-1.5000
0 -1.2990
0
0 0.7500 2.2500
0
0.5000 1.4142 0.4330 1.2247 0.7071 0.7500 0.2500 2.0000
0.5000 1.4142 -0.4330 -1.2247 0.7071 0.7500 0.2500 2.0000
-1.0000 1.4142
0
0 -1.4142
0 1.0000 2.0000
-0.5000 -1.4142 0.4330 1.2247 0.7071 0.7500 0.2500 2.0000
-0.5000 -1.4142 -0.4330 -1.2247 0.7071 0.7500 0.2500 2.0000
1.0000 -1.4142
0
0 -1.4142
0 1.0000 2.0000
0
0
0
0
0
0
0
0
The X0X matrix for a 3 factor UNFSD is:








X0X = 







13
0
0
0
0
0
0
12
12
12
0
12
0
0
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
0
0
7.50
2.12
0
0
0
0
0
0
0
0
2.12
6.00
0
0
0
0
0
0
0
0
0
0
6.00
2.12
−2.12
0
12
0
0
0
0
0
2.12
22.50
7.50
6.00
12
0
0
0
0
0
−2.12
7.50
22.50
6.00
12
0
0
0
0
0
0
6.00
6.00
24.00
















52
For a 4 factor UNFSD, the structure of the expanded design matrix X is:
x2
x3
x4
x1 x2
x1 x3
x1 x4
x2 x3
x2 x4
x3 x4
x21
x22
x23
x24
0
1.7321
1.7321
0.5774
0.5774
-1.1547
0.5774
0.5774
-1.1547
0
0
-1.7321
-1.7321
-0.5774
-0.5774
1.1547
-0.5774
-0.5774
1.1547
0
0
0
0
0
1.6330
1.6330
1.6330
0.4082
0.4082
0.4082
-1.2247
0
0
0
-1.6330
-1.6330
-1.6330
-0.4082
-0.4082
-0.4082
1.2247
0
0
0
0
0
0
0
1.5811
1.5811
1.5811
1.5811
0
0
0
0
0
0
-1.5811
-1.5811
-1.5811
-1.5811
0
0
1.7321
-1.7321
0.5774
-0.5774
0
0.5774
-0.5774
0
0
0
1.7321
-1.7321
0.5774
-0.5774
0
0.5774
-0.5774
0
0
0
0
0
0
1.6330
-1.6330
0
0.4082
-0.4082
0
0
0
0
0
1.6330
-1.6330
0
0.4082
-0.4082
0
0
0
0
0
0
0
0
0
1.5811
-1.5811
0
0
0
0
0
0
0
0
1.5811
-1.5811
0
0
0
0
0
0
0.9428
0.9428
-1.8856
0.2357
0.2357
-0.4714
0
0
0
0
0.9428
0.9428
-1.8856
0.2357
0.2357
-0.4714
0
0
0
0
0
0
0
0
0.9129
0.9129
-1.8257
0
0
0
0
0
0
0
0.9129
0.9129
-1.8257
0
0
0
0
0
0
0
0
0.6455
0.6455
0.6455
-1.9365
0
0
0
0
0
0
0.6455
0.6455
0.6455
-1.9365
0
4.0000
1.0000
1.0000
1.0000
1.0000
0
1.0000
1.0000
0
0
4.0000
1.0000
1.0000
1.0000
1.0000
0
1.0000
1.0000
0
0
0
0
3.0000
3.0000
0.3333
0.3333
1.3333
0.3333
0.3333
1.3333
0
0
3.0000
3.0000
0.3333
0.3333
1.3333
0.3333
0.3333
1.3333
0
0
0
0
0
2.6667
2.6667
2.6667
0.1667
0.1667
0.1667
1.5000
0
0
0
2.6667
2.6667
2.6667
0.1667
0.1667
0.1667
1.5000
0
0
0
0
0
0
0
2.5000
2.5000
2.5000
2.5000
0
0
0
0
0
0
2.5000
2.5000
2.5000
2.5000
0
x0 x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
-1
1
-1
0
1
-1
0
0
-2
-1
1
-1
1
0
-1
1
0
0
0
The X0X matrix for a 4 factor UNFSD is:

21
 0

 0

 0

 0

 0


 0

X0X =  0

 0

 0

 0

 20

 20

 20
20
0
20
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
20
20
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20 0
0
0
0
0
0
0
0
0
0 14.67 4.71 3.65
0
0
0
0
0
0
0 4.71 11.33 2.58
0
0
0
0
0
0
0 3.65 2.58 10.00 0
0
0
0
0
0
0
0
0
0 11.33 2.58
0
4.71 −4.71 0
0
0
0
0
2.58 10.00
0
3.65 −3.65 0
0
0
0
0
0
0
10.00 2.58 2.58 −5.16
0
0
0
0
4.71 3.65 2.58 44 14.67 11.33
0
0
0
0 −4.71 −3.65 2.58 14.67 44.00 11.33
0
0
0
0
0
0 −5.16 11.33 11.33 47.33
0
0
0
0
0
0
0 10.00 10.00 10.00

20
0 

0 

0 

0 

0 


0 

0 

0 

0 

0 

10.00 

10.00 

10.00 
50.00
By examining the X0X matrix of a response surface design in a spherical design
region, it can be determined whether or not a design is symmetric. A design is
symmetric if any permutation (relabeling) of the design variables yields the same
53
X0X matrix. In particular, a design is symmetric, if X0X has the form:


N
φ 01
φ 02
β J 0K

 φ1
Diag (β)
φ 03
φ 04

X0X = 
0

 φ2
φ3
Diag (γ)
φ5
0
β JK
φ4
φ5
δ IK + γ JK J K
where φ1 is a K × 1 zero matrix, φ2 is a
K
2
× 1 zero matrix, φ3 is a
matrix, φ4 is a K × K zero matrix, φ5 is a K ×
K
2
K
2
× K zero
zero matrix, JK is a K × 1 unit
column vector, IK is a K × K identity matrix, Diag(β) is a K × K diagonal matrix
with elements β on the diagonal, and Diag(γ) is a
elements γ on the diagonal.
K
2
×
K
2
diagonal matrix with
The CCDs, BBDs, and a hybrid 311B design are symmetric designs because the
X0X matrix of a CCD, a BBD, or a hybrid 311B design has the block matrix form:


N
φ 01
β J 0K

Diag (di )
φ 02
X0X =  φ1
0
β JK
φ2
δ IK + γ JK J K
where
φ1 is a (K +
K
2
) × 1 zero matrix,
φ2 is a K × (K +
K
2
) zero matrix,
JK is a K × 1 unit column vector,
IK is a K × K identity matrix,
Diag(di ) is a diagonal matrix such that
di =
and β, δ, and γ are defined as:
β, for 1 ≤ i ≤ K
γ, for K + 1 ≤ i ≤ K +
K
2
54
Design
CCD
BBD
311B
β
f + 2rs α2
f rα2
f + 2α2 − n2s
δ
2rs α4
f (r − λ) α4
2α4 − α2
γ
f
f λ α4
f − α2
Further simplified, the X0X matrix is:
0
XX =
A11
A21
A12
A22
where
A11
=
A12
=
A21
=
A22
=
N
φ01
φ1 Diag (di )
β J0K
,
φ02
β JK φ2 ,
,
[δ IK + γJK J0K ] .
The (X0X)−1 Matrix for Symmetric Designs
We can find a general form of the (X0X)−1 matrix for CCDs, BBDs, and a hybrid
311B design. This will provide useful information when studying the A, G, and IV
optimality criteria. By using the block matrix inversion formula (Graybill [30] and
Searle [50]):
0
(X X)
where
−1
=
A11
A21
A12
A22
55
A11
−1
= (A11 − A12 A−1
22 A21 ) ,
A12
22
= − A−1
11 A12 A ,
A21
11
= − A−1
22 A21 A ,
A22
−1
= (A22 − A21 A−1
11 A12 ) .
Then, determine the components of A11 , A12 , A21 , and A22 through substitution and matrix inversion. A11 , A12 , A21 , and A22 are combined to form the
(X0X)−1 matrix for a CCD, a BBD based on a BIBD, and a hybrid 311B design.
The (X0X)−1 matrix for a CCD
• The Components of A11 :
(i) The matrix A−1
22 is of the form
A−1
22
=
(aIK + bJK J0K )−1
=
1
a
b
JK J0K
IK −
a + Kb
(2.2)
where a = 2rs α4 and b = f , for more information on Equation 2.2, see Graybill
[30].
Hence,
A−1
22
=
1
2rs α4
f
0
IK −
JK JK .
2rs α4 + Kf
(ii) Then, pre-multiplying A−1
22 by A12 :
A12 A−1
22
=
=
=
f + 2rs α2 J0K
f
0
IK −
JK JK
φ02
2rs α4
2rs α4 + Kf
0 0 Kf
f + 2rs α2
JK
JK
−
0
4
4
φ2
2rs α
2rs α + Kf φ02
f + 2rs α2
J0K
.
2rs α4 + Kf φ02
56
(iii) Then, post-multiplying by A21 :
A12 A−1
22 A21
=
=
(f + 2rs α2 )2 J0K JK φ 2
0
4
φ
2rs α + Kf
2 0 2 2
(f + 2rs α )
K φ1
4
2rs α + Kf φ1 φ?
where φ? is a 1 × 1 zero matrix. Then,
A11
A11 − A12 A−1
22 A21
=
N−
=
2rs
=
K(f +2rs α2 )2
2rs α4 +Kf
φ1
−1
φ01
Diag (di )
Kf +2rs α4
α4 N +Kf N −K(f +2r
s
!−1
φ01
Diag d1i
α2 ) 2
φ1
• The Components of A22 :
!
.
(i) The matrix A21 A−1
11 is of the form
A21 A−1
11
=
=
2
(f + 2rs α )
JK φ 2
f + 2rs α2 JK φ 2 .
N
1
N
φ1
φ01
Diag ( d1i )
(ii) Then, post-multiplying by A12 :
A21 A−1
11 A12
=
=
(f + 2rs α2 )2 JK φ 2
N
(f + 2rs α2 )2
JK J0K .
N
J0K
φ02
(iii) Therefore,
A22
=
=
=
−1
(A22 − A21 A−1
11 A12 )
−1
(f + 2rs α2 )2
0
4
0
f JK JK + 2rs α IK −
JK JK
N
−1
N f − (f + 2rs α2 )2
0
4
JK JK
(same form as in Equation 2.2)
2rs α IK +
N
57
where a = 2rs α4 and b =
N f −(f +2rs α2 )2
.
N
Thus,
A
22
=
N f − (f + 2rs α2 )2
1
0
IK −
JK JK .
2rs α4
2rs α4 N + Kf N − K(f + 2rs α2 )2
• The Components of A12 :
(i) The matrix A12 A22 is of the form
A12 A
22
=
=
=
N f − (f + 2rs α2 )2
f + 2rs α2 J0K
0
JK JK
IK −
φ02
2rs α4
2rs α4 N + Kf N − K(f + 2rs α2 )2
0 0 Kf N − K(f + 2rs α2 )2
f + 2rs α2
JK
JK
−
φ02
2rs α4
2rs α4 N + Kf N − K(f + 2rs α2 )2 φ02
0 N (f + 2rs α2 )
JK
.
4
2
2
φ02
2rs α N + Kf N − K(f + 2rs α )
(ii) Then pre-multiplying by −A−1
11 :
A12
=
=
=
22
−A−1
11 A12 A
1
0 N (f + 2rs α2 )
φ01
JK
N
−
1
4
2
2
φ
Diag
(
)
φ02
2rs α N + Kf N − K(f + 2rs α )
1
di
f + 2rs α2
J0K
.
−
2rs α4 N + Kf N − K(f + 2rs α2 )2 φ02
• The Components of A21 :
(i) The matrix −A−1
22 A21 is of the form
−A−1
22 A21
=
=
=
f + 2rs α2
f
0
J
φ
−
I
−
J
J
K
2
K
K
K
2rs α4
2rs α4 + Kf
f + 2rs α2 Kf
JK φ 2 −
JK φ 2
−
2rs α4
2rs α4 + Kf
f + 2rs α2 J
φ
.
−
K
2
2rs α4 + Kf
58
(ii) Then, post-multiplying by A11 :
A21
11
−A−1
22 A21 A
=
f + 2rs α2 J
φ
−
K
2
2rs α4 + Kf
=
=
−
"
2rs
Kf +2rs α4
α4 N +Kf N −K(f +2r
s
α2 ) 2
φ1
φ01
Diag ( d1i )
#
f + 2rs α2
J
φ
.
K
2
2rs α4 N + Kf N − K(f + 2rs α2 )2
By combining the resulting forms of A11 , A12 , A21 , and A22 , the (X0X)−1 matrix for a CCD is:
0
(X X)
−1
=
A11
A21
A12
A22
where
"
φ01
2rs
s
φ1
Diag d1i
"
#
f +2rs α2
0
− 2rs α4 N +Kf
J
N −K(f +2rs α2 )2 K
,
φ02
A11
=
A12
=
A21
=
A012
=
h
A22
=
Kf +2rs α4
α4 N +Kf N −K(f +2r
α2 ) 2
#
,
i
f +2rs α2
− 2rs α4 N +Kf
J
φ
,
2
N −K(f +2rs α2 )2 K
N f − (f + 2rs α2 )2
1
0
IK −
JK JK .
2rs α4
2rs α4 N + Kf N − K(f + 2rs α2 )2
The (X0X)−1 matrix for a BBD
For the (X0X)−1 matrix of a BBD based on a BIBD, apply the same procedure
that was used on a CCD. This provides the following form of (X0X)−1 matrix for a
BBD:
0
(X X)
−1
=
A11
A21
A12
A22
59
where
A11
=
A12
=
A21
=
A22
=
"
#
φ01
,
φ1
Diag ( d1i )
#
"
2
− f (r−λ)α4 N +f fλαrα4 KN −K(f rα2 )2 J0K
,
φ02
h
i
2
− f (r−λ)α4 N +f fλαrα4 KN −K(f rα2 )2 JK φ2 ,
f λα4 N − (f rα2 )2
1
0
IK −
JK JK .
f (r − λ)α4
f (r − λ)α4 N + f λα4 KN − K(f rα2 )2
Kf λα4 +f (r−λ)α4
f (r−λ)α4 N +f λα4 KN −K(f rα2 )2
The (X0X)−1 matrix for a hybrid 311B
For the (X0X)−1 matrix of a hybrid 311B design, let f be the number of points
for the factorial portion of the design, ns is the number of points for the star or axial
portion, and α is the value of the star distance where α =
√
K. Then, apply the same
procedure that was used on a CCD. This provides the following form of the (X 0X)−1
matrix for a hybrid 311B design:
0
(X X)
−1
=
A11
A21
A12
A22
where
A11
=
A12
=
A21
=
A22
=
#
φ01
,
φ1
Diag ( d1i )
#
"
+2α2 −n2s
0
J
− (2α4 −α2 )N +(ff−α
2 )KN −K(f +2α2 −n2 ) K
s
,
φ02
h
i
+2α2 −n2s
,
− (2α4 −α2 )N +(ff−α
J
φ
2
2 )KN −K(f +2α2 −n2 ) K
s
1
(f − α2 )N − (f + 2α2 − n2s )2
0
IK −
JK JK .
2α4 − α2
(2α4 − α2 )N + (f − α2 )KN − K(f + 2α2 − n2s )
"
K(f −α2 )+2α4 −α2
(2α4 −α2 )N +(f −α2 )KN −K(f +2α2 −n2s )
60
In general, the closed-form of the (X0X)−1 matrix for a CCD, a BBD based on
a BIBD, and a hybrid 311B design can be written in an analogous form to the form
given by Borkowski [2]. That is,

(X0X)−1


=
where
α11
φ1
φ01 Diag d1i
α12 JK φ2
α11
=
α12
=
α22
=
α12 J0K
φ02
1
δ
[IK − α22 JK J0K ]
γK + δ
,
δN + γKN − Kβ 2
β
,
−
δN + γKN − Kβ 2
γN − β 2
,
δN + γKN − Kβ 2
and
K
(K +
) × 1 zero matrix,
2
K
K × (K +
) zero matrix,
2
φ1
=
φ2
=
JK
=
K × 1 unit column vector,
IK
=
K × K identity matrix,
and Diag (di ) is a diagonal matrix such that
di =
β, for 1 ≤ i ≤ K
γ, for K + 1 ≤ i ≤ K +
and define β, δ, and γ as:
Design
CCD
BBD
311B
β
f + 2rs α2
f rα2
f + 2α2 − n2s
δ
2rs α4
f (r − λ) α4
2α4 − α2
K
2
γ
f
f λ α4
f − α2



61
Because the X0X matrices of SCDs, 310s, 311As, and UNFSDs for K = 3, and
SCDs, PBCDs, 416As, 416Bs, 416Cs, and UNFSDs for K = 4 are not in an acceptable
block matrix form, they are not symmetric. Thus, there is not a simple closed-form
of the (X0X)−1 matrix for these spherical response surface designs.
The |X0X| for Symmetric Designs
For each reduced model, let K = number of design variables, N = the total
number of design points, l = number of linear terms, c = number of cross-product
terms, and q = number of quadratic terms in the model. Borkowski and Valeroso [9]
found a closed-form for |X0X| for CCDs and BBDs based on BIBDs using the structure
of the X0X matrices. Using their method, a closed-form of |X0X| for the hybrid 311B
design was found. This closed-form of the |X0X| will provide useful information when
studying the D optimality criterion. For these designs:
0
l c q
|X X| = N β γ δ
h
i
q
2
(N γ − β )
1+
Nδ
and where β, δ, and γ are defined as:
Design
CCD
BBD
311B
β
f + 2rs α2
f rα2
f + 2α2 − n2s
δ
2rs α4
f (r − λ) α4
2α4 − α2
γ
f
f λ α4
f − α2
For more information on |X0X| criterion and some related matters, see Box and
Draper [14].
62
Spherical Prediction Variance Properties
The average spherical prediction variance Vρ is the expected value of the scaled
prediction variance function N V (x) assuming a uniform distribution on the spherical
o
n P
surface Sρ where V (x) = f 0 (x)(X0X)−1 f (x), x ∈ Sρ , and Sρ = x : ki=1 x2i = ρ2 .
Thus,
N
Vρ =
ωρ
Z
N
V (x)dx =
ωρ
Sρ
where the surface area of Sρ , denoted ωρ =
and
Γ
K
2
=



K−2
2
Z
√
π
(2.3)
Sρ
R
!
(K−2)(K−4)···(3)(1)
2(K−1)/2
f 0 (x)(X0X)−1 f (x) dx
dx =
Sρ
√
2ρK−1 ( π)K
K
Γ( 2 )

for K even. 
for K odd.

.
For more information on spherical surface area, see Courant [20].
To evaluate the integral in ( 2.3), Borkowski [4] converted rectangular coordinates
to hyperspherical coordinates and evaluated
R
Sρ
V (x) dx as a spherical surface inte-
gral. For more information on spherical surface integrals, see Edwards [26] and Buck
[17].
63
Borkowski [4] used the following hyperspherical representation of x = (x 1 , . . . , xK ):
x1 (θ) = ρcosθ1
x2 (θ) = ρsinθ1 cosθ2
x3 (θ) = ρsinθ1 sinθ2 cosθ3
..
.
..
.
..
.
(2.4)
xK−2 (θ) = ρsinθ1 . . . sinθK−3 cosθK−2
xK−1 (θ) = ρsinθ1 . . . sinθK−3 sinθK−2 cosθK−1
xK (θ) = ρsinθ1 . . . sinθK−3 sinθK−2 sinθK−1 .
That is, a constant ρ corresponds to a sphere centered at the origin. Note that ρ =
qP
k
2
i=1 xi (θ), θi is the angle between the vector x and the axis xi for i = 1, . . . , K −1,
and θ = (ρ, θ1 , . . . , θK−1 ) such that ρ ≥ 0, 0 ≤ θi ≤ π for i = 1, . . . , K − 2 and
0 ≤ θK−1 ≤ 2π. The geometric meanings of ρ, θi for i = 1, . . . , K − 1 are shown in
Figure 1 for K = 3.
Figure 1. Spherical Coordinates for K = 3.
x1
6
ρ
@
R
@
θ1
- x3
θ@
2
x2
q (x1 , x2 , x3 )
@
@ ρ cos θ1
ρ sin θ @
1
64
Then,
Z
V (x)dx =
Sρ
Z
√
0
dv
Z
π
···
0
Z
π
0
Z
2π
V (θ)
0
p
|D| dθ
(2.5)
where dv is the number of design variables in model and |D| is the Jacobian of the
transformation (see Courant [20]). The Jacobian |D| associated with this transformation (Edwards [27]) is given by
D = ρ2(K−1)
K−2
Y
sin2(K−1−t) θt .
(2.6)
t=1
Hence,
p
|D| = ρ(K−1)
K−2
Y
sin(K−1−t) θt .
(2.7)
t=1
This spherical prediction variance provides useful information when studying the
IV optimality criterion in this dissertation. For more information on spherical prediction variance properties, see Borkowski [4, 5], Giovannitti-Jensen and Myers [29],
and Myers, Vining, Giovannitti-Jensen and Myers [46].
The results of the research related to the optimality criteria and the design criteria
comparisons for the full second-order model for 3 and 4 factor response surface designs
in a spherical design region based on D, A, G, and IV criteria will be presented in
Chapter 3.
65
CHAPTER 3
OPTIMALITY CRITERIA FOR A SPHERICAL RESPONSE SURFACE
DESIGNS
Optimality Criteria for the Full Second Order Model
In this research, one and three center point CCDs, BBDs, SCDs, UNFSDs, 310,
311A, and 311B designs are considered for K = 3 design variables and one and three
center point CCDs, BBDs, SCDs, PBCDs, UNFSDs, 416A, 416B designs and one and
two center point 416C designs are considered for K = 4 design variables. The four
optimality criteria D, A, G, and IV criteria are computed for the proposed secondorder model in ( 2.1) assuming a spherical response surface design region. The results
are shown in Table 9 and Table 10.
Table 9 and Table 10 indicate the following general results:
1. Replicating star points (increasing rs ) tends to reduce the D, A, and G criteria
and increase the IV criterion for the CCDs when K = 3 and 4 factors, and for
PBCD when K = 4 factors. Similar results are true of the SCDs for the A and
IV criteria. The SCD exceptions are for the D criterion when K = 3 factors
and for the G criterion when K = 3 and 4 factors.
2. Increasing center points (increasing n0 ) tends to reduce the D and G criteria
except for the G criterion of the CCDs when K = 3 and 4 factors whether or
66
not star points are replicated, and for the G criterion of the BBD when K = 4
factors. Increasing n0 , however, tends to improve the A and IV criteria.
Table 9. The Optimality Criteria for K = 3 Design Variables.
Designs
CCD
BBD
SCD
310
311A
311B
UNFSD
rs
1
2
1
2
–
–
1
2
1
2
–
–
–
–
–
–
–
–
–
n0
1
1
3
3
1
3
1
1
3
3
0
1
3
1
3
1
3
1
3
N
15
21
17
23
13
15
11
17
13
19
10
11
13
11
13
11
13
13
15
D-Eff
71.1296
67.3113
70.0495
68.5948
69.5854
67.3104
59.0785
56.6631
55.7945
56.5859
62.1772
60.6397
55.0194
67.6003
63.8425
70.9973
67.0507
69.5913
67.3162
A-Eff
32.4011
24.6659
50.3343
41.7389
35.5007
52.1694
28.1641
22.1162
32.8879
29.7209
36.9127
45.7457
47.1490
37.4090
50.6899
37.8798
50.9072
34.0475
48.6477
G-Eff
66.6667
47.6190
89.2039
76.4730
76.9140
66.6588
32.7923
33.3844
27.7473
29.8702
47.3893
45.0198
38.9577
78.6243
69.0153
90.9091
77.4084
76.9231
66.6770
IV -criterion
17.5556
23.1576
9.4271
11.1987
16.3622
9.2957
17.0840
22.4519
12.1843
13.3923
14.3356
10.6710
9.6415
14.4549
9.2126
14.4290
9.2126
16.3622
9.6418
The results of these tables suggest replication affects the different criteria in very
different ways. That is, what improves one criterion may be detrimental to a different
criterion.
67
Table 10. The Optimality Criteria for K = 4 Design Variables.
Designs
CCD
BBD
SCD
PBCD
416A
416B
416C
UNFSD
rs
1
2
1
2
–
–
1
2
1
2
1
2
1
2
–
–
–
–
–
–
–
–
n0
1
1
3
3
1
3
1
1
3
3
1
1
3
3
1
3
1
3
1
2
1
3
N
25
33
27
35
25
27
17
25
19
27
21
29
23
31
17
19
17
19
16
17
21
23
D-Eff
76.7266
73.4893
76.4417
74.5552
76.7262
76.4413
65.0312
61.5916
62.6073
61.3629
69.8808
66.4403
68.6527
66.8769
70.0185
67.1423
73.5228
68.9424
74.9411
73.8686
72.4056
71.1331
A-Eff
31.6484
25.1869
52.2876
44.1210
31.6483
52.2874
30.1982
24.2526
37.8002
33.8624
31.0800
24.4177
44.5206
37.6562
39.0902
52.7264
52.3632
58.0290
40.8478
52.9251
36.6220
48.0865
G-Eff
60.0000
45.4545
95.2381
81.4780
60.0000
95.2376
29.3713
32.6890
26.2796
30.2676
44.2317
39.3544
40.3854
36.8154
74.3053
69.1018
70.0683
62.8626
77.4937
72.9368
71.4286
67.5553
IV -criterion
33.6111
42.4875
17.1000
20.1736
33.6112
17.1001
29.4667
37.7778
19.4222
21.6000
31.3200
40.3332
17.9473
21.0704
23.9666
15.1685
17.0991
13.9233
24.4033
16.9110
30.2400
16.7645
The results for the G-criterion should not be surprising. To reduce the maximum
prediction variance, an additional point should be at or near the point where the
maximum prediction variance occurs. Usually this is on the boundary of the design
region and not near the center point. Thus, for most designs, the G-efficiency will
not be improved by replication of center points.
68
The results of replicating star-points and center points for the CCD in a spherical
design region are consistent with the results for the CCD in a hypercube design region
(Borkowski [3]).
Although some efficiencies may decrease when replicating star-points or increasing
the number of center points, experimenters may be willing to sacrifice design efficiency
to gain pure error degrees of freedom for a lack-of-fit test.
Design Criteria Comparison Ranking
In this section, the four optimality criteria (D, A, G, and IV ) comparisons for
seven 3-factor designs (CCDs, BBDs, SCDs, 310s, 311As, 311Bs, UNFSDs) and eight
4-factor designs (CCDs, BBDs, SCDs, PBCDs, 416As, 416Bs, 416Cs, UNFSDs) for
the full second order model will be summarized. For the D, A, and G criteria, larger
values imply a better design (on a per point basis), while for the IV criterion, a
smaller value implies a better design. It is important to stress that the comparison is
on a ’per point basis’, or, in other words, the optimality criteria are based on functions
that are scaled by the design size N . Thus, the experimenter hopes that any gains in
the prediction variance properties are not offset by increased sample size.
For the comparison ranking for K = 3, each entry in Table 11 and Table 12
contains the row rank that ranges from 1 (’best’) to 7 (’worst’). The rank represents
that design’s rank relative to the other 6 designs. For K = 4, each entry in Table 13
and Table 14 contains the row rank that ranges from 1 (’best’) to 8 (’worst’). The
69
rank also represents that design’s rank relative to the other 7 designs. In case of ties,
average ranks are shown.
Table 11. Design Optimality Criteria Comparison Ranking for K = 3, n0 = 1.
Designs
CCD
BBD
SCD
310
311A
311B UNFSD
Criterion (N =15) (N =13) (N =11,17) (N =11) (N =11) (N =11) (N =13)
D
1
4
7
6
5
2
3
A
6
4
7
1
3
2
5
G
5
4
7
6
2
1
3
IV
7
4.5
6
1
3
2
4.5
Table 12. Design Optimality Criteria Comparison Ranking for K = 3, n0 = 3.
Designs
CCD
BBD
SCD
310
311A
311B UNFSD
Criterion (N =17) (N =15) (N =13,19) (N =13) (N =13) (N =13) (N =15)
D
1
3
6
7
5
4
2
A
4
1
7
6
3
2
5
G
1
5
7
6
3
2
4
IV
4
3
7
5
1.5
1.5
6
Based on the one center point results in Table 11, the CCD is the superior design
for the D criterion. The 310 design is the superior design for the A and IV criteria.
The 311B design is the superior design for the G criterion. However, the 311B design
is robust with respect to all four criteria and requires only 11 experimental runs,
while the SCD is inefficient with respect to all four criteria. Thus, if resources are
limited, the 311B design is recommended. The large variability of the ranks for the
70
Table 13. Design Optimality Criteria Comparison Ranking for K = 4, n0 = 1.
Designs
CCD
BBD
SCD
PBCD
416A
416B
416C UNFSD
Criterion (N =25) (N =25) (N =17,25) (N =21) (N =17) (N =17) (N =16) (N =21)
D
1.5
1.5
8
7
6
4
3
5
A
5.5
5.5
8
7
3
1
2
4
G
5.5
5.5
8
7
2
4
1
3
IV
7.5
7.5
4
6
2
1
3
5
CCD and the 310 design indicates the potential inconsistency of design efficiencies
across multiple criteria.
Based on the three center point results in Table 12, the CCD is the superior design
for the D and G criteria, the BBD is the superior design for the A criterion, and the
311A and 311B designs are the superior designs for the IV criterion. However, if there
are only enough resources to run a 15-point design, choosing between the UNFSD
and the BBD will depend on the criterion. If there are only enough resources to run
a 13-point design, the 311B design is recommended specifically because it is robust
with respect to the A, G, and IV criteria. The 310 design and (once again) the SCD
are consistently inefficient across the four criteria.
For any number of center points, the four factor CCD and BBD are both rotatable
designs and have identical efficiencies values. This is reflected in the identical rankings
in Table 13 and Table 14. Based on the one center point results in Table 13, they
are also the superior designs for the D criterion. The 416B design, however, is the
superior design for the A and IV criteria. The 416C design is the superior design for
71
the G criterion. If the D criterion is considered, and there are not enough resources to
run the 25-point CCD or BBD, the 17-point 416C design is robust to all four criteria
and requires 9 fewer runs than the CCD and BBD. Similarly, the 416B design is the
best 17-point design, and the UNFSD is the best 21-point design.
Table 14. Design Optimality Criteria Comparison Ranking for K = 4, n0 = 2, 3.
Designs
CCD
BBD
SCD
PBCD
416A
416B
416C UNFSD
Criterion (N =27) (N =27) (N =19,27) (N =23) (N =19) (N =19) (N =17) (N =23)
(n0 = 3) (n0 = 3) (n0 = 3) (n0 = 3) (n0 = 3) (n0 = 3) (n0 = 2) (n0 = 3)
D
1.5
1.5
8
6
7
5
3
4
A
4.5
4.5
8
7
3
1
2
5
G
1.5
1.5
8
7
4
6
3
5
IV
5.5
5.5
8
7
2
1
4
3
Based on the multiple center point results in Table 14, the CCD and BBD are
the superior designs for the D and G criteria. The 416B design is the superior design
for the A and IV criteria. If there are not enough resources to run the 27-point CCD
and BBD, the 416C design is robust with respect to the D and G criteria and requires
10 fewer runs than the CCD and BBD. If the A criterion is considered, and if there
are not enough resources to run the 19-point 416B design, the 416C design is robust
and requires 2 fewer runs. In general, the CCD and BBD are robust with respect to
the D and G criteria and the 416B, 416C, and 416A designs are robust with respect
to the A and IV criteria. As in Table 13, the UNFSD is better than the PBCD for
23-point designs.
72
Tables 11, 12, 13, and 14 indicate the consistently poor performance of the SCDs
and PBCDs across the four criteria. The reason for this is based on the factorial points
used in these designs. Because the SCD uses a Resolution III or IV fractional factorial
design, this portion of the design aliases main effects with two-factor interactions
(Resolution III) or it aliases two-factor interactions with other two-factor interactions
(Resolution IV). Thus, the factorial points have D, A and G efficiencies equal to 0 and
the IV criterion equal to ∞. Addition of center and star points permits estimation
of all effects but very inefficiently. The PBCD is based on a 12-run Plackett-Burman
design which has a very complex alias structure (Lin and Draper [36]) but, like the
SCD, is inefficient at estimation of two-factor interactions and remains inefficient
when center points and star points are added to form the PBCD.
The results given in Table 11, 12, 13, and 14 are of practical value only if the
second-order model given in ( 2.1) is appropriate. This, however, is not often the case
as discussed later in Chapter 4.
73
VIFs and the Design Criteria
The variance inflation factor (VIF) for the ith regression coefficient is defined as:
VIFi =
1
,
1 − Ri2
where Ri2 is the coefficient of multiple determination of the regression produced by an
independent variable xi against the all other independent variables xj where (j 6= i).
Thus, higher values of VIF indicate increased multicollinearity, while a value of VIF i
close to one indicates no linear relationship between an independent variable xi and
all other independent variables xj , (j 6= i) (Sen and Srivastava [51]).
In this section, the VIFs for 3 and 4 factor spherical response surface designs are
given in Table 15 and Table 16. In addition, Tables 17 to 25 contain mean VIFs,
optimality criteria values, and ranks for 3 and 4 factor spherical response surface
designs of equal design sizes N . Specifically, there are 4 columns in Tables 17 to 25
that contain the D, A, G, and IV criteria values and the column rank that range from
1 (’best’) to the number of designs that are compared (’worst’). The rank represents
an optimality criteria’s rank (D, A, G, or IV ) relative to the other designs. In the
mean of VIFs column, the rank represents that average VIF’s rank relative to the
other designs’ average VIFs.
By comparing the mean VIF ranks to the optimality criterion ranks, we can
perform a quick exploratory analysis of potential relationship between VIFs and optimality criteria. We would expect a large VIF (VIF > 5) or several moderately large
VIFs (3 < VIF < 5) to adversely affect optimality criteria. For example,
74
1. When several of the VIFs are moderately large, it will (1) be reflected in a
larger trace (σ 2 (X0 X)−1 ), and, hence, a smaller A-efficiency and (2) negatively
affect the prediction variances at all points in the design space, and therefore
increase the IV -criterion values. However, it is unclear how large the impact
larger VIFs may have on the D and G-efficiencies.
2. When all of the VIFs are small, we would expect the D, A, and G efficiencies
would be high and the IV criterion to be low.
Tables 15 and Table 16 support the supposition that the A optimality criterion
tends to be higher when VIFs are smaller and IV optimality criterion tends to be lower
when VIFs are smaller. Differences between mean VIF ranks and A or IV criteria
ranks typically occur when the A or IV optimality criteria values are relatively close.
For example, in Table 18, the A-efficiency values for the 310, 311A, and 311B designs
(A-eff = 47.1, 50.7, and 50.9, respectively) are relatively close and are ranked 3, 2,
and 1, respectively. Thus, it is not surprising that when using a crude measure like
the mean VIF, the ranks may change (ranked 1, 2, and 3, respectively). However,
these Tables also indicate that there is no obvious relationship between the mean
VIFs and D and G optimality criteria.
75
Table 15. VIFs for the 3-Factor Response Surface Designs.
CCD
SCD
(rs , n0 )
Term
x1
x2
x3
x12
x13
x23
x21
x22
x23
Mean
(1, 1)
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
2.01869
2.01869
2.01869
1.3396
(2, 1)
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
3.37319
3.37319
3.37319
1.7911
(rs , n0 )
(1, 3)
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.18674
1.18674
1.18674
1.0622
(2, 3)
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.62609
1.62609
1.62609
1.2087
(1, 1)
1.66671
1.66671
1.66671
1.66671
1.66671
1.66671
2.05589
2.05589
2.05589
1.7964
(2, 1)
1.33335
1.33335
1.33335
1.33335
1.33335
1.33335
3.40632
3.40632
3.40632
2.0243
(1, 3)
1.66671
1.66671
1.66671
1.66671
1.66671
1.66671
1.21880
1.21880
1.21880
1.5174
(2, 3)
1.33335
1.33335
1.33335
1.33335
1.33335
1.33335
1.65789
1.65789
1.65789
1.4415
Table 15. cont’d
BBD
310
n0
Term
x1
x2
x3
x12
x13
x23
x21
x22
x23
Mean
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.34615
1.34615
1.34615
1.1154
311A
n0
3
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.01111
1.01111
1.01111
1.0037
0
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.12120
1.12120
1.14855
1.0434
n0
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00456
1.00456
1.00559
1.0016
3
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.02947
1.02947
1.03612
1.0106
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.49716
1.49716
1.56818
1.1736
Table 15. cont’d
Term
x1
x2
x3
x12
x13
x23
x21
x22
x23
Mean
311B
UNFSD
n0
n0
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.81810
1.81810
1.81818
1.2727
3
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.13957
1.13957
1.13960
1.0465
1
1.00000
1.00000
1.00000
1.11111
1.11111
1.11111
1.90385
1.90385
2.03419
1.3528
3
1.00000
1.00000
1.00000
1.11111
1.11111
1.11111
1.19444
1.19444
1.20000
1.1025
3
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.05048
1.05048
1.05769
1.0176
76
Table 16. VIFs for the 4-Factor Response Surface Designs.
Term
x1
x2
x3
x4
x12
x13
x14
x23
x24
x34
x21
x22
x23
x24
Mean
(1, 1)
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
2.21000
2.21000
2.21000
2.21000
1.3457
CCD
SCD
(rs , n0 )
(2, 1)
(1, 3)
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
1.00000 1.00000
3.73011 1.25000
3.73011 1.25000
3.73011 1.25000
3.73011 1.25000
1.7800
1.0714
(rs , n0 )
(2, 1)
(1, 3)
1.50000 2.00000
1.50000 2.00000
1.00000 1.00000
1.50000 2.00000
1.50000 2.00000
1.00000 1.00000
1.50000 2.00000
1.00000 1.00000
1.50000 2.00000
1.00000 1.00000
3.76125 1.27796
3.76125 1.27796
3.76125 1.27796
3.76125 1.27796
2.0032
1.5080
(2, 3)
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.75089
1.75089
1.75089
1.75089
1.2145
(1, 1)
2.00000
2.00000
1.00000
2.00000
2.00000
1.00000
2.00000
1.00000
2.00000
1.00000
2.24081
2.24081
2.24081
2.24081
1.7831
(2, 3)
1.50000
1.50000
1.00000
1.50000
1.50000
1.00000
1.50000
1.00000
1.50000
1.00000
1.78125
1.78125
1.78125
1.78125
1.4375
Table 16. cont’d
Term
x1
x2
x3
x4
x12
x13
x14
x23
x24
x34
x21
x22
x23
x24
Mean
(1, 1)
1.29464
1.29464
1.29464
1.29464
1.36607
1.36607
1.36607
1.36607
1.36607
1.36607
2.22232
2.22232
2.22232
2.22232
1.5903
PBCD
BBD
416A
(rs , n0 )
(2, 1)
(1, 3)
1.19318 1.29464
1.19318 1.29464
1.19318 1.29464
1.19318 1.29464
1.28409 1.36607
1.28409 1.36607
1.28409 1.36607
1.28409 1.36607
1.28409 1.36607
1.28409 1.36607
3.74346 1.26114
3.74346 1.26114
3.74346 1.26114
3.74346 1.26114
1.9608
1.3157
n0
n0
(2, 3)
1.19318
1.19318
1.19318
1.19318
1.28409
1.28409
1.28409
1.28409
1.28409
1.28409
1.76390
1.76390
1.76390
1.76390
1.3952
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
2.21000
2.21000
2.21000
2.21000
1.3457
3
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.25000
1.25000
1.25000
1.25000
1.0714
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.56775
1.56775
1.56775
1.66570
1.1692
3
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.08486
1.08486
1.08486
1.09951
1.0253
77
Table 16. cont’d
416B
416C
n0
Term
x1
x2
x3
x4
x12
x13
x14
x23
x24
x34
x21
x22
x23
x24
Mean
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.10674
1.10674
1.10674
1.11590
1.0312
UNFSD
n0
3
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00431
1.00431
1.00431
1.00468
1.0013
1
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.36904
1.36904
1.36904
1.50654
1.1153
n0
2
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.09111
1.09111
1.09111
1.14699
1.0300
1
1.00000
1.00000
1.00000
1.00000
1.22225
1.18055
1.12500
1.18056
1.12500
1.12499
2.33929
2.33931
2.50448
2.61161
1.4824
3
1.00000
1.00000
1.00000
1.00000
1.22225
1.18055
1.12500
1.18056
1.12500
1.12499
1.38587
1.38588
1.40353
1.39266
1.1804
Table 17. Mean VIFs, Criteria Values, and Ranks for 3-Factor, 11-Point Designs.
Design
SCD
310
311A
311B
D-eff
59.0785
4
60.6397
3
67.6003
2
70.9973
1
A-eff
28.1641
4
45.7457
1
37.4090
3
37.8798
2
G-eff
32.7923
4
45.0198
3
78.6243
2
90.9091
1
IV -criterion
17.0814
4
10.6710
1
14.4549
3
14.4290
2
Mean of VIFs
1.7964
4
1.0016
1
1.1736
2
1.2727
3
78
Table 18. Mean VIFs, Criteria Values, and Ranks for 3-Factor, 13-Point Designs.
Design
BBD
SCD
310
311A
311B
UNFSD
D-eff
69.5854
2
55.7945
5
55.0194
6
63.8425
4
67.0507
3
69.5913
1
A-eff
35.5007
4
32 8879
6
47.1490
3
50.6899
2
50.9072
1
34.0475
5
G-eff
76.9140
3
27.7473
6
38.9577
5
69.0153
4
77.4084
1
76.9231
2
IV -criterion
16.3622
5.5
12.1843
4
9.6415
3
9.2126
1.5
9.2126
1.5
16.3622
5.5
Mean of VIFs
1.1154
4
1.5174
6
1.0106
1
1.0176
2
1.0465
3
1.3528
5
Table 19. Mean VIFs, Criteria Values, and Ranks for 3-Factor, 15-Point Designs.
Design
CCD
BBD
UNFSD
D-eff
71.1296
1
67.3104
3
67.3162
2
A-eff
32.4011
3
52.1694
1
48.6477
2
G-eff
66.6667
2
66.6588
3
66.6770
1
IV -criterion
17.5556
3
9.2957
1
9.6418
2
Mean of VIFs
1.3396
3
1.0037
1
1.1025
2
Table 20. Mean VIFs, Criteria Values, and Ranks for 4-Factor, 17-Point Designs.
Design
SCD
416A
416B
416C
D-eff
65.0312
4
70.0185
3
73.5228
2
73.8686
1
A-eff
30.1982
4
39.0902
3
52.3632
2
52.9251
1
G-eff
29.3713
4
74.3053
1
70.0683
3
72.9368
2
IV -criterion
29.4667
4
23 9666
3
17.0991
2
16.9110
1
Mean of VIFs
1.7831
4
1.1692
3
1.0312
2
1.0300
1
79
Table 21. Mean VIFs, Criteria Values, and Ranks for 4-Factor, 19-Point Designs.
Design
SCD
416A
416B
D-eff
62.6073
3
67.1423
2
68.9424
1
A-eff
37.8002
3
52.7264
2
58.0290
1
G-eff
26.2796
3
69.1018
1
62.8626
2
IV -criterion
19.4222
3
15.1685
2
13.9233
1
Mean of VIFs
1.5080
3
1.0253
2
1.0013
1
Table 22. Mean VIFs, Criteria Values, and Ranks for 4-Factor, 21-Point Designs.
Design
PBCD
UNFSD
D-eff
69.8808
2
72.4056
1
A-eff
31.0800
2
36.6220
1
G-eff
44.2317
2
71.4286
1
IV -criterion
31.3200
2
30.2400
1
Mean of VIFs
1.5903
2
1.4824
1
Table 23. Mean VIFs, Criteria Values, and Ranks for 4-Factor, 23-Point Designs.
Design
PBCD
UNFSD
D-eff
68.6527
2
71.1331
1
A-eff
44.5206
2
48.0865
1
G-eff
40.3854
2
67.5553
1
IV -criterion
17.9473
2
16.7645
1
Mean of VIFs
1.3157
2
1.1804
1
Table 24. Mean VIFs, Criteria Values, and Ranks for 4-Factor, 25-Point Designs.
Design
CCD
BBD
SCD
D-eff
76.7266
1.5
76.7262
1.5
61.5916
3
A-eff
31.6484
1.5
31.6483
1.5
24.2526
3
G-eff
60.0000
1.5
60.0000
1.5
32.6890
3
IV -criterion
33.6111
1.5
33.6112
1.5
37.7778
3
Mean of VIFs
1.3457
1.5
1.3457
1.5
2.0032
3
Table 25. Mean VIFs, Criteria Values, and Ranks for 4-Factor, 27-Point Designs.
Design
CCD
BBD
SCD
D-eff
76.4417
1.5
76.4413
1.5
61.3629
3
A-eff
52.2876
1.5
52.2874
1.5
33.8624
3
G-eff
95.2381
1.5
95.2376
1.5
30.2676
3
IV -criterion
17.1000
1.5
17.1001
1.5
21.6000
3
Mean of VIFs
1.0714
1.5
1.0714
1.5
1.4375
3
80
Reduced Models
In this section, the set of reduced models for 3 and 4 design variables will be
introduced. The results of the research related to the robustness of these response
surface designs: CCDs, BBDs, SCDs, UNFSDs, and 310, 311A, and 311B designs
for K = 3 and CCDs, BBDs, SCDs, PBCDs, UNFSDs, and 416A, 416B, and 416C
designs for K = 4 across reduced models of the second-order model will be presented
in Chapter 4. Specifically, a comparison of design optimality criteria based on the
D, A, G, and IV criteria across the set of reduced models for K = 3 and K = 4
design variables cases in the spherical region will be provided. Table 26 contains the
44 models considered when K = 3 and Table 27 contains the 224 models considered
when K = 4. In the Tables, the 1’s and 0’s in the Terms in Model columns indicate,
respectively, the presence or absence of that term in the reduced model. The column
p indicates the number of model parameters, the column dv indicates the number of
design variables present in the model, and the columns l, c, and q indicate the number
of linear, crossproduct and quadratic terms in the model, respectively.
Note that these designs possess a hierarchical structure. That is, (i) an interaction
xi xj term is in the model only if the xi or xj or both terms are also in the model and
(ii) a quadratic x2i term is in the model only if the xi term is also in the model. A
formal discussion of different hierarchical structures will be discussed in Chapter 5.
81
Table 26. Reduced Models (K = 3).
Terms in Model
model
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
x1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
1
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x21
1
0
1
0
0
1
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
1
1
1
0
1
1
1
1
0
0
1
1
1
1
0
1
0
0
1
0
1
0
0
1
1
1
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
x23
1
1
1
1
1
0
1
1
0
1
0
1
0
1
1
1
0
1
0
0
1
0
1
0
1
1
0
1
0
1
0
1
1
1
0
0
0
1
0
1
0
0
1
0
x1 x2
1
1
0
1
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
1
1
1
1
1
0
1
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
0
0
0
0
0
x2 x3
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
0
1
0
1
1
0
0
0
1
0
1
0
0
1
1
0
1
0
0
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
p = # of model parameters and dv = # of design variables appearing in the reduced model
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
82
Table 27. Reduced Models (K = 4).
model
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
x1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
1
1
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
1
0
0
0
1
1
1
1
1
0
0
1
1
0
0
0
0
0
0
0
0
1
1
1
0
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
1
0
0
0
1
0
1
0
0
0
0
0
0
Terms in Model
x1 x4 x2 x3 x2 x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
0
1
1
1
1
1
1
1
1
0
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
1
0
0
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
0
1
1
1
1
1
0
0
1
x3 x4
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
0
0
1
1
1
1
0
0
1
0
1
1
1
1
0
1
1
1
0
1
1
1
1
0
0
0
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
0
1
1
1
1
0
1
x2
1
1
0
1
0
0
1
1
1
0
0
0
1
1
0
1
0
1
1
1
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
1
1
0
0
0
0
0
0
0
1
0
0
1
0
1
0
0
1
0
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
x2
2
1
1
1
0
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
0
1
0
0
0
0
1
0
0
1
1
1
1
1
1
1
1
0
0
1
1
1
0
0
0
0
0
1
0
1
0
1
0
1
0
1
1
1
0
0
1
0
1
0
1
1
1
0
0
0
0
0
0
0
0
0
x2
3
1
1
1
1
1
1
1
1
1
0
1
0
0
1
1
1
1
1
1
1
1
0
0
0
1
0
1
0
1
1
1
1
1
0
1
1
0
1
1
1
1
0
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
1
x2
4
1
1
1
1
1
0
1
1
1
1
1
1
0
1
0
1
1
1
1
1
1
0
1
0
1
0
1
1
1
0
1
0
1
1
1
1
1
1
0
1
1
0
1
0
1
1
0
1
0
1
0
1
1
1
0
1
1
0
1
0
1
0
1
1
1
1
1
0
0
1
0
1
0
1
1
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
83
Table 27. cont’d
model
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
p
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
x1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
Terms in Model
x1 x4 x2 x3 x2 x4
0
0
1
0
0
1
0
0
1
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
1
0
1
1
0
1
0
0
1
1
0
0
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
1
1
1
0
0
1
0
0
1
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
1
0
1
1
0
1
0
0
1
0
0
1
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
0
0
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
1
1
0
1
0
0
1
0
0
1
1
0
0
1
0
0
1
0
0
x3 x4
1
1
1
0
0
1
1
0
1
1
1
0
1
1
1
1
1
1
1
0
0
1
1
0
1
1
0
1
1
1
0
1
1
1
0
1
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
0
1
1
1
0
0
1
1
0
1
1
0
1
1
0
1
1
1
1
1
1
1
1
x2
1
0
1
1
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
x2
2
1
0
1
0
1
1
1
1
0
0
0
0
0
1
0
0
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
1
0
0
1
0
1
0
1
1
0
1
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
x2
3
1
0
0
1
1
1
1
1
0
0
1
0
1
1
1
1
0
1
1
1
1
0
1
1
0
0
0
0
1
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
1
1
1
1
1
1
1
0
0
1
1
0
1
1
1
1
1
0
0
0
1
0
0
0
0
0
0
1
0
x2
4
0
1
0
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
1
1
0
1
0
1
1
0
1
0
1
0
1
1
0
1
0
0
1
1
1
1
1
1
0
1
0
1
0
0
0
1
0
1
0
0
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
l
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
c
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
q
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
84
Table 27. cont’d
model
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
0
0
1
0
0
0
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
0
0
1
1
0
0
1
1
0
0
1
0
0
0
0
0
1
1
1
0
0
1
1
0
0
0
0
0
1
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
0
0
0
1
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
Terms in Model
x1 x4 x2 x3 x2 x4
1
0
1
1
0
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
0
0
1
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
1
1
1
0
0
1
1
0
0
0
0
1
0
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3 x4
0
0
0
0
1
1
0
0
0
1
0
1
1
1
0
0
1
1
1
1
1
0
0
1
0
1
1
0
0
1
1
0
0
0
1
1
0
1
1
1
1
1
1
0
0
0
0
1
1
0
1
0
0
1
1
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
1
0
0
x2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
2
0
0
0
1
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
3
0
1
0
0
1
1
1
0
1
0
0
0
1
0
1
0
1
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
1
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
x2
4
1
0
1
0
1
0
1
1
1
0
0
1
0
1
0
1
0
1
1
1
1
1
1
0
1
0
0
0
0
1
0
1
0
1
0
0
0
0
0
1
0
1
0
1
0
0
1
1
1
0
0
0
1
0
0
0
0
1
0
1
0
1
0
1
0
0
0
1
0
1
0
0
1
0
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
85
Optimality Criteria for Reduced Models
In this dissertation, for each of the spherical response surface designs considered,
robustness was quantified by calculating the D, A, G, and IV criteria for sets of
reduced models of the second-order model in ( 2.1). Recall that
D − efficiency
=
A − efficiency
=
G − efficiency
=
IV − criterion
=
|X0 X|1/p
N
p
100
trace [N (X0X)−1 ]
p
100
2
Nσ
bmax
100
2
N σave
where X is the design matrix, p is the number of model parameters, and N is the
2
design size. σave
is the average of f 0 (x)(X0 X)−1 f (x) over the spherical design region.
Thus, the IV -criterion involves integration over spherical surfaces which was discussed
2
in Chapter 2. σ
bmax
is the maximum of f 0 (x)(X0 X)−1 f (x) approximated over the set
of spherical surface candidate points SK . In this research, for K = 3 and 4 design
variables, the G-criterion values require computation of
2
σ
bmax
= max f 0 (x)(X0X)−1 f (x) ,
x∈SK
where SK = {x(θ) : ρ = 0, 0.1, . . . , 1 and θi = 0, 0.1π, . . . , 1.9π for i = 1, . . . , K − 1},
x(θ) = (x1 (θ), . . . , xK (θ)) and the hyperspherical coordinates xi (θ) are defined in
2.4. Thus, SK is a set of equispaced points on concentric spheres of equispaced radii
2
in K-dimensions. Hence, for K = 3 and 4 design variables, σ
bmax
is the maximum of
86
f 0 (x)(X0 X)−1 f (x) approximated over the set of 4,400 and 88,000 candidate points in
S3 and S4 , respectively.
D and A-efficiencies represent the percentage of the number of runs required by
a hypothetical orthogonal design to achieve the same |X0 X| and trace [N (X0X)−1 ]
(Mitchell [41]). In the dissertation, these design optimality criteria are used to compare the spherical response surface designs across the set of reduced models. The
values of the four criteria were calculated using Matlab software (Mathworks [40]).
Note that the 310, and 311A designs, and the UNFSDs for K = 3 and the 416A,
416B, and 416C designs, and the UNFSDs for K = 4 are nonsymmetric with respect
to an optimality criterion. That is, the value of the criterion is not necessarily unique
over the set of permutations of the design variables for any particular reduced model,
or, equivalently, relabeling the design variables may yield multiple optimality criterion
values for certain reduced models. For example, for UNFSDs when K = 4, there are
4! = 24 permutations of x1 , x2 , x3 , and x4 (or, there are 24 ways to assign factors to
the columns of the design matrix).
Thus, from the X0X matrices of these nonsymmetric designs given in Chapter 2,
it can be concluded that for K = 3, there are 3 unique permutations for 310 and 311A
designs: (x1 , x2 , x3 ), (x1 , x3 , x2 ), (x3 , x1 , x2 ). That is, if the x1 and x2 columns in the
310 and 311A designs are switched, the designs remain the same with respect to an
optimality criterion for all reduced models. This is not the case if the x1 and x3 (or,
x2 and x3 ) columns are switched. For the UNFSDs there are 6 unique permutations:
87
(x1 , x2 , x3 ), (x1 , x3 , x2 ), (x2 , x1 , x3 ), (x2 , x3 , x1 ), (x3 , x1 , x2 ), (x3 , x2 , x1 ). That is, if the
x1 and x2 (or, x1 and x3 , or x2 and x3 ) columns are switched, the criterion values are
not unique for certain reduced models. For K = 4, there are only 4 unique permutations for the 416A, 416B, and 416C designs: (x1 , x2 , x3 , x4 ), (x1 , x2 , x4 , x3 ), (x1 , x4 , x2 ,
x3 ), (x4 , x1 , x2 , x3 ). That is, if either pairs of the x1 and x2 , x1 and x3 , or x2 and
x3 columns are switched, the designs remain the same with respect to an optimality
criterion. This is not the case if the x1 and x4 (or, x2 and x4 , or x3 and x4 ) columns
are switched. For the UNFSDs, however, there are 24 unique permutations. That is,
if any pairs of the x1 , x2 , x3 , and x4 columns are switched, the criterion values are
not necessarily unique for any particular reduced model.
Therefore, to make a fair comparison of the symmetric to nonsymmetric designs,
optimality criteria (D, A, G, and IV ) will be calculated for all relevant permutations
of the design variables for the nonsymmetric designs and the minimum values of D,
A, and G and the maximum value of IV are chosen from the set of permutations
of the design variables. Then these D, A, G, and IV optimality criteria values that
were selected to represent these nonsymmetric designs will be used to evaluate the
robustness of those designs and in comparisons with other response surface designs.
88
The Robustness of the Response Surface Designs
In the next chapter, the robustness properties of the spherical response surface
designs for the sets of reduced models for 3 and 4 factors based on D, A, G, and
IV criteria will be studied. For economy, the D, A, G, and IV criteria are denoted
simply as D, A, G, and IV .
To study the effects of removing squared terms from models, the 44 models for
K = 3 and the 224 models for K = 4 have been partitioned in subsets of models called
“Q-paths”. A Q-path has the property that any two models in the same Q-path differ
only by the number of squared terms (q) it contains. Or, in other words, each model
in a Q-path includes the same xi and xi xj terms.
For K = 3, the Q-paths will be labelled with letters A, B, b, C, c, D, E, F, G, H,
I, J, K, L and for K = 4, the Q-paths will be labelled with letters A, B, b, b1, C, c,
c1, D, d, d1, E, e, e1, F, f, G, H, I, i, J, j, K, K1, k, L, L1, l, l1, M, M1, m, m1, N,
O, P, Q, q, R, R1, r, S, s, T, U, V, W, X as shown in Table 28 and Table 29.
For example the “C” Q-path for K = 3 contains the five models:
(1) y = β0 +
(2) y = β0 +
(3) y = β0 +
3
X
i=1
3
X
i=1
3
X
i=1
βi xi + β23 x2 x3 + β11 x21 + β22 x22 + β33 x23
βi xi + β23 x2 x3 + β22 x22 + β33 x23
βi xi + β23 x2 x3 + β22 x22
89
(4) y = β0 +
3
X
βi xi + β23 x2 x3 + β33 x23
i=1
(5) y = β0 +
3
X
βi xi + β23 x2 x3 .
i=1
For the models above, there are 2 ways to form the “C” Q-path:
I. (1) → (2) → (3) → (5),
or II. (1) → (2) → (4) → (5).
That is, each model in the “C” Q-path has the same linear and cross-product terms
and different squared terms.
Similarly, the “c” Q-path for K = 3 contains the models:
(6) y = β0 +
(7) y = β0 +
3
X
i=1
3
X
βi xi + β23 x2 x3 + β11 x21 + β22 x22
βi xi + β23 x2 x3 + β11 x21 .
i=1
The “c” Q-path can be formed by (1) → (6) → (7) → (5). Notice that (1) and (5)
were already used to form the “C” Q-path, and each model can be used only once to
form Q-paths. Thus, the lower-case letter ”c” is used as a label to indicate it is also
related to the upper-case “C” Q-path.
In this dissertation, upper-case and lower-case letters (including some with subscripts) are used to label Q-paths and indicate their relationships to each other (e.g.,
the “C” and “c” Q-path).
90
Table 28. Q-Paths for K = 3.
Q-Path
A
B
b
C
C
c
D
E
F
F
G
G
H
I
J
K
L
Model
1
2
4
9
3
5
10
17
6
11
7
12
18
27
19
13
20
14
21
28
35
8
15
22
16
23
29
24
26
31
37
32
34
39
25
30
36
33
38
41
40
42
43
44
p
10
9
8
7
9
8
7
6
8
7
8
7
6
5
6
7
6
7
6
5
4
8
7
6
7
6
5
6
6
5
4
5
5
4
6
5
4
5
4
3
4
3
3
2
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
x2 x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
0
x21
1
0
0
0
1
0
0
0
1
0
1
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
1
1
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
1
1
0
0
1
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
1
0
0
0
0
0
0
x23
1
1
1
0
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
1
0
1
1
0
1
1
0
0
1
0
0
1
1
0
1
1
0
1
1
0
1
0
1
0
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
1
1
p = # of model parameters and dv = # of design variables appearing in the reduced model
l
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
c
3
3
3
3
2
2
2
2
2
2
1
1
1
1
1
1
1
0
0
0
0
3
3
3
2
2
2
2
1
1
1
1
2
2
1
1
1
0
0
0
1
1
0
0
q
3
2
1
0
3
2
1
0
2
1
3
2
1
0
1
2
1
3
2
1
0
2
1
0
2
1
0
1
2
1
0
1
1
0
2
1
0
2
1
0
1
0
1
0
91
Table 29. Q-Paths for K = 4.
Q-Path Model
A
1
2
4
10
22
B
3
5
11
23
42
b
6
13
24
b1
12
C
7
14
25
43
68
C
8
16
27
45
69
c
15
26
44
c1
28
D
17
29
46
70
100
D
18
31
48
72
101
D
19
33
50
74
102
d
30
47
71
d
32
49
73
d1
34
52
d2
51
E
35
53
75
103
p x1
15 1
14 1
13 1
12 1
11 1
14 1
13 1
12 1
11 1
10 1
13 1
12 1
11 1
12 1
13 1
12 1
11 1
10 1
9
1
13 1
12 1
11 1
10 1
9
1
12 1
11 1
10 1
11 1
12 1
11 1
10 1
9
1
8
1
12 1
11 1
10 1
9
1
8
1
12 1
11 1
10 1
9
1
8
1
11 1
10 1
9
1
11 1
10 1
9
1
11 1
10 1
10 1
11 1
10 1
9
1
8
1
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
1
1
1
0
0
0
0
x2 x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
0
x2 x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3 x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
0
0
0
1
1
1
1
x21
1
0
0
0
0
1
0
0
0
0
1
1
1
1
1
0
0
0
0
1
0
0
0
0
1
1
1
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
1
1
1
1
1
0
1
0
0
0
x22
1
1
0
0
0
1
1
0
0
0
1
1
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
1
1
0
1
0
1
1
1
0
0
x23
1
1
1
0
0
1
1
1
0
0
1
0
0
0
1
1
1
0
0
1
1
1
0
0
1
0
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
0
0
1
0
0
0
0
1
1
1
1
0
x24
1
1
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
0
0
0
0
0
0
1
1
0
1
1
1
1
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c
6
6
6
6
6
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
q
4
3
2
1
0
4
3
2
1
0
3
2
1
2
4
3
2
1
0
4
3
2
1
0
3
2
1
2
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
3
2
1
3
2
1
3
2
2
4
3
2
1
92
Table 29. cont’d
Q-Path Model
E
36
56
79
106
139
e
54
77
105
e
55
76
104
e1
78
e1
80
F
57
81
107
141
174
F
108
f
82
109
140
G
83
110
142
175
200
H
9
20
37
58
I
21
38
59
84
i
39
60
J
40
61
85
111
J
41
63
87
112
j
62
86
K
64
88
113
143
K
66
91
116
145
K1
65
90
115
144
p x1
11 1
10 1
9
1
8
1
7
1
10 1
9
1
8
1
10 1
9
1
8
1
9
1
9
1
10 1
9
1
8
1
7
1
6
1
8
1
9
1
8
1
7
1
9
1
8
1
7
1
6
1
5
1
13 0
12 0
11 0
10 0
12 0
11 0
10 0
9
0
11 0
10 0
11 0
10 0
9
0
8
0
11 0
10 0
9
0
8
0
10 0
9
0
10 0
9
0
8
0
7
0
10 0
9
0
8
0
7
0
10 0
9
0
8
0
7
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x4
1
1
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
x2 x3
1
1
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
x2 x4
0
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3 x4
0
0
0
0
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x21
1
0
0
0
0
1
1
1
1
0
0
1
0
1
1
1
1
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
1
1
0
0
0
0
0
0
1
1
0
1
1
1
1
1
0
0
0
1
0
0
1
1
0
0
0
1
0
0
0
1
0
0
0
1
1
1
0
0
0
1
0
0
0
1
0
1
0
0
0
1
0
0
0
1
0
0
0
x23
1
1
1
0
0
1
0
0
1
1
1
0
1
1
1
0
0
0
1
1
1
0
1
1
1
0
0
1
1
0
0
1
1
0
0
1
0
1
1
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
x24
1
1
1
1
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
1
1
1
1
1
1
1
0
1
1
1
0
1
1
1
0
0
0
1
1
1
0
1
1
1
0
0
0
1
1
1
0
1
1
1
0
1
1
1
0
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
l
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
0
0
0
0
0
6
6
6
6
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
q
4
3
2
1
0
3
2
1
3
2
1
2
2
4
3
2
1
0
2
3
2
1
4
3
2
1
0
3
2
1
0
3
2
1
0
2
1
3
2
1
0
3
2
1
0
2
1
3
2
1
0
3
2
1
0
3
2
1
0
93
Table 29. cont’d
Q-Path Model
k
89
114
k
92
117
L
94
120
148
177
L
95
123
151
178
L
96
125
153
179
L1
93
118
146
176
l
121
150
l
122
149
l
124
152
l
126
154
l1
119
147
M
128
157
182
202
M
158
M1
127
155
180
201
m
183
m1
156
181
N
159
184
203
215
O
67
97
129
P
98
130
160
P
99
132
161
P
131
P
133
p x1
9
0
8
0
9
0
8
0
9
0
8
0
7
0
6
0
9
0
8
0
7
0
6
0
9
0
8
0
7
0
6
0
9
0
8
0
7
0
6
0
8
0
7
0
8
0
7
0
8
0
7
0
8
0
7
0
8
0
7
0
8
0
7
0
6
0
5
0
7
0
8
0
7
0
6
0
5
0
6
0
7
0
6
0
7
0
6
0
5
0
4
0
10 0
9
0
8
0
9
0
8
0
7
0
9
0
8
0
7
0
8
0
8
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1
0
0
0
0
1
0
0
0
0
0
0
1
1
1
0
0
0
1
1
1
0
1
x1 x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
x2 x3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
x2 x4
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
x3 x4
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
1
1
0
0
0
0
0
1
1
1
1
0
1
1
0
0
0
0
1
1
1
1
1
1
0
0
0
1
0
x21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
1
1
1
1
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
1
1
0
1
0
1
1
1
1
1
0
0
0
1
1
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x23
1
0
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
1
0
1
0
1
1
0
0
0
1
1
0
0
1
1
0
1
1
0
0
1
0
0
1
0
0
1
1
0
1
0
x24
0
0
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
0
0
0
0
0
0
0
0
0
1
1
1
0
1
1
1
1
0
0
0
0
1
1
1
0
1
1
0
1
1
0
1
0
0
0
1
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
3
3
3
3
4
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
l
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
c
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
5
5
5
4
4
4
4
4
4
4
4
q
2
1
2
1
3
2
1
0
3
2
1
0
3
2
1
0
3
2
1
0
2
1
2
1
2
1
2
1
2
1
3
2
1
0
2
3
2
1
0
1
2
1
3
2
1
0
2
1
0
2
1
0
2
1
0
1
1
94
Table 29. cont’d
Q-Path Model
Q
135
163
186
Q
136
165
187
Q
137
167
188
Q
164
Q
166
Q
168
q
134
162
185
q
138
169
189
R
172
194
206
R
195
R1
170
190
204
R1
171
192
205
R1
191
R1
193
r
173
196
207
S
197
208
216
S
209
s
198
210
217
s
211
T
212
218
221
U
199
213
V
214
219
W
220
222
X
223
224
p x1
8 0
7 0
6 0
8 0
7 0
6 0
8 0
7 0
6 0
7 0
7 0
7 0
8 0
7 0
6 0
8 0
7 0
6 0
7 0
6 0
5 0
6 0
7 0
6 0
5 0
7 0
6 0
5 0
6 0
6 0
7 0
6 0
5 0
6 0
5 0
4 0
5 0
6 0
5 0
4 0
5 0
5 0
4 0
3 0
6 0
5 0
5 0
4 0
4 0
3 0
3 0
2 0
x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
0
0
0
0
0
0
1
1
1
0
0
1
0
0
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x4
1
1
1
1
1
1
0
0
0
1
1
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
x2 x3
0
0
0
1
1
1
0
0
0
0
1
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2 x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
x3 x4
1
1
1
0
0
0
1
1
1
1
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
x21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x23
1
1
0
1
1
0
1
1
0
0
0
0
1
0
0
1
0
0
1
0
0
1
1
0
0
1
0
0
1
1
1
1
0
1
0
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
x24
1
0
0
1
0
0
1
0
0
1
1
1
1
1
0
1
1
0
1
1
0
0
1
1
0
1
1
0
0
0
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
4
4
3
3
2
2
1
1
l
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
0
0
0
3
3
2
2
1
1
0
0
q
2
1
0
2
1
0
2
1
0
1
1
1
2
1
0
2
1
0
2
1
0
1
2
1
0
2
1
0
1
1
2
1
0
2
1
0
1
2
1
0
1
2
1
0
1
0
1
0
1
0
1
0
95
Figure 2. Example of the D-Efficiency Plot (Plotting Symbol = Q-Path).
To observe the impact of removing squared terms, efficiencies will be plotted as
groups of models defined by the Q-paths. The efficiencies of models in the same
Q-path will be connected in each plot according to the number of model parameters.
For example, see Figure 2 which is a plot of the D-efficiencies of the rs = 1, n0 = 1,
3-factor CCD. The plotting symbol is the Q-path label. Moving right to left along
any connected Q-path indicates removal of squared terms from the model.
In addition, to study the effects of removing cross-product terms from models,
the 44 models for K = 3 and the 224 models for K = 4 have been partitioned in
subsets of models called “C-paths”. A C-path has the property that any two models
96
in the same path differ only by the number of cross-product terms (c) it contains. Or,
in other words, each model in a C-path includes the same xi and x2i terms.
For K = 3, the C-paths will be labelled with letters A, B, b, C, c, D, E, e, F, f,
G, g, H, I, J, K and for K = 4, the C-paths will be labelled with letters A, a, B, b,
b1, C, c, c1, D, d, d1, E, e, e1, F, f, f1, f2, G, g, g1, H, h, h1, h2, I, i, i1, J, j, j1, j2,
K, k, k1, k2, L, l, l1, l2, M, N as shown in Table 30 and Table 31.
For example the “C” C-path for K = 3 contains the four models:
(1) y = β0 +
3
X
βi xi + β12 x1 x2 + β13 x1 x3 + β23 x2 x3 + β33 x23
i=1
(2) y = β0 +
3
X
βi xi + β13 x1 x3 + β23 x2 x3 + β33 x23
i=1
(3) y = β0 +
3
X
βi xi + β23 x2 x3 + β33 x23
i=1
(4) y = β0 +
3
X
βi xi + β33 x23 .
i=1
The “C” C-path can be formed by (1) → (2) → (3) → (4). That is, each model in
the “C” C-path has the same linear and squared terms and different cross-product
terms. The “c” C-path for K = 3 contains the models:
(5) y = β0 +
3
X
βi xi + +β13 x1 x3 + β23 x2 x3 + β22 x22
i=1
(6) y = β0 +
3
X
βi xi + +β23 x2 x3 + β22 x22 .
i=1
The “c” C-path can be formed by (1) → (5) → (6). Notice that (1) was already
used to form the “C” C-path and each model can be used only once to form C-paths.
97
Thus, the lower-case letter “c” is used as a label to indicate it is also related to the
upper-case “C” C-path.
Similar to Q-paths, the upper-case and lower-case letters (including some with
subscripts) are used to label C-paths and indicate their relationships to each other
(e.g., the “C” and “c” C-path).
Table 30. C-Paths for K = 3.
C-Path
A
B
b
C
c
D
E
e
F
f
G
g
H
I
J
K
Model
1
3
7
14
2
5
12
21
6
13
4
10
18
28
11
19
9
17
27
35
8
16
26
25
33
15
23
32
30
38
22
29
37
36
41
24
31
20
34
40
43
39
42
44
p
10
9
8
7
9
8
7
6
8
7
8
7
6
5
7
6
7
6
5
4
8
7
6
6
5
7
6
5
5
4
6
5
4
4
3
6
5
6
5
4
3
4
3
2
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
1
0
0
1
1
0
0
1
0
1
1
0
0
1
0
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
0
1
0
0
1
0
0
x2 x3
1
1
1
0
1
1
1
0
1
1
1
1
1
0
1
1
1
1
1
0
1
1
0
1
0
1
1
0
1
0
1
1
0
1
0
1
0
1
1
1
0
1
1
0
x2
1
1
1
1
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
x2
2
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
0
0
0
0
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
x2
3
1
1
1
1
1
1
1
1
0
0
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
0
0
0
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
3
3
3
2
2
3
3
3
2
2
3
3
3
3
2
1
3
2
1
p = # of model parameters and dv = # of design variables appearing in the reduced model
l
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
1
1
1
1
1
1
c
3
2
1
0
3
2
1
0
2
1
3
2
1
0
2
1
3
2
1
0
3
2
1
1
0
3
2
0
1
0
3
2
1
1
0
2
1
1
2
1
0
2
1
0
q
3
3
3
3
2
2
2
2
2
2
1
1
1
1
1
1
0
0
0
0
2
2
2
2
2
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
0
0
0
98
Table 31. C-Paths for K = 4.
C-Path Model
A
1
3
7
18
35
57
83
A
17
a
8
19
36
B
2
5
14
31
53
82
110
B
29
b
6
15
32
55
81
b
16
33
56
b
30
b1
34
b1
54
C
4
11
25
48
75
109
142
C
46
c
13
26
49
78
107
c
12
28
52
c
27
50
79
c
51
80
c
47
c1
76
c1
77
c1
108
p x1
15 1
14 1
13 1
12 1
11 1
10 1
9
1
12 1
13 1
12 1
11 1
14 1
13 1
12 1
11 1
10 1
9
1
8
1
11 1
13 1
12 1
11 1
10 1
9
1
12 1
11 1
10 1
11 1
11 1
10 1
13 1
12 1
11 1
10 1
9
1
8
1
7
1
10 1
12 1
11 1
10 1
9
1
8
1
12 1
11 1
10 1
11 1
10 1
9
1
10 1
9
1
10 1
9
1
9
1
8
1
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
1
0
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
x1 x4
1
1
1
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
0
0
0
1
1
1
1
1
0
1
1
1
0
0
0
0
1
1
1
0
0
0
1
1
1
1
1
1
1
1
1
0
0
0
x2 x3
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
0
0
1
1
1
0
1
0
1
1
1
1
0
0
0
0
1
1
1
0
0
1
1
1
1
1
1
1
1
0
0
0
0
x2 x4
1
1
1
1
1
0
0
1
1
1
0
1
1
1
1
1
0
0
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
0
1
1
1
1
1
0
1
0
1
1
1
0
x3 x4
1
1
1
1
1
1
0
1
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
0
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
1
1
1
1
x21
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
1
0
1
1
x22
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
x23
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
0
1
0
1
x24
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
1
0
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l c
4 6
4 5
4 4
4 3
4 2
4 1
4 0
4 3
4 4
4 3
4 2
4 6
4 5
4 4
4 3
4 2
4 1
4 0
4 3
4 5
4 4
4 3
4 2
4 1
4 4
4 3
4 2
4 3
4 3
4 2
4 6
4 5
4 4
4 3
4 2
4 1
4 0
4 3
4 5
4 4
4 3
4 2
4 1
4 5
4 4
4 3
4 4
4 3
4 2
4 3
4 2
4 3
4 2
4 2
4 1
q
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
99
Table 31. cont’d
C-Path Model
D
10
23
43
72
103
140
175
D
70
d
24
44
73
105
141
d
45
74
106
d
71
d1
104
E
22
42
69
102
139
e
100
e
68
101
e1
174
e1
200
F
9
21
40
64
66
94
127
159
F
93
f
96
128
f
95
f1
41
f2
65
G
20
38
61
88
118
155
184
G
90
g
39
62
89
122
g
63
g
91
120
g
92
121
g
125
157
p x1
12 1
11 1
10 1
9
1
8
1
7
1
6
1
9
1
11 1
10 1
9
1
8
1
7
1
10 1
9
1
8
1
9
1
8
1
11 1
10 1
9
1
8
1
7
1
8
1
9
1
8
1
6
1
5
1
13 0
12 0
11 0
10 0
10 0
9
0
8
0
7
0
9
0
9
0
8
0
9
0
11 0
10 0
12 0
11 0
10 0
9
0
8
0
7
0
6
0
9
0
11 0
10 0
9
0
8
0
10 0
9
0
8
0
9
0
8
0
8
0
7
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
1
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
1
0
1
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
1
x1 x4
1
1
1
0
0
0
0
1
1
1
0
0
0
1
1
1
1
0
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
1
0
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
x2 x3
1
1
1
1
0
0
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
1
1
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
1
1
1
0
0
1
0
0
0
0
0
0
x2 x4
1
1
1
1
1
0
0
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
0
1
1
1
0
0
1
1
1
1
1
0
0
0
1
0
0
1
1
1
1
1
1
1
1
0
0
1
1
1
1
0
1
1
0
1
0
0
0
x3 x4
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
0
1
1
1
1
0
0
0
1
1
1
1
0
1
1
1
1
1
1
1
0
1
0
0
0
0
1
1
1
1
1
1
1
0
1
1
1
1
1
0
1
1
1
1
0
0
x21
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
1
1
0
0
x23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
x24
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
4
4
4
4
3
4
4
4
4
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
l c
4 6
4 5
4 4
4 3
4 2
4 1
4 0
4 3
4 5
4 4
4 3
4 2
4 1
4 4
4 3
4 2
4 3
4 2
4 6
4 5
4 4
4 3
4 2
4 3
4 4
4 3
4 1
4 0
3 6
3 5
3 4
3 3
3 3
3 2
3 1
3 0
3 2
3 2
3 1
3 2
3 4
3 3
3 6
3 5
3 4
3 3
3 2
3 1
3 0
3 3
3 5
3 4
3 3
3 2
3 4
3 3
3 2
3 3
3 2
3 2
3 1
q
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
100
Table 31. cont’d
C-Path Model
g
123
g
124
g
126
g
158
g1
119
156
H
37
59
85
113
116
148
180
203
H
146
h
114
117
147
181
h1
153
182
h1
60
h1
86
h1
87
h1
149
h1
150
h1
151
h1
152
h1
154
h1
183
h2
115
I
58
84
111
143
177
201
215
I
176
i
145
178
i
179
202
i
112
i1
144
J
67
98
135
170
198
212
J
134
j
99
136
j1
137
172
197
j2
138
171
j2
173
p x1
8
0
8
0
8
0
7
0
8
0
7
0
11 0
10 0
9
0
8
0
8
0
7
0
6
0
5
0
7
0
8
0
8
0
7
0
6
0
7
0
6
0
10 0
9
0
9
0
7
0
7
0
7
0
7
0
7
0
6
0
8
0
10 0
9
0
8
0
7
0
6
0
5
0
4
0
6
0
7
0
6
0
6
0
5
0
8
0
7
0
10 0
9
0
8
0
7
0
6
0
5
0
8
0
9
0
8
0
8
0
7
0
6
0
8
0
7
0
7
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
0
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
0
0
0
0
1
1
0
1
1
0
0
0
0
0
0
0
0
1
1
1
0
1
0
0
0
0
0
0
1
0
1
1
1
1
0
1
x1 x4
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
1
1
0
0
1
0
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
1
1
1
0
1
0
1
1
1
0
0
0
0
1
1
0
0
0
1
1
1
x2 x3
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
0
0
0
0
0
0
x2 x4
1
1
0
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
0
0
0
1
1
1
0
0
1
1
0
0
1
1
1
1
1
0
0
0
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
0
x3 x4
0
0
0
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
0
0
1
1
0
1
1
0
0
0
0
1
1
1
1
1
1
1
0
1
1
0
0
0
0
1
1
1
1
1
1
0
1
0
0
1
0
0
1
1
0
x21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
0
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x23
1
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x24
1
0
0
1
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
0
0
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
4
3
3
4
4
4
4
4
4
3
3
3
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
3
3
3
4
4
4
4
4
3
4
4
4
3
2
2
3
4
4
4
4
3
3
3
3
l
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c
2
2
2
1
2
1
6
5
4
3
3
2
1
0
2
3
3
2
1
2
1
5
4
4
2
2
2
2
2
1
3
6
5
4
3
2
1
0
2
3
2
2
1
4
3
5
4
3
2
1
0
3
4
3
3
2
1
3
2
2
q
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
101
Table 31. cont’d
C-Path Model
K
97
130
164
192
210
218
K
131
163
191
211
K
193
k
132
165
k
133
166
k1
167
195
209
k1
168
194
208
k2
162
190
k2
169
k2
196
L
129
160
186
204
217
221
L
185
l
161
187
l
188
206
l1
189
207
216
l2
205
M
199
214
220
223
N
213
219
222
224
p x1
9 0
8 0
7 0
6 0
5 0
4 0
8 0
7 0
6 0
5 0
6 0
8 0
7 0
8 0
7 0
7 0
6 0
5 0
7 0
6 0
5 0
7 0
6 0
7 0
6 0
8 0
7 0
6 0
5 0
4 0
3 0
6 0
7 0
6 0
6 0
5 0
6 0
5 0
4 0
5 0
6 0
5 0
4 0
3 0
5 0
4 0
3 0
2 0
x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
x4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x1 x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x1 x3
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
1
1
1
1
1
0
0
1
1
1
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
0
0
0
0
0
0
x1 x4
1
1
1
1
0
0
1
1
0
0
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
1
1
0
0
0
1
0
0
0
x2 x3
1
1
0
0
0
0
1
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2 x4
1
1
1
0
0
0
1
1
1
0
0
1
1
1
1
1
1
0
1
1
0
1
1
0
0
1
1
1
1
0
0
1
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
x3 x4
1
1
1
1
1
0
1
1
1
1
1
0
0
0
0
1
0
0
1
0
0
1
1
1
0
1
1
1
1
1
0
1
0
0
1
0
1
0
0
1
1
1
1
0
1
1
1
0
x21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x22
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x23
0
0
0
0
0
0
1
1
1
1
1
1
1
0
0
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x24
1
1
1
1
1
1
0
0
0
0
0
0
0
1
1
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
p = # of model parameters and dv = # of design variables appearing in the reduced model
dv
4
4
4
3
2
2
4
4
3
2
3
4
4
4
4
4
4
3
4
4
3
3
3
3
3
4
4
4
3
2
2
3
4
4
4
4
3
3
3
3
4
3
2
1
4
3
2
1
l
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
c
5
4
3
2
1
0
4
3
2
1
2
4
3
4
3
3
2
1
3
2
1
3
2
3
2
5
4
3
2
1
0
3
4
3
3
2
3
2
1
2
3
2
1
0
3
2
1
0
q
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
102
Figure 3. Example of the D-Efficiency Plot (Plotting Symbol = C-Path).
To observe the impact of removing cross-product terms, efficiencies will be plotted
as groups defined by the C-paths. The efficiencies of models in the same C-path will
be connected in each plot according to the number of model parameters. For example,
see Figure 3 which is a plot of the D-efficiencies of the rs = 1, n0 = 1, 3-factor CCD.
The plotting symbol is the C-path label. Moving right to left along any connected
C-path indicates removal of cross-product terms from the model.
The results of the research related to the robustness of the response surface designs
assuming a spherical design region across the set of reduced models for 3 and 4
design variables will be presented in Chapter 4. Specifically, a comparison of design
optimality criteria based on the D, A, G, and IV criteria will be provided.
103
CHAPTER 4
ROBUSTNESS OF SPHERICAL RESPONSE SURFACE DESIGNS
The Robustness of 3-Factor Response Surface Designs
In this section, three-factor CCDs, BBDs, SCDs, UNFSDs, 310, 311A, and 311B
response surface designs in a spherical design region are considered. Analysis of all
four criteria (D, A, G, and IV ) will be restricted to Q-paths and C-paths in which
all 3 design variables (dv = 3) appear for the set of 44 reduced models. Only paths
for which dv = 3 are considered because they represent the reduced models most
likely to occur in practice. These models are most likely because the experimenter a
priori selects design variables that are known to affect or are very likely to affect the
response.
Tables of D, A, G, and IV criteria values and minimum, maximum, median, and
mean changes in criteria values for the three factor CCDs, BBDs, SCDs, UNFSDs,
310, 311A, and 311B response surface designs are given in Appendix A and B, respectively. Discussion of changes in D, A, G, and IV criteria will correspond to their
mean change.
104
The Central Composite Designs (CCDs)
The CCDs with rs = 1, 2 axial point replicates and with n0 = 1, 3 center points
are examined for K = 3 design variables. For a summary of the number of Q-paths
and C-paths that increase (“↑”) or decrease (“↓”) or indicate no change (“=”) for the
D, A, G, and IV criteria when a squared or a cross-product term is removed from
the model, see Table 32.
In Table 32, the “criterion” column indicates the four criteria examined, the r s
and n0 columns indicate the number of star and center points considered, respectively,
and the dv column indicates the number of design variables present in the model. The
notation dv = 3 → 2 → 1 indicates that the number of design variables is reduced
from 3 to 2 to 1 when a squared or cross-product term is removed from the model.
The column “Q” indicates the number of Q-paths that increase or decrease or do not
change when a squared term is removed as the model is reduced from 3 to 2 squared
terms (3 → 2), from 2 to 1 squared term (2 → 1), and from 1 to no squared term
(1 → 0). The column “C” indicates the number of C-paths that increase or decrease
or do not change when a cross-product term is removed as the model is reduced from
3 to 2 cross-product terms (3 → 2), from 2 to 1 cross-product term (2 → 1), and
from 1 to no cross-product term (1 → 0). When an individual “↑” or “↓” appears, it
indicates that all of Q or C-paths increase or decrease when a squared or cross-product
term is removed from the model. An “=” indicates that all of the Q or C-paths that
do not change when a squared or cross-product term is removed from the model.
105
The “↑(#)” notation indicates the number (#) of Q or C-paths that increase when
a squared or cross-product term is removed from the model (e.g., “↑(3)” means that
three of the Q or C-paths increase when a squared or cross-product term is removed
from the model). Similar meanings are also applied to “↓(#)” and “=(#)”.
The ↑, ↓, ↑ (#), and ↓ (#) notation will be used throughout this dissertation to
describe the relationship between optimality criteria and reduced models for all of
the three and four factor response surface designs studied.
For the CCDs, the Q-paths are depicted in Figures 4, 8, 12, and 16 for the D,
A, G and IV criteria, respectively, and the C-Paths are shown in Figures 6, 10, 14,
18 for the D, A, G and IV criteria, respectively. For changes in the D, A, G, and
IV criteria that result when squared terms are removed from the model, see Figures
5, 9, 13, and 17, respectively, and for changes in the D, A, G, and IV criteria that
result when cross-product terms are removed from the model, see Figures 7, 11, 15,
and 19, respectively.
In figures that reveal changes in D, A, G, or IV criteria values, a horizontal
reference line at zero is included. A point above the zero line indicates an increase
in the mean, minimum, median, or maximum criterion value when q or c is reduced
from 3 to 2 (3 → 2), 2 to 1 (2 → 1), or 1 to 0 (1 → 0). Similarly, a point below the
zero line indicates a decrease in the mean, minimum, median, or maximum criterion
value when q or c is reduced from 3 to 2 (3 → 2), 2 to 1 (2 → 1), or 1 to 0 (1 → 0).
106
Table 32. The Optimality Criteria Across the Reduced Models for the CCD (K = 3).
Q
Criterion
D
A
G
IV
C
rs
1
n0
1
dv
3
1, 2
3→2
↑
2→1
↑
↑
1→0
↑(1) ↓(10)
↑
2
1
3
1, 2
↑
↑(1) ↓(11)
↑
↓
↑(2) ↓(2)
1
3
3
1, 2
↑(3) ↓(1)
↑(1) ↓(11)
↑
↑(1) ↓(10)
↑
2
3
3
1, 2
↑(3) ↓(1)
↓
↑
↓
↑(2) ↓(2)
1
1
3
1, 2
↑
↑
↑
↑
↑
2
1
3
1, 2
↑
↑
↑
↑(6) ↓(5)
↑(3) ↓(1)
1
3
3
1, 2
↑
↑
↑
↑
↑
2
3
3
1, 2
↑
↑
↑
↑(3) ↓(8)
↑(2) ↓(2)
1
1
3
1, 2
↑
↓
↓
↑(4) ↓(7)
↑(2) ↓(2)
2
1
3
1, 2
↑
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
1
3
3
1, 2
↓
↓
↓
↑(4) ↓(7)
↑(2) ↓(2)
2
3
3
1, 2
↓
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
1
1
3
1, 2
↓
↓
↓
↓
↓
2
1
3
1, 2
↓
↓
↓
↓
↓
1
3
3
1, 2
↓
↓
↓
↓
↓
2
3
3
1, 2
↓
↓
↓
↓
↓
dv
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3→2
↑
2→1
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑(6) ↓(1)
↓
↑(9) ↓(1)
↑
↓
↑
↑
↓
↑
↑
↓
↑
↓
↓
↓
↑(1) ↓(6)
↓
↓
↓
↓
↓
↑(2) ↓(5)
↓
↓
↓
↓
↓(4) =(6)
↓
↑(1) ↓(1)
↓(4) =(6)
↓
↑(1) ↓(1)
↓(4) =(6)
↓
↑(1) ↓(1)
↓(4) =(6)
↑
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
1→0
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑(1) ↓(1)
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
107
The results based on D, A, G, and IV criteria for the 3-factor CCDs are summarized as follows:
For D, paths with dv = 3 (Figures 4, 5, 6, and 7):
1. Removal of an x2i term can either increase or decrease D.
2. Removal of an xi xj term increases D.
3. D tends to be lower when center points are replicated.
4. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
5. Within a C-path, there is more variability when star points are replicated while
the variability is unaffected by replication of center points.
6. The change in D decreases as q decreases, i.e., the paths in Figure 4 are concave
down, and there is a downward trend in Figure 5. When two models differ by
one xi xj term, the change in D when an x2i term is removed is larger for the
model having one less xi xj term (e.g., compare the models in Q-path “A” to
the models in Q-path “B” to the models in Q-path “C”, etc.).
7. The change in D increases as c decreases (see Figures 6 and 7). When two
models differ by one x2i term, the change in D when an xi xj term is removed
is the same (e.g., compare the models in C-path “A” to the models in C-path
“B” to the models in c-path “C”, etc.).
108
Notice that there are some cases of D > 100% (see Figures 4 and 6). This will be
discussed later in this chapter.
For A, paths with dv = 3 (Figures 8, 9, 10, and 11):
1. Removal of an x2i or an xi xj term can either increase or decrease A.
2. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
3. Within a C-path, there is more variability when star points are replicated while
the variability is unaffected by replication of center points.
4. The change in A decreases as q decreases. When two models differ by one xi xj
term, the change in A when an x2i term is removed is larger for the model having
one less xi xj term.
5. The change in A increases as c decreases. When two models differ by one x2i
term, the change in A when an xi xj term is removed is similar.
For G, paths with dv = 3 (Figures 12, 13, 14, and 15):
1. Removal of an x2i or an xi xj term can either increase or decrease G.
2. G tends to be lower when star points are replicated.
3. Within a Q-path, there is less variability when center points are replicated while
the variability is unaffected by replication of star points.
109
4. There is more variability within a C-path when star points or center points are
replicated.
5. When n0 = 1, the change in G drops as q decreases from 3 → 2 to 2 → 1, and
then increases (see Figure 13). When n0 = 3, the change in G increases as q
decreases. When two models differ by one xi xj term, there is no pattern to the
change in G when an x2i term is removed.
6. The change in G drops as c decreases from 3 → 2 to 2 → 1, and then increases
(see Figure 15). When two models differ by one x2i term, the change in G when
an xi xj term is removed is similar for both models when c decreases from 4 to
3 to 2 to 1. However, when c decreases from 1 to 0, there is no pattern to the
change in G for both models.
For IV , paths with dv = 3 (Figures 16, 17, 18, and 19):
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
2. IV tends to be higher when star points are replicated and lower when center
points are replicated.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
110
4. The variability within a C-path is unaffected by replication of star or center
points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is similar.
6. The change in IV seems fairly constant as c decreases. When two models differ
by one x2i term, the change in IV when an xi xj term is removed is similar.
111
Figure 4. D-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path).
112
Figure 5. The Change in D-Efficiency Plots by Reduction of Squared Terms in Model
for 3 Factor CCDs.
113
Figure 6. D-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path).
114
Figure 7. The Change in D-Efficiency Plots by Reduction of Cross-Product Terms in
Model for 3 Factor CCDs.
115
Figure 8. A-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path).
116
Figure 9. The Change in A-Efficiency Plots by Reduction of Squared Terms in Model
for 3 Factor CCDs.
117
Figure 10. A-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path).
118
Figure 11. The Change in A-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 3 Factor CCDs.
119
Figure 12. G-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path).
120
Figure 13. The Change in G-Efficiency Plots by Reduction of Squared Terms in
Model for 3 Factor CCDs.
121
Figure 14. G-Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path).
122
Figure 15. The Change in G-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 3 Factor CCDs.
123
Figure 16. IV -Efficiency Plots for 3 Factor CCDs (Plotting Symbol = Q-Path).
124
Figure 17. The Change in IV -Efficiency Plots by Reduction of Squared Terms in
Model for 3 Factor CCDs.
125
Figure 18. IV -Efficiency Plots for 3 Factor CCDs (Plotting Symbol = C-Path).
126
Figure 19. The Change in IV -Efficiency Plots by Reduction of Cross-Product Terms
in Model for 3 Factor CCDs.
127
The Box-Behnken Designs (BBDs)
The BBDs with n0 = 1, 3 center points are examined for K = 3 design variables.
For a summary of the number of Q-paths and C-paths that increase (“↑”) or decrease
(“↓”) or indicate no change (“=”), see Table 33.
Table 33. The Optimality Criteria Across the Reduced Models for the BBD (K = 3).
C
dv
3→2
2→1
1→0
3
↑
↑
↑
2
↑
3→2→1
↓
↓
D
3
3
↑
↑
↓
3
↑
↑
↑
1, 2
↑
↑
2
↑
3→2→1
↓
↓
1
3
↑
↑
↑
3
↑(2) ↓(5)
↑(2) ↓(8)
↑(1) ↓(3)
1, 2
↑
↑
2
↑(2) ↓(1)
3→2→1
↓
↓
A
3
3
↑
↑
↑
3
↑(4) ↓(3)
↑(6) ↓(4)
↑(2) ↓(2)
1, 2
↑
↑
2
↑
3→2→1
↓
↓
1
3
↓
↓
↑
3
↓
↓
↑(1) ↓(3)
1, 2
↓
↑
2
↑(1) ↓(2)
3→2→1
↓
↓
G
3
3
↓
↓
↑
3
↓
↓
↑(1) ↓(3)
1, 2
↓
↑
2
↑(1) ↓(2)
3→2→1
↓
↓
1
3
↓
↓
↓
3
↓
↓
↓
1, 2
↓
↓
2
↓
3→2→1
↓
↓
IV
3
3
↓
↓
↓
3
↓
↓(4) =(6)
↓
1, 2
↓
↓
2
↓
3→2→1
↑(1) ↓(1)
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
Criterion
n0
1
dv
3
1, 2
3→2
↑
Q
2→1
↑
↑
1→0
↑
↑
For Q-paths, see Figures 20, 24, 28, and 32 for the D, A, G, and IV criteria,
respectively. For C-paths, see Figures 22, 26, 30, and 34 for the D, A, G, and IV
criteria, respectively. For changes in the D, A, G, and IV criteria that result when
squared terms are removed from the model, see Figures 21, 25, 29, and 33, respectively,
128
and for changes in the D, A, G, and IV criteria that result when cross-product terms
are removed from the model, see Figures 23, 27, 31, and 35, respectively. The results
based on D, A, G, and IV criteria for the 3-factor BBDs are summarized as follows:
For D, paths with dv = 3 (Figure 20, 21, 22, and 23):
1. Removal of an x2i or an xi xj term increases D.
2. D tends to be lower when center points are replicated.
3. There is less variability within a Q-path or a C-path when center points are
replicated.
4. When n0 = 1, the change in D drops as q decreases from 3 → 2 to 2 → 1, and
then increases (see Figure 21). When n0 = 3, the change in D increases as q
decreases. When two models differ by one xi xj term, the change in D when an
x2i term is removed is very similar.
5. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D when an xi xj term is removed is very similar.
For A, paths with dv = 3 (Figure 24, 25, 26, and 27):
1. Removal of an x2i term increases A.
2. Removal of an xi xj term can either increase or decrease A.
3. There is less variability within a Q-path or a C-path when center points are
replicated.
129
4. When n0 = 1, the change in A drops as q decreases from 3 → 2 to 2 → 1,
and then increases (see Figure 25). When n0 = 3, the change in A increases
as q decreases. When two models differ by one xi xj term, the change in A is
larger for the model having one less xi xj term when q decreases from 1 to 0.
Otherwise, the change in A is similar.
5. When n0 = 1, the change in A drops as c decreases from 3 → 2 to 2 → 1, and
then increases (see Figure 27). When n0 = 3, the change in A increases as c
decreases. When two models differ by one x2i term, the change in A when an
xi xj term is removed is similar.
For G, paths with dv = 3 (Figure 28, 29, 30, and 31):
1. Removal of an x2i or an xi xj term can either increase or decrease G.
2. G tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. The change in G drops as q decreases from 3 → 2 to 2 → 1, and then increases
(see Figure 29). When two models differ by one xi xj term, there is no pattern
to the change in G when q decreases from 1 to 0. Otherwise, the change in G
when an x2i term is removed is similar.
130
6. The change in G drops as c decreases from 3 → 2 to 2 → 1, and then increases
(see Figure 31). When two models differ by one x2i term, the change in G when
an xi xj term is removed is similar (except for the models in C-paths “C” and
“D” are compared).
For IV , paths with dv = 3 (Figure 32, 33, 34, and 35):
1. Removal of an x2i or an xi xj term decreases IV . However, the decrease in IV
is smaller when an xi xj is removed.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. There is more variability within a C-path when center points are replicated.
5. When n0 = 1, the change in IV decreases as q decreases. When n0 = 3, the
change in IV seems to be constant as q decreases. When two models differ by
one xi xj term, the change in IV when an x2i term is removed is very similar.
6. The change in IV looks fairly constant as c decreases. When two models differ
by one x2i term, the change in IV when an xi xj term is removed is very similar.
131
Figure 20. D-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path).
Figure 21. The Change in D-Efficiency Plots by Reduction of Squared Terms in
Model for 3 Factor BBDs.
132
Figure 22. D-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path).
Figure 23. The Change in D-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 3 Factor BBDs.
133
Figure 24. A-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path).
Figure 25. The Change in A-Efficiency Plots by Reduction of Squared Terms in Model
for 3 Factor BBDs.
134
Figure 26. A-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path).
Figure 27. The Change in A-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 3 Factor BBDs.
135
Figure 28. G-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path).
Figure 29. The Change in G-Efficiency Plots by Reduction of Squared Terms in
Model for 3 Factor BBDs.
136
Figure 30. G-Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path).
Figure 31. The Change in G-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 3 Factor BBDs.
137
Figure 32. IV -Efficiency Plots for 3 Factor BBDs (Plotting Symbol = Q-Path).
Figure 33. The Change in IV -Efficiency Plots by Reduction of Squared Terms in
Model for 3 Factor BBDs.
138
Figure 34. IV -Efficiency Plots for 3 Factor BBDs (Plotting Symbol = C-Path).
Figure 35. The Change in IV -Efficiency Plots by Reduction of Cross-Product Terms
in Model for 3 Factor BBDs.
139
The Small Composite Designs (SCDs)
The SCDs with rs = 1, 2 axial point replicates and with n0 = 1, 3 center points
are examined for K = 3 design variables. For a summary of the number of Q-paths
and C-paths that increase (“↑”) or decrease(“↓”) or indicate no change (“=”), see
Table 34. For SCDs, plots of the D, A, G, and IV criteria and plots of the change in
the D, A, G, and IV criteria are given in Appendix B. The results based on D, A,
G, and IV criteria for the 3-factor SCDs are summarized as follows:
For D, paths with dv = 3:
1. D can either increase or decrease when q decreases from 3 to 2. For all other
cases, removal of an x2i term decreases D.
2. Removal of an xi xj term increases D.
3. D tends to be lower when center points are replicated.
4. Within a Q-path, there is slightly more variability when star points are replicated and slightly less variability when center points are replicated.
5. Within a C-path, there is slightly more variability when star points are replicated while the variability is unaffected by replication of center points.
6. The change in D decreases as q decreases. When two models differ by one xi xj
term, the change in D when an x2i term is removed is similar for both models
(except when n0 = 1).
140
Table 34. The Optimality Criteria Across the Reduced Models for the SCD (K = 3).
Q
Criterion
D
A
G
IV
C
rs
1
n0
1
dv
3
1, 2
3→2
↑
2→1
↓
↑
1→0
↓
↑(2) ↓(2)
2
1
3
1, 2
↑
↓
↑(1) ↓(1)
↓
↑(2) ↓(2)
1
3
3
1, 2
↓
↓
↑(1) ↓(1)
↓
↑(2) ↓(2)
2
3
3
1, 2
↑(1) ↓(3)
↓
↑(1) ↓(3)
↓
↑(2) ↓(2)
1
1
3
1, 2
↑
↑
↑
↑(1) ↓(10)
↑(2) ↓(2)
2
1
3
1, 2
↑
↑(10) ↓(2)
↑
↑(1) ↓(10)
↑(2) ↓(2)
1
3
3
1, 2
↑
↑(1) ↓(11)
↑
↑(1) ↓(10)
↑(2) ↓(2)
2
3
3
1, 2
↑
↑(1) ↓(11)
↑(1) ↓(1)
↑(1) ↓(10)
↑(2) ↓(2)
1
1
3
1, 2
↑(1) ↓(3)
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
2
1
3
1, 2
↑(1) ↓(3)
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
1
3
3
1, 2
↓
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
2
3
3
1, 2
↓
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
1
1
3
1, 2
↓
↓
↓
↓
↓
2
1
3
1, 2
↓
↓
↓
↓
↓
1
3
3
1, 2
↓
↓
↓
↓
↓
2
3
3
1, 2
↓
↓
↓
↓
↓
dv
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3→2
↑
2→1
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↑
↓
↑
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
1→0
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
141
7. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D is larger for the model having one less x2i term when c
decreases from 1 to 0. Otherwise, the change in D when an xi xj term is removed
is similar.
For A, paths with dv = 3:
1. A increases when q decreases from 3 to 2. Otherwise, removal of an x2i term
can either increase or decrease A.
2. Removal of an xi xj term increases A.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. Within a C-path, there is more variability when star points are replicated (except for the “A” path) and more variability when center points are replicated.
5. The change in A decreases as q decreases. When two models differ by one xi xj
term and n0 = 1, the change in A is larger for the model having one less xi xj
term when q decreases from 3 to 2. Otherwise, the change in A is similar for
both models (except when the models in Q-paths “C” and “D” are compared).
However, when n0 = 3, the change in A is similar for both models (except when
the models in C-paths “C” and “D” are compared).
142
6. The change in A increases as c decreases. When two models differ by one x2i
term, the change in A is larger for the model having one less x2i term when c
decreases from 2 to 1 to 0. Otherwise, the change in A is similar.
For G, paths with dv = 3:
1. Removal of an x2i term decreases G (except for the Q-path “D” in which G can
either increase or decrease).
2. Removal of an xi xj term can either increase or decrease G.
3. Within a Q-path, there is more variability when star points are replicated and
slightly less variability when center points are replicated.
4. There is more variability within a C-path when star points or center points are
replicated.
5. When n0 = 1, the change in G drops as q decreases from 3 → 2 to 2 → 1, and
then increases. When n0 = 3, the change in G increases as q decreases. When
two models differ by one xi xj term, the change in G when an x2i term is removed
is similar (except when the models in Q-paths “C” and “D” are compared).
6. The change in G drops as c decreases from 3 → 2 to 2 → 1, and then increases.
When two models differ by one x2i term, the change in G when an xi xj term
is removed is smaller for the model having one less x2i term (except when c
decreases from 1 to 0).
143
For IV , paths with dv = 3:
1. IV decreases as q or c decreases. The decrease in IV , however, is smaller when
an xi xj term is removed.
2. IV tends to be higher when star points are replicated and lower when center
points are replicated.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. Within a C-path, there is more variability when center points are replicated
while the variability is unaffected by replication of star points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is similar.
6. The change in IV looks fairly constant as c decreases. When two models differ
by one x2i term, the change in IV when an xi xj term is removed is similar.
The Uniform Shell Designs (UNFSDs)
The UNFSDs with n0 = 1, 3 center points are examined for K = 3 design variables. For a summary of the number of Q-paths and C-paths that increase (“↑”) or
decrease (“↓”) or indicate no change (“=”), see Table 35. For UNFSDs, plots of the
D, A, G, and IV criteria and plots of the change in the D, A, G, and IV criteria are
144
given in Appendix B. The results based on D, A, G, and IV criteria for the 3-factor
UNFSDs are summarized as follows:
Table 35. The Optimality Criteria Across the Reduced Models for the UNFSDs
(K = 3).
Q
2→1
↑(8) ↓(4)
↑
1→0
↑(1) ↓(10)
↑
↑(2) ↓(2)
↑(1) ↓(11)
↑
↓
↑
3
1, 2
↑
↑
↑
↑
↑
3
3
1, 2
↑
↑
↑
↑(10) ↓(1)
↑
1
3
1, 2
↑(1) ↓(3)
↓
↓
↑(9) ↓(2)
↑
3
3
1, 2
↓
↑(3) ↓(9)
↓
↑(8) ↓(3)
↑
1
3
1, 2
↓
↓
↓
↓
↓
3
3
1, 2
↓
↓
↓
↓
↓
Criterion
n0
1
dv
3
1, 2
3→2
↑
D
3
3
1, 2
1
A
G
IV
dv
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3→2
↑
C
2→1
↑
↑
↓
↑
↑(6) ↓(1)
↓
↑(9) ↓(1)
↑
↓
↑
↑(2) ↓(5)
↓
↓
↑(2) ↓(5)
↓
↑(1) ↓(9)
↓
↓
↓
↓
↓
↓
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’indicates the number of Q or C-paths with criterion values that do not change.
For D, paths with dv = 3:
1. Removal of an x2i term can either increase or decrease D.
2. Removal of an xi xj term increases D.
3. D tends to be lower when center points are replicated.
1→0
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
145
4. There is less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D decreases as q decreases. When two models differ by one xi xj
term, the change in D is similar for both models (except for n0 = 1, when the
models in Q-paths “C” and “D” are compared).
7. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D is similar (except when c decreases from 1 to 0).
For A, paths with dv = 3:
1. When n0 = 1, removal of an x2i term increases A. When n0 = 3, removal of an
x2i term increases A for all but one case.
2. Removal of an xi xj term increases A (except for C-path “A” when n0 = 1).
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected when center points are replicated.
5. The change in A decreases as q decreases. When two models differ by one xi xj
term and n0 = 1, the change in A when an x2i term is removed is larger for the
model having one less xi xj term. However, when n0 = 3, the change in A is
similar (except when the models in Q-paths “C” and “D” are compared).
146
6. The change in A increases as c decreases. When two models differ by one x2i
term, the change in A is larger for the model having one less x2i term when c
decreases from 1 to 0. Otherwise, the change in A is similar.
For G, paths with dv = 3:
1. Removal of an x2i or an xi xj term can either increase or decrease G.
2. G tends to be lower when center points are replicated.
3. The variability within a Q-path is unaffected by replication of center points.
4. There is more variability within a C-path when center points are replicated.
5. When n0 = 1, the change in G drops as q decreases from 3 → 2 to 2 → 1, and
then increases. When n0 = 3, the change in G increases as q decreases. When
two models differ by one xi xj term, the change in G when an x2i term is removed
is similar (except when the models in Q-paths “C” and “D” are compared).
6. The mean change in G decreases as c decreases from 3 → 2 to 2 → 1, and then
increases. When two models differ by one x2i term, the change in G is similar
when c decreases from 3 to 2 to 1. However, the change in G is slightly smaller
for the model having one less x2i term when c decreases from 1 to 0 (except
when the models in C-paths “C” and “D” are compared).
For IV , paths with dv = 3:
147
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is very similar.
6. The change in IV decreases as c decreases from 3 → 2 to 2 → 1, and then
slightly increases. When two models differ by one x2i term, the change in IV
when an xi xj term is removed is very similar.
The Hybrid 310 Designs (310s)
The 310 designs with n0 = 0, 1, and 3 center points are examined for K = 3
design variables. For a summary of the number of Q-paths and C-paths that increase
(“↑”) or decrease (“↓”) or indicate no change (“=”), see Table 36. For 310 designs,
plots of the D, A, G, and IV criteria and plots of the change in the D, A, G, and IV
criteria are given in Appendix B. The results based on D, A, G, and IV criteria for
the 3-factor 310 designs are summarized as follows:
For D, paths with dv = 3:
1. Removal of an x2i term increases D.
148
Table 36. The Optimality Criteria Across the Reduced Models for the 310s (K = 3).
C
dv
3→2
2→1
3
↑
↑
2
3→2→1
↓
1
3
↑
↑
↑
3
↑
↑
D
1, 2
↑
↑
2
3→2→1
↓
3
3
↑
↑
↑
3
↑
↑
1, 2
↑
↑
2
3→2→1
↓
0
3
↑
↑
↑
3
↑(2) ↓(5)
↑(3) ↓(7)
1, 2
↑
↑
2
3→2→1
↓
1
3
↑
↑
↑
3
↑(2) ↓(5)
↑(3) ↓(7)
A
1, 2
↑
↑
2
3→2→1
↓
3
3
↑
↑
↑
3
↑(5) ↓(2)
↑(8) ↓(2)
1, 2
↑
↑
2
3→2→1
↓
0
3
↓
↑(1) ↓(9)
↑
3
↓
↑(1) ↓(9)
1, 2
↓
↑
2
3→2→1
↓
1
3
↓
↑(1) ↓(9)
↑
3
↓
↑(1) ↓(9)
G
1, 2
↓
↑
2
3→2→1
↓
3
3
↓
↑(1) ↓(9)
↑
3
↓
↑(1) ↓(9)
1, 2
↓
↑
2
3→2→1
↓
0
3
↓
↓
↓
3
↓
↓
1, 2
↓
↓
2
3→2→1
↓
1
3
↓
↓
↓
3
↓
↓
IV
1, 2
↓
↓
2
3→2→1
↓
3
3
↓
↓
↓
3
↓
↓
1, 2
↓
↓
2
3→2→1
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path riterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
Criterion
n0
0
dv
3
1, 2
3→2
↑
Q
2→1
↑
↑
1→0
↑
↑
1→0
↑(3) ↓(1)
↑
↓
↑(3) ↓(1)
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(1) ↓(3)
↑(1) ↓(2)
↓
↑(1) ↓(3)
↑
↓
↑(3) ↓(1)
↑
↓
↑(1) ↓(3)
↑(1) ↓(2)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↓
↑(1) ↓(3)
↓
↓
↑(1) ↓(3)
↓
↓
↑(1) ↓(3)
↓
↓
2. Removal of an xi xj term can either increase or decrease D.
3. D tends to be lower when center points are replicated.
4. There is less variability within a Q-path when center points are replicated.
149
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D increases as q decreases. When two models differ by one xi xj
term, there is no pattern to the change in D when q decreases from 1 to 0.
Otherwise, the change in D when an x2i term is removed is similar.
7. The change in D increases as c decreases from 3 → 2 to 2 → 1, and then
decreases. When two models differ by one x2i term, the change in D when an
xi xj term is removed is similar.
For A, paths with dv = 3:
1. Removal of an x2i term increases A.
2. Removal of an xi xj term can either increase or decrease A.
3. There is less variability within a Q-path or C-path when center points are
replicated.
4. The change in A increases as q decreases. When two models differ by one xi xj
term, there is no pattern to the change in A when q decreases from 1 to 0.
Otherwise, the change in A when an x2i term is removed is similar (except when
the models in Q-paths “C” and “D” are compared).
5. The change in A increases as c decreases from 3 → 2 to 2 → 1, and then
decreases. When two models differ by one x2i term, the change in A when an
150
xi xj term is removed is similar (except when the models in C-paths “C” and
“D” are compared).
For G, paths with dv = 3:
1. G decreases as q decreases from 3 to 2 and increases as q decreases from 1 to 0.
Otherwise, removal of an x2i term can either increase or decrease G.
2. G decreases as c decreases from 3 to 2. Otherwise, removal of an xi xj term can
either increase or decrease G.
3. G tends to be lower when center points are replicated.
4. There is less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in G increases as q decreases. When two models differ by one xi xj
term, the change in G is similar when q decreases from 3 to 2. However, there
is no pattern to the change in G when q decreases from 2 to 1 to 0.
7. The change in G increases as c decreases. When two models differ by one x2i
term, the change in G is similar when c decreases from 3 to 2. However, there
is no pattern to the change in G when c decreases from 2 to 1 to 0.
For IV , paths with dv = 3:
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
151
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. When n0 = 0, the change in IV drops as q decreases from 3 → 2 to 2 → 1, and
then increases. When n0 = 1 or 3, the change in IV increases as q decreases.
When two models differ by one xi xj term, the change in IV when an x2i term
is removed is very similar (except for the designs having n0 = 0 or 1, when the
models in Q-paths “B” and “C” and Q-paths “C” and “D” are compared).
6. The change in IV increases as c decreases from 3 → 2 to 2 → 1, and then
decreases. When two models differ by one x2i term, the change in IV when an
xi xj term is removed is very similar (except for the designs having n0 = 0 or
1, when the models in C-paths “B” and “C” and C-paths “C” and “D” are
compared).
7. The change in IV when an x2i term is removed increases when center points are
replicated.
8. The change in IV when an xi xj term is removed is similar when center points
are replicated.
152
The Hybrid 311A Designs (311As)
The 311A designs with n0 = 1, 3 center points are examined for K = 3 design
variables. For a summary of the number of Q-paths and C-paths that increase (“↑”)
or decrease (“↓”) or indicate no change (“=”), see Table 37. For 311A designs, plots
of the D, A, G, and IV criteria and plots of the change in the D, A, G, and IV
criteria are given in Appendix B. The results based on D, A, G, and IV criteria for
the 3-factor 311A designs are summarized as follows:
Table 37. The Optimality Criteria Across the Reduced Models for the 311As (K = 3).
Q
2→1
↑(8) ↓(4)
↑
1→0
↑
↑
↑(1) ↓(3)
↑(1) ↓(11)
↑
↑(10) ↓(1)
↑
3
1, 2
↑
↑
↑
↑
↑
3
3
1, 2
↑
↑
↑
↑
↑
1
3
1, 2
↓
↑(3) ↓(9)
↓
↑
↑
3
3
1, 2
↓
↑(3) ↓(9)
↓
↑
↑
1
3
1, 2
↓
↓
↓
↓
↓
3
3
1, 2
↓
↓
↓
↓
↓
Criterion
n0
1
dv
3
1, 2
3→2
↑
D
3
3
1, 2
1
A
G
IV
dv
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3→2
↑
C
2→1
↑
↑
↓
↑
↑(4) ↓(3)
↓
↑(6) ↓(4)
↑
↓
↑
↓
↓
↑(2) ↓(8)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
For D, paths with dv = 3:
1→0
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(2) ↓(2)
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↑(1) ↓(3)
↓
↓
↓
↓
↓
153
1. D can either increase or decrease when an x2i term is removed. When n0 = 3,
the changes in D are consistently small.
2. Removal of an xi xj term increases D.
3. D tends to be lower when center points are replicated.
4. There is less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D decreases as q decreases from 3 → 2 to 2 → 1, and then
increases. When two models differ by one xi xj term, the change in D when an
x2i term is removed is similar (except when the models in Q-paths “C” and “D”
are compared).
7. When n0 = 1, the change in D increases as c decreases from 3 → 2 to 2 → 1,
and then decreases. When n0 = 3, the change in D increases as c decreases.
When two models differ by one x2i term, the change in D when an xi xj term
is removed is similar (except when the models in C-paths “C” and “D” are
compared).
For A, paths with dv = 3:
1. Removal of an x2i term increases A.
154
2. Removal of an xi xj term tends to increase A when q = 0 or 1 (i.e., in C-paths
“C”, “D”, “F”, and “G”) or decrease A otherwise (i.e., in C-paths “A”, “B”,
and “E”).
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected when center points are replicated.
5. The change in A decreases as q decreases from 3 → 2 to 2 → 1, and then
increases. When two models differ by one xi xj term and n0 = 1, the change in
A when an x2i term is removed is larger for the model having one less xi xj term
(except when the models in Q-paths “C” and “D” are compared). However,
when n0 = 3, the change in A is similar (except when the models in Q-paths
“C” and “D” are compared).
6. When n0 = 1, the change in A increases as c decreases from 3 → 2 to 2 → 1,
and then decreases. When n0 = 3, the change in A increases as c decreases.
When two models differ by one x2i term, the change in A when an xi xj term
is removed is similar (except when the models in C-paths “B” and “C” and
C-paths “C” and “D” are compared).
For G, paths with dv = 3:
1. G decreases when q decreases from 3 to 2 and G increases when q decreases
from 1 to 0. Otherwise, removal of an x2i term can either increase or decrease
G.
155
2. G decreases when c decreases from 3 to 2. Otherwise, removal of an xi xj term
can either increase or decrease G.
3. G tends to be lower when center points are replicated.
4. The variability within a Q-path or C-path is unaffected by replication of center
points.
5. The change in G increases as q decreases. When two models differ by one xi xj
term, the change in G is similar when q decreases from 3 to 2 to 1 (except when
the models in Q-paths “B” and “C” and Q-paths “C” and “D” are compared).
However, there is no pattern to the change in G when q decreases from 1 to 0.
6. The change in G increases as c decreases. When two models differ by one x2i
term, the change in G is similar when c decreases from 3 to 2 to 1 (except when
the models in C-paths “B” and “C” and C-paths “C” and “D” are compared).
However, there is no pattern to the change in G when c decreases from 1 to 0.
For IV , paths with dv = 3:
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
156
4. The variability within a C-path is unaffected by replication of center points.
5. When n0 = 1, the change in IV decreases as q decreases. When n0 = 3, the
change in IV is fairly constant as q decreases. When two models differ by one
xi xj term, the change in IV when an x2i term is removed is similar (except when
n0 = 1 and the models in Q-paths “B” and “C” and Q-paths “C” and “D” are
compared).
6. The change in IV increases as c decreases from 3 → 2 to 2 → 1, and then
decreases. When two models differ by one x2i term, the change in IV when an
xi xj term is removed is similar.
The Hybrid 311B Designs (311Bs)
The 311B designs with n0 = 1, 3 center points are examined for K = 3 design
variables. For a summary of the number of Q-paths and C-paths that increase (“↑”)
or decrease (“↓”) or indicate no change (“=”), see Table 38. For 311B designs, plots
of the D, A, G, and IV criteria and plots of the change in the D, A, G, and IV
criteria are given in Appendix B. The results based on D, A, G, and IV criteria for
the 3-factor 311B designs are summarized as follows:
For D, paths with dv = 3:
1. When n0 = 1, D increases when q decreases from 3 to 2. For all other cases,
removal of an x2i term decreases D (except on Q-path “D”).
2. Removal of an xi xj term increases D.
157
Table 38. The Optimality Criteria Across the Reduced Models for the 311Bs (K = 3).
Q
2→1
↑(1) ↓(11)
↑
1→0
↓
↑
↑(1) ↓(3)
↓
↑
↓
↑
3
1, 2
↑
↑
↑
↑
↑
3
3
1, 2
↑
↑
↑
↑(5) ↓(6)
↑
1
3
1, 2
↑(1) ↓(3)
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
3
3
1, 2
↓
↓
↓
↑(1) ↓(10)
↑(2) ↓(2)
1
3
1, 2
↓
↓
↓
↓
↓
3
3
1, 2
↓
↓
↓
↓
↓
Criterion
n0
1
dv
3
1, 2
3→2
↑
D
3
3
1, 2
1
A
G
IV
dv
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3
2
3→2→1
3→2
↑
C
2→1
↑
↑
↓
↑
↑(6) ↓(1)
↓
↑(9) ↓(1)
↑
↓
↑
↑(1) ↓(6)
↓
↓
↑(1) ↓(6)
↓
↓
↓
↓
↓
↓
↓
↓
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
1→0
↑
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑(1) ↓(1)
↑
↑
↑(1) ↓(1)
↑(2) ↓(2)
↓
↑(1) ↓(1)
↑(3) ↓(1)
↓
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
3. D tends to be lower when center points are replicated.
4. There is less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D decreases as q decreases. When two models differ by one xi xj
term, the change in D is similar (except when n0 = 1 and the models in Q-paths
“C” and “D” are compared).
158
7. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D is similar for both models (except when c decreases from
1 to 0).
For A, paths with dv = 3:
1. When n0 = 1, removal of an x2i term increases A. When n0 = 3, removal of an
x2i term can either increase or decrease A.
2. Removal of an xi xj term increases A (except for C-path “A” when n0 = 1).
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected when center points are replicated.
5. The change in A decreases as q decreases. When two models differ by one xi xj
term and n0 = 1, the change in A when an x2i term is removed is larger for the
model having one less xi xj term. However, when n0 = 3, the change in A is
similar (except when the models in Q-paths “C” and “D” are compared).
6. The change in A increases as c decreases. When two models differ by one x2i
term, the change in A is similar when c decreases from 3 to 2 to 1. However,
the change in A is larger for the model having one less x2i term when c decreases
from 1 to 0.
For G, paths with dv = 3:
159
1. Removal of an x2i term decreases G (except for Q-path “D” in which removal
of an x2i term can either increase or decrease G).
2. Removal of an xi xj term can either increase or decrease G.
3. G tends to be lower when center points are replicated.
4. There is less variability within a Q-path when center points are replicated.
5. There is more variability within a C-path when center points are replicated.
6. When n0 = 1, the change in G decreases as q decreases from 3 → 2 to 2 → 1,
and then increases. When n0 = 3, there is very little change in G as q decreases
from 3 → 2 to 2 → 1, but then increases. When two models differ by one xi xj
term, the change in G is similar (except when the models in Q-paths “C” and
“D” are compared).
7. The change in G decreases slightly as c decreases from 3 → 2 to 2 → 1, and
then increases. When two models differ by one x2i term, the change in G when
an xi xj term is removed is similar (except when (i) the models in C-paths “C”
and “D” are compared or (ii) when n0 = 1 and the models in C-paths “A” and
“B” are compared).
For IV , paths with dv = 3:
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
160
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. When n0 = 1, the change in IV decreases as q decreases. When n0 = 3, the
change in IV is fairly constant as q decreases. When two models differ by one
xi xj term, the change in IV when an x2i term is removed is very similar.
6. The change in IV is constant as c decreases. Thus, when two models differ by
one x2i term, the change in IV when an xi xj term is removed is also constant.
161
The Robustness of 4-Factor Response Surface Designs
In this section, four-factor CCDs, BBDs, SCDs, PBCDs, UNFSDs, 416A, 416B,
and 416C response surface designs in a spherical design region are considered. Tables
and plots that summarize Q-path and C-path patterns for the set of 224 reduced
models across the D, A, G, and IV criteria will be given. Only paths in which all 4
design variables (dv = 4) appear for the set of 224 reduced models will be analyzed and
discussed because they represent the reduced models most likely to occur in practice.
These models are most likely because the experimenter a priori selects design variables
that are known to affect or are very likely to affect the response.
Tables of D, A, G, and IV criteria values and minimum, maximum, median,
and mean changes in criteria values for the four factor CCDs, BBDs, SCDs, PBCDs,
UNFSDs, 416A, 416B, and 416C response surface designs are given in Appendix A
and C, respectively. Discussion of changes in D, A, G, and IV criteria will correspond
to their mean change.
The Central Composite Designs (CCDs)
The CCDs with rs = 1, 2 axial point replicates and with n0 = 1, 3 center points
are examined for K = 4 design variables. For a summary of the number of Q-paths
and C-paths that increase (“↑”) or decrease (“↓”) or indicate no change (“=”), see
Table 39. The results based on D, A, G, and IV criteria for the 4-factor CCDs are
summarized as follows:
162
Table 39. The Optimality Criteria Across the Reduced Models for the CCD (K = 4).
Q
Criterion rs
1
n0
1
dv
4
3
2
1
4→3
↑
3→2
↑
↑
2→1
1→0
dv
6→5
5→4
4→3
↑(1) ↓(42)
↓
4
↑
↑
↑
↑
↑(7) ↓(4)
4→3
↑
↑
↑
↑
↑
4→3→2
↑
↑
↑
4→3→2→1
3
2
1
4
↑
↑(1) ↓(28)
↓
↓
4
↑
↑
↑
3
↑(2) ↓(2) ↑(1) ↓(11)
↓
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
D
1
3
4 ↑(9) ↓(2) ↑(1) ↓(28)
↓
↓
4
↑
↑
↑
3
↑
↑(9) ↓(3) ↑(4) ↓(7)
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
2
3
4 ↑(7) ↓(4)
↓
↓
↓
4
↑
↑
↑
3
↑(1) ↓(3) ↑(1) ↓(11)
↓
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
1
4
↑
↑
↑
↑
4
↑(3) ↓(2) ↑(7) ↓(3) ↑(13) ↓(6)
3
↑
↑
↑
4→3
↑(3) ↓(1) ↑(3) ↓(1) ↑(3) ↓(1)
2
↑
↑
4→3→2
↑(2) ↓(1) ↑(2) ↓(1)
1
↑
4→3→2→1
3
2
1
4
↑
↑
↑
↑(3) ↓(25)
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2)
3
↑
↑
↑(2) ↓(9)
4→3
↑
↑
↑
2
↑
↑(2) ↓(1)
4→3→2
↑
↑
1
↑
4→3→2→1
3
A
1
3
4
↑
↑
↑
↑
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2)
3
↑
↑
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
2
3
4
↑
↑
↑
↑(2) ↓(26)
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2)
3
↑
↑(10) ↓(2) ↑(1) ↓(10)
4→3
↑
↑
↑
2
↑
↑(1) ↓(2)
4→3→2
↑
↑
1
↑
4→3→2→1
3
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
C
3→2
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑(13) ↓(5)
↑(4) ↓(4)
↓
↓
↑
↑(16) ↓(2)
↑(6) ↓(2)
↓
↓
↑
↑(16) ↓(2)
↑(6) ↓(2)
↓
↓
↑
↑(16) ↓(2)
↑(6) ↓(2)
↓
↓
↑
2→1
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑(7) ↓(4)
↑(2) ↓(6)
↓
↓
↑
↑(10) ↓(1)
↑(2) ↓(6)
↓
↓
↑
↑(10) ↓(1)
↑(2) ↓(6)
↓
↓
↑
↑(10) ↓(1)
↑(2) ↓(6)
↓
↓
↑
1→0
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑(2) ↓(2)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
For D, paths with dv = 4 (Figures 36, 38, 48, and 51):
1. When n0 = 1, D increases as q decreases from 4 to 3. For all other cases,
removal of an x2i term tends to decrease D.
2. Removal of an xi xj term increases D.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
163
Table 39. cont’d
Q
Criterion
rs
1
n0
1
dv
4
3
2
1
4→3 3→2 2→1
↑
↓
↓
↓
↓
↓
C
1→0
↑(21) ↓(7)
↑(10) ↓(1)
↑
↑
dv
6→5
5→4
4→3
3→2
2→1
4
↓
↑(1) ↓(9)
↓
↓
↓
4→3
↓
↓
↑(1) ↓(3)
↓
↓
4→3→2
↓
↓
↓
↓
4→3→2→1
↓
↓
3
↑(1) ↓(2)
↓
2
1
4
↑
↓
↓
↑(3) ↓(25)
4
↓
↑(2) ↓(8) ↑(1) ↓(18) ↑(1) ↓(17)
↓
3
↓
↓
↑(1) ↓(10)
4→3
↓
↓
↑(2) ↓(2)
↑(1) ↓(7)
↓
2
↓
↑(1) ↓(2)
4→3→2
↓
↓
↓
↓
1
↑
4→3→2→1
↓
↓
3
↑(1) ↓(2)
↓
G
1
3
4
↓
↓
↓
↑(21) ↓(7)
4
↓
↑(1) ↓(9)
↓
↓
↓
3
↓
↓
↑(10) ↓(1)
4→3
↓
↓
↑(1) ↓(3)
↓
↓
2
↓
↑
4→3→2
↓
↓
↓
↓
1
↑
4→3→2→1
↓
↓
3
↑(1) ↓(2)
↓
2
3
4
↓
↓
↓
↑(5) ↓(23)
4
↓
↑(2) ↓(8) ↑(1) ↓(18) ↑(3) ↓(15) ↑(1) ↓(10)
3
↓
↓
↑(1) ↓(10)
4→3
↓
↓
↑(1) ↓(3)
↓
↓
2
↓
↑(1) ↓(2)
4→3→2
↓
↓
↓
↓
1
↑
4→3→2→1
↓
↓
3
↑(1) ↓(2)
↓
1
1
4
↓
↓
↓
↓
4
↓
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
↓
1
↓
4→3→2→1
↓
↓
3
↓
↓
2
1
4
↓
↓
↓
↓
4
↓
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
↓
1
↓
4→3→2→1
↓
↓
3
↓
↓
IV
1
3
4
↓
↓
↓
↓
4
↓
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
↓
1
↓
4→3→2→1
↓
↓
3
↓
↓
2
3
4
↓
↓
↓
↓
4
↓
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
↓
1
↓
4→3→2→1
↓
↓
3
↓
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
1→0
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
4. Within a C-path, there is more variability when star points are replicated while
the variability is unaffected by replication of center points.
5. The change in D decreases as q decreases, i.e., the paths in Figure 36 are concave
down, and there is a downward trend in Figure 38. When two models differ by
one xi xj term and rs = 1, n0 = 1, the change in D is larger for the model having
one less xi xj term when q decreases from 4 to 3. Otherwise, the change in D
is similar. For rs = 1, n0 = 3, the change in D when an x2i term is removed is
164
similar. For rs = 2, the change in D when an x2i term is removed tends to be
larger for the model having one less xi xj term.
6. The change in D increases as c decreases (see Figures 51). When two models
differ by one x2i term, the change in D is larger for the model having one less
x2i term when c decreases from 1 to 0. Otherwise, the change in D is similar.
For A, paths with dv = 4 (Figures 39, 41, 52, and 55):
1. A increases with the removal of an x2i term. The only exception is when q
decreases from 1 to 0 and rs = 2 in which A can either increase or decrease.
2. Removal of an xi xj term increases A (except for C-paths “A” and “a” for all
cases and for C-paths “B”, “b”, “b1”, “f”, and “f1” when rs = 1, n0 = 1).
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. Within a C-path, there is more variability when star points are replicated while
the variability is unaffected by replication of center points.
5. The change in A decreases as q decreases. When two models differ by one xi xj
term, the change in A when an x2i term is removed is slightly larger for the
model having one less xi xj term.
6. When rs = 1, the change in A increases as c decreases from 6 → 5 to 5 → 4
to 4 → 3 to 3 → 2 to 2 → 1, and then decreases. However, when rs = 2, the
165
change in A increases as c decreases. When two models differ by one x2i term,
the change in A when an xi xj term is removed is similar (except when rs = 2
and the models in C-paths “C” and “D” are compared)
For G, paths with dv = 4 (Figures 42, 44, 56, and 59):
1. When n0 = 1, G increases as q decreases from 4 to 3. Otherwise, removal of
an x2i term can either increase or decrease G. When n0 = 3, G decreases as q
decreases from 4 to 3 to 2 to 1. Otherwise, removal of an x2i term can either
increase or decrease G.
2. When rs = 1, removal of an xi xj term decreases G (except for C-path “E”).
When rs = 2, G decreases as c decreases from 6 to 5 to 4 to 3. Otherwise,
removal of an xi xj term can either increase or decrease G.
3. G tends to be lower when star points are replicated.
4. Within a Q-path, there is less variability when star points are replicated and
less variability when center points are replicated.
5. Within a C-path, there is no consistent decreasing patterns and less variability
when rs = 2. When rs = 1, the variability is unaffected by replication of center
points (with the exception of C-path “A”). However, when rs = 2, there is more
variability when center points are replicated especially for C-path “A”.
166
6. When n0 = 1, the change in G drops as q decreases from 4 → 3 to 3 → 2,
and then looks fairly constant as q decreases from 3 → 2 to 2 → 1, and then
increases. When n0 = 3, the change in G is fairly constant as q decreases from
4 → 3 to 3 → 2 to 2 → 1, and then increases.
7. When two models differ by one xi xj term:
(a) For the rs = 1, n0 = 1 case, the change in G is smaller for the model having
one less xi xj term when q decreases from 4 to 3 and the change in G is
similar when q decreases from 3 to 2 to 1. Otherwise, there is no pattern
to the change in G.
(b) For the rs = 2, n0 = 1 case, there is no pattern to the change in G when q
decreases from 4 to 3. Otherwise, the change in G is similar (except when
the models in Q-paths “F” and “G” are compared).
(c) For the rs = 1, n0 = 3 case, there is no pattern to the change in G when q
decreases from 1 to 0. Otherwise, the change in G is similar.
(d) For the rs = 2, n0 = 3 case, the change in G is similar (except when the
models in Q-paths “F” and “G” are compared).
8. The mean change in G drops as c decreases from 6 → 5 to 5 → 4 to 4 → 3,
and then increases as c decreases from 4 → 3 to 3 → 2, and then decreases as c
decreases from 3 → 2 to 2 → 1, and then increases.
For IV , paths with dv = 4 (Figures 45, 47, 60, and 63):
167
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
2. IV tends to be higher when star points are replicated and lower when center
points are replicated.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. The variability within a C-path is unaffected by replication of star or center
points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is similar.
6. The change in IV is constant as c decreases. Thus, when two models differ by
one x2i term, the change in IV when an xi xj term is removed is also constant.
168
Figure 36. D-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
Q-Path).
169
Figure 37. D-Efficiency Plots for 4 Factor CCDs for dv = 1, 2, and 3 (Plotting Symbol
= Q-Path).
170
Figure 38. The Change in D-Efficiency Plots by Reduction of Squared Terms in
Model for 4 Factor CCDs.
171
Figure 39. A-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
Q-Path).
172
Figure 40. A-Efficiency Plots for 4 Factor CCDs for dv = 1, 2, and 3 (Plotting Symbol
= Q-Path).
173
Figure 41. The Change in A-Efficiency Plots by Reduction of Squared Terms in Model
for 4 Factor CCDs.
174
Figure 42. G-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
Q-Path).
175
Figure 43. G-Efficiency Plots for 4 Factor CCDs for dv = 1, 2, and 3 (Plotting Symbol
= Q-Path).
176
Figure 44. The Change in G-Efficiency Plots by Reduction of Squared Terms in
Model for 4 Factor CCDs.
177
Figure 45. IV -Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
Q-Path).
178
Figure 46. IV -Efficiency Plots for 4 Factor CCDs for dv = 1, 2, and 3 (Plotting
Symbol = Q-Path).
179
Figure 47. The Change in IV -Efficiency Plots by Reduction of Squared Terms in
Model for 4 Factor CCDs.
180
Figure 48. D-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
C-Path).
181
Figure 49. D-Efficiency Plots for 4 Factor CCDs for dv = 4 → 3 and 4 → 3 → 2
(Plotting Symbol = C-Path).
182
Figure 50. D-Efficiency Plots for 4 Factor CCDs for dv = 3, 4 → 3 and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path).
183
Figure 51. The Change in D-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 4 Factor CCDs.
184
Figure 52. A-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
C-Path).
185
Figure 53. A-Efficiency Plots for 4 Factor CCDs for dv = 4 → 3 and 4 → 3 → 2
(Plotting Symbol = C-Path).
186
Figure 54. A-Efficiency Plots for 4 Factor CCDs for dv = 3, 4 → 3 and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path).
187
Figure 55. The Change in A-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 4 Factor CCDs.
188
Figure 56. G-Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
C-Path).
189
Figure 57. G-Efficiency Plots for 4 Factor CCDs for dv = 4 → 3 and 4 → 3 → 2
(Plotting Symbol = C-Path).
190
Figure 58. G-Efficiency Plots for 4 Factor CCDs for dv = 3, 4 → 3 and 4 → 3 → 2 → 1
(Plotting Symbol = C-Path).
191
Figure 59. The Change in G-Efficiency Plots by Reduction of Cross-Product Terms
in Model for 4 Factor CCDs.
192
Figure 60. IV -Efficiency Plots for 4 Factor CCDs for dv = 4 (Plotting Symbol =
C-Path).
193
Figure 61. IV -Efficiency Plots for 4 Factor CCDs for dv = 4 → 3 and 4 → 3 → 2
(Plotting Symbol = C-Path).
194
Figure 62. IV -Efficiency Plots for 4 Factor CCDs for dv = 3, 4 → 3 and 4 → 3 →
2 → 1 (Plotting Symbol = C-Path).
195
Figure 63. The Change in IV -Efficiency Plots by Reduction of Cross-Product Terms
in Model for 4 Factor CCDs.
196
The Box-Behnken Designs (BBDs)
The BBDs with n0 = 1, 3 center points are examined for K = 4 design variables.
For a summary of the number of Q-paths and C-paths that increase (“↑”) or decrease
(“↓”) or indicate no change (“=”), see Table 40.
Table 40. The Optimality Criteria Across the Reduced Models for the BBD (K = 4).
Q
Criterion n0
1
dv
4
3
2
1
4→3
↑
3→2
↑
↑
2→1
1→0
dv
6→5
5→4
4→3
↑(1) ↓(42)
↓
4
↑
↑
↑
↑
↑(7) ↓(4)
4→3
↑
↑
↑
↑
↑
4→3→2
↑
↑
↑
4→3→2→1
3
D
3
4 ↑(9) ↓(2) ↑(1) ↓(28)
↓
↓
4
↑
↑
↑
3
↑
↑(9) ↓(3) ↑(4) ↓(7)
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4
↑
↑
↑
↑
4
↑(3) ↓(2) ↑(7) ↓(3) ↑(13) ↓(6)
3
↑
↑
↑
4→3
↑(3) ↓(1) ↑(3) ↓(1) ↑(3) ↓(1)
2
↑
↑
4→3→2
↑(2) ↓(1) ↑(2) ↓(1)
1
↑
4→3→2→1
3
A
3
4
↑
↑
↑
↑
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2)
3
↑
↑
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4
↑
↓
↓
↑(21) ↓(7)
4
↓
↑(1) ↓(9)
↓
3
↓
↓
↑(10) ↓(1)
4→3
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
G
3
4
↓
↓
↓
↑(21) ↓(7)
4
↓
↑(1) ↓(9)
↓
3
↓
↓
↑(10) ↓(1)
4→3
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
1
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
IV
3
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
C
3→2
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑(13) ↓(5)
↑(4) ↓(4)
↓
↓
↑
↑(16) ↓(2)
↑(6) ↓(2)
↓
↓
↑
↓
↓
↓
↓
↑(1) ↓(2)
↓
↓
↓
↓
↑(1) ↓(2)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
2→1
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑(7) ↓(4)
↑(2) ↓(6)
↓
↓
↑
↑(10) ↓(1)
↑(2) ↓(6)
↓
↓
↑
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
1→0
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑(2) ↓(2)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
↓
↓
In Chapter 2, it was shown that the D, A, G, and IV criteria for the 4 factor
BBDs are identical to the 4 factor CCDs having rs = 1. Thus, the two plots in the
197
first column of Figures 36 to 63 are also the corresponding plots for the 4 factor BBDs
having n0 = 1 and 3, respectively. The results based on D, A, G, and IV criteria for
the 4-factor BBDs are summarized as follows:
For D, paths with dv = 4:
1. When n0 = 1, D increases as q decreases from 4 to 3. For all other cases,
removal of an x2i term tends to decrease D.
2. Removal of an xi xj term increases D.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. The change in D decreases as q decreases. When two models differ by one xi xj
term and n0 = 1, the change in D is slightly larger for the model having one less
xi xj term when q decreases from 4 to 3. Otherwise, the change in D is similar.
6. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D is larger for the model having one less x2i term when c
decreases from 1 to 0. Otherwise, the change in D is similar.
For A, paths with dv = 4:
1. Removal of an x2i term increases A.
2. Removal of an xi xj term increases A (except for C-paths “A” and “a” for all
cases and for C-paths “B”, “b”, “b1”, “f”, and “f1” when n0 = 1).
198
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. The change in A decreases as q decreases. When two models differ by one xi xj
term, the change in A when an x2i term is removed is slightly larger for the
model having one less xi xj term.
6. The change in A increases slightly as c decreases from 6 → 5 to 5 → 4 to 4 → 3
to 3 → 2 to 2 → 1, and then decreases. When two models differ by one x2i term,
the change in A when an xi xj term is removed is similar.
For G, paths with dv = 4:
1. When n0 = 1, G increases as q decreases from 4 to 3. Otherwise, removal of
an x2i term can either increase or decrease G. When n0 = 3, G decreases as q
decreases from 4 to 3 to 2 to 1. Otherwise, removal of an x2i term can either
increase or decrease G.
2. Removal of an xi xj term decreases G (except for C-path “E”).
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points
(with the exception of C-path “A”).
5. When n0 = 1, the change in G drops as q decreases from 4 → 3 to 3 → 2,
and then looks fairly constant as q decreases from 3 → 2 to 2 → 1, and then
199
increases. When n0 = 3, the change in G looks fairly constant as q decreases
from 4 → 3 to 3 → 2 to 2 → 1, and then increases.
6. When two models differ by one xi xj term:
(a) For the n0 = 1 case, the change in G is smaller for the model having one
less xi xj term when q decreases from 4 to 3 and the change in G is similar
when q decreases from 3 to 2 to 1. Otherwise, there is no pattern to the
change in G.
(b) For the n0 = 3 case, there is no pattern to the change in G when q decreases
from 1 to 0. Otherwise, the change in G is similar.
7. There is no pattern to the mean change in G as c decreases.
For IV , paths with dv = 4:
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is similar.
200
6. The change in IV is constant as c decreases. Thus, when two models differ by
one x2i term, the change in IV when an xi xj term is removed is also constant.
The Small Composite Designs (SCDs)
The SCDs with rs = 1, 2 axial point replicates and with n0 = 1, 3 center points
are examined for K = 4 design variables. For a summary of the number of Q-paths
and C-paths that increase (“↑”) or decrease (“↓”) or indicate no change (“=”), see
Table 41. For SCDs, plots of the D, A, G, and IV criteria and plots of the change in
the D, A, G, and IV criteria are given in Appendix C. The results based on D, A,
G, and IV criteria for the 4-factor SCDs are summarized as follows:
For D, paths with dv = 4:
1. When n0 = 1, D increases as q decreases from 4 to 3 (except for Q-path “A”
when rs = 1). For all other cases, removal of an x2i term tends to decrease D
(with the exception of Q-path “G” when rs = 2, n0 = 3).
2. Removal of an xi xj term increases D.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. Within a C-path, there is more variability when star points are replicated while
the variability is unaffected by replication of center points.
201
Table 41. The Optimality Criteria Across the Reduced Models for the SCD (K = 4).
Q
Criterion rs
1
D
A
n0
1
dv
4→3
3→2
4 ↑(10),↓(1)
↓
3
↑(2) ↓(2)
2
1
C
2→1
↓
↑(1) ↓(11)
↑
1→0
↓
↓
↑
↑
2
1
4
3
2
1
↑
↓
↑(1) ↓(3)
↓
↓
↑(1) ↓(1)
↓
↓
↑(1) ↓(2)
↑
1
3
4
3
2
1
↓
↓
↑(1) ↓(3)
↓
↓
↑
↓
↓
↑
↑
2
3
4
3
2
1
↑(1) ↓(10)
↓
↓
↓
↓
↑(1) ↓(1)
↓
↓
↑(1) ↓(2)
↑
1
1
4
3
2
1
↑
↑
↑
↑(28) ↓(15) ↑(3) ↓(25)
↑
↑(1) ↓(10)
↑
↑(1) ↓(2)
↑
2
1
4
3
2
1
↑
↑
↑
↑(4) ↓(39)
↓
↑(1) ↓(11) ↑(1) ↓(10)
↑(1) ↓(1) ↑(1) ↓(2)
↑
1
3
4
3
2
1
↑
↑(17) ↓(12) ↑(4) ↓(39) ↑(1) ↓(27)
↑
↑(4) ↓(8) ↑(1) ↓(10)
↑
↑(1) ↓(2)
↑
2
3
4
3
2
1
↑
↑(25) ↓(4) ↑(1) ↓(42)
↓
↑
↑(1) ↓(11)
↓
↑(1) ↓(1) ↑(1) ↓(2)
↑
dv
6→5 5→4
4→3
4
↑
↑
↑
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
4
↑
↑
↑
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
4
↑
↑
↑
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
4
↑
↑
↑
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
4
↑ ↑(3) ↓(7) ↑(15) ↓(4)
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
4
↑ ↑(9) ↓(1) ↑(18) ↓(1)
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
4
↑ ↑(5) ↓(5) ↑(13) ↓(6)
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
4
↑
↑
↑
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
3→2
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑(17) ↓(1)
↑(6) ↓(2)
↓
↓
↑
↑(17) ↓(1)
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
2→1
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(4) ↓(4)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
1→0
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
5. The change in D decreases as q decreases. When two models differ by one xi xj
term and n0 = 1, the change in D is similar (except when the models in Qpaths “E” and “F” and Q-paths “F” and “G” are compared). When n0 = 3,
the change in D when an x2i term is removed is similar.
6. The change in D decreases as c decreases from 6 → 5 to 5 → 4, and then
increases as c decreases from 5 → 4 to 4 → 3 to 3 → 2 to 2 → 1. When c
decreases from 2 → 1 to 1 → 0, the change in D decreases when rs = 1 and
202
Table 41. cont’d
Q
Criterion
rs
1
n0
1
dv
4
3
2
1
4→3
↑(1) ↓(10)
3→2 2→1
1→0
↓
↓
↑(1) ↓(27)
↓
↓
↑(1) ↓(10)
↓
↑(1) ↓(2)
↑
dv
6→5 5→4
4→3
4
↑
↓
↑(8) ↓(11)
4→3
↑
↓
↓
4→3→2
↓
↓
4→3→2→1
3
2
1
4
↑(2) ↓(9)
↓
↓
↑(1) ↓(27)
4
↑
↓
↑(8) ↓(11)
3
↓
↓
↑(1) ↓(10)
4→3
↑
↓
↓
2
↓
↑(1) ↓(2)
4→3→2
↓
↑(1) ↓(2)
1
↑
4→3→2→1
3
G
1
3
4
↓
↓
↓
↑(2) ↓(26)
4
↑
↓
↑(8) ↓(11)
3
↓
↓
↑(1) ↓(10)
4→3
↑
↓
↓
2
↓
↑(1) ↓(2)
4→3→2
↓
↓
1
↑
4→3→2→1
3
2
3
4
↓
↓
↓
↑(1) ↓(27)
4
↑
↓
↑(8) ↓(11)
3
↓
↓
↑(1) ↓(10)
4→3
↑
↓
↓
2
↓
↑(1) ↓(2)
4→3→2
↓
↑(1) ↓(2)
1
↑
4→3→2→1
3
1
1
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
2
1
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
IV
1
3
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
2
3
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
C
3→2
2→1
1→0
↑(6) ↓(12)
↑
↑(3) ↓(1)
↑(2) ↓(6)
↑(3) ↓(5)
↑
↓
↓
↑
↓
↓
↑(1) ↓(1)
↑(1) ↓(2)
↓
↑(6) ↓(12) ↑(10) ↓(1) ↑(3) ↓(1)
↑(2) ↓(6)
↑(3) ↓(5)
↑
↓
↓
↑
↓
↓
↑
↑(1) ↓(2)
↓
↑(6) ↓(12)
↑
↑
↑(2) ↓(6)
↑(3) ↓(5)
↑
↓
↓
↑
↓
↓
↑(1) ↓(1)
↑(1) ↓(2)
↓
↑(6) ↓(12)
↑
↑
↑(2) ↓(6)
↑(3) ↓(5)
↑
↓
↓
↑
↓
↓
↑
↑(1) ↓(2)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
increases when rs = 2. When two models differ by one x2i term, the change in
D is slightly larger for the model having one less x2i term when c decreases from
1 to 0. Otherwise, the change in D is similar.
For A, paths with dv = 4:
1. A increases as q decreases from 4 to 3, and when n0 = 1, continues to increase
as q decreases from 3 to 2. Otherwise, A can either increase or decrease with
removal of an x2i term.
203
2. A can either increase or decrease as c decreases from 5 to 4 to 3. Otherwise, A
increases with removal of an xi xj term (except for C-path “A”).
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. Within a C-path, there is slightly more variability when star points are replicated while the variability is unaffected by replication of center points (with the
exception of C-path “A” for both cases).
5. The change in A decreases as q decreases. When two models differ by one xi xj
term, the change in A is larger for the model having one less xi xj term when q
decreases from 4 to 3. Otherwise, the change in A is similar (except when the
models in Q-paths “E” and “F” and Q-paths “F” and “G” are compared).
6. The change in A decreases as c decreases from 6 → 5 to 5 → 4, and then
increases as c decreases from 5 → 4 to 4 → 3. A changes slightly as c decreases
from 4 → 3 to 3 → 2. A increases as c decreases from 3 → 2 to 2 → 1, and
then decreases.
7. When two models differ by one x2i term, the change in A is similar when c
decreases from 6 to 5 to 4 and from 3 to 2. Otherwise, the change in A is larger
for the model having one less x2i term. (except when the models in C-paths “D”
and “E” are compared).
204
For G, paths with dv = 4:
1. G tends to decrease with removal of an x2i term (except for Q-paths “F” and
“G”).
2. G increases as c decreases from 6 to 5 and decreases as c decreases from 5 to 4.
Otherwise, removal of an xi xj term can either increase or decrease G.
3. The variability within a Q-path is unaffected by replication of star or center
points (with the exception of Q-paths “F” and “G”).
4. Within a C-path, there is slightly more variability when star points are replicated while the variability is unaffected by replication of center points (with the
exception of C-path “A”).
5. When n0 = 1, the change in G drops as q decreases from 4 → 3 to 3 → 2,
and then looks fairly constant as q decreases from 3 → 2 to 2 → 1, and then
increases. When n0 = 3, the change in G looks fairly constant as q decreases
from 4 → 3 to 3 → 2 to 2 → 1, and then increases.
6. When two models differ by one xi xj term, the change in G when an x2i term
is removed is similar (except when the models in Q-paths “F” and “G” are
compared).
7. The change in G drops as c decreases from 6 → 5 to 5 → 4, and then increases
slightly as c decreases from 5 → 4 to 4 → 3. A either increase or decreases
205
slightly as c decreases from 4 → 3 to 3 → 2, but then increases as c decreases
from 3 → 2 to 2 → 1. The change in G as c decreases from 2 → 1 to 1 → 0
decreases sharply when rs = 1 and increases sharply when rs = 2.
8. When two models differ by one x2i term, the change in G when an xi xj term is
removed tends to be slightly smaller for the model having one less x2i term.
For IV , paths with dv = 4:
1. IV decreases as q or c decreases. However, the decrease in IV when an xi xj
term is removed is smaller.
2. IV tends to be higher when star points are replicated and lower when center
points are replicated.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. The variability within a C-path is unaffected by replication of star or center
points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is similar.
6. The change in IV decreases as c decreases from 6 → 5 to 5 → 4, and then
increases as c decreases from 5 → 4 to 4 → 3, and then decreases as c decreases
from 4 → 3 to 3 → 2, and then increases as c decreases from 3 → 2 to 2 → 1,
206
and then decreases. When two models differ by one x2i term, the change in IV
when an xi xj term is removed is similar.
The Plackett-Burman Composite Designs
(PBCDs)
The PBCDs with rs = 1, 2 axial point replicates and with n0 = 1, 3 center points
are examined for K = 4 design variables. For a summary of the number of Q-paths
and C-paths that increase (“↑”) or decrease (“↓”) or indicate no change (“=”), see
Table 42. For PBCDs, plots of the D, A, G, and IV criteria and plots of the change
in the D, A, G, and IV criteria are given in Appendix C. The results based on D, A,
G, and IV criteria for the 4-factor PBCDs are summarized as follows:
For D, paths with dv = 4:
1. When n0 = 1, D increases as q decreases from 4 to 3. For all other cases,
removal of an x2i term tends to decrease D (with the exception of Q-path “G”
when n0 = 3 and Q-path “F” when rs = 2, n0 = 3).
2. Removal of an xi xj term increases D.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. Within a C-path, there is more variability when star points are replicated while
the variability is unaffected by replication of center points.
207
Table 42. The Optimality Criteria Across the Reduced Models for the PBCD (K =
4).
Q
Criterion rs n0 dv
1
1
4
3
2
1
4→3
↑
C
3→2
2→1
↑(1) ↓(28)
↓
↑
↑(4) ↓(8)
↑
1→0
dv
6→5
5→4
4→3
3→2
2→1
1→0
↓
4
↑
↑
↑
↑
↑
↑
↑(1) ↓(10)
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
↑
4→3→2
↑
↑
↓
↓
↑
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
2
1
4
↑
↓
↓
↓
4
↑
↑
↑
↑
↑
↑
3
↑(1) ↓(3)
↓
↓
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑
↑(1) ↓(2)
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
D
1
3
4 ↑(1) ↓(10)
↓
↓
↓
4
↑
↑
↑
↑
↑
↑
3
↑(2) ↓(2) ↑(1) ↓(11) ↑(1) ↓(10)
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑
↑
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
2
3
4 ↑(2) ↓(9)
↓
↓
↓
4
↑
↑
↑
↑
↑
↑
3
↑(1) ↓(3)
↓
↓
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑
↑(1) ↓(2)
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
1
1
4
↑
↑
↑
↑(18) ↓(10)
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2) ↑(16) ↓(2) ↑(10) ↓(1) ↑(3) ↓(1)
3
↑
↑
↑(9) ↓(2)
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑
↑
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
2
1
4
↑
↑
↑(26) ↓(17) ↑(1) ↓(27)
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2) ↑(16) ↓(2) ↑(10) ↓(1) ↑(3) ↓(1)
3
↑
↑(6) ↓(6) ↑(1) ↓(10)
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑
↑(1) ↓(2)
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
A
1
3
4
↑
↑
↑(42) ↓(1) ↑(5) ↓(23)
4
↑
↑
↑
↑
↑
↑
3
↑
↑
↑(3) ↓(8)
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑
↑
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
2
3
4
↑
↑
↑(4) ↓(39) ↑(1) ↓(27)
4
↑
↑
↑
↑
↑
↑
3
↑
↑(4) ↓(8) ↑(1) ↓(10)
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑(1) ↓(1) ↑(1) ↓(2)
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
5. The change in D decreases as q decreases. When two models differ by one xi xj
term, the change in D is similar (except when n0 = 1 and the models in Q-paths
“E” and “F” and Q-paths “F” and “G” are compared).
6. The change in D increases as c decreases. The only exception is a slight decrease
in the change in D as c decreases from 4 → 3 to 3 → 2 when rs = 1. When two
models differ by one x2i term, the change in D is slightly larger for the model
208
Table 42. cont’d
Q
Criterion
rs
1
n0
1
dv
4
3
2
1
C
4→3
3→2 2→1
1→0
dv
6→5
5→4
4→3
3→2
↑(3) ↓(8)
↓
↓
↑(2) ↓(26)
4
↑
↑
↑
↑(10) ↓(8)
↓
↓
↑(3) ↓(8)
4→3
↑
↑(3) ↓(1)
↑
↑(2) ↓(6)
↓
↑(1) ↓(2)
4→3→2
↑
↑
↓
↑
4→3→2→1
↓
3
↑(1) ↓(2)
2
1
4
↑(7) ↓(4)
↓
↓
↑(1) ↓(27)
4
↑
↑
↑(17) ↓(2) ↑(8) ↓(10)
3
↓
↓
↑(1) ↓(10)
4→3
↑
↑
↑
↓
2
↓
↑(1) ↓(2)
4→3→2
↑
↑
↓
1
↑
4→3→2→1
↓
3
↑
G
1
3
4
↓
↓
↓
↑(4) ↓(24)
4
↑
↑
↑
↑(9) ↓(9)
3
↓
↓
↑(4) ↓(7)
4→3
↑
↑(3) ↓(1)
↑
↑(2) ↓(6)
2
↓
↑(1) ↓(2)
4→3→2
↑
↑
↓
1
↑
4→3→2→1
↓
3
↑(1) ↓(2)
2
3
4
↓
↓
↓
↑(1) ↓(27)
4
↑
↑
↑
↑(9) ↓(9)
3
↓
↓
↑(1) ↓(10)
4→3
↑
↑
↑
↓
2
↓
↑(1) ↓(2)
4→3→2
↑
↑
↓
1
↑
4→3→2→1
↓
3
↑
1
1
4
↓
↓
↓
↓
4
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
1
↓
4→3→2→1
↓
3
↓
2
1
4
↓
↓
↓
↓
4
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
1
↓
4→3→2→1
↓
3
↓
IV
1
3
4
↓
↓
↓
↓
4
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
1
↓
4→3→2→1
↓
3
↓
2
3
4
↓
↓
↓
↓
4
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
1
↓
4→3→2→1
↓
3
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
2→1
↑(7) ↓(4)
↓
↓
↓
↓
↑(8) ↓(3)
↑(1) ↓(7)
↓
↓
↓
↑(7) ↓(4)
↓
↓
↓
↓
↑(9) ↓(2)
↑(1) ↓(7)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
1→0
↑(2) ↓(2)
↑(2) ↓(2)
↑(1) ↓(2)
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑(2) ↓(1)
↑(1) ↓(1)
↑(3) ↓(1)
↑(2) ↓(2)
↑(1) ↓(2)
↑(1) ↓(1)
↑
↑
↑(2) ↓(1)
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
having one less x2i term when c decreases from 1 to 0. Otherwise, the change in
D is similar.
For A, paths with dv = 4:
1. A increases as q decreases from 4 to 3 to 2. Otherwise, A can either increase or
decrease with removal of an x2i term.
2. A tends to increases with removal of an xi xj term (except for C-paths “A” and
“a” when n0 = 1).
209
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. Within a C-path, there is slightly more variability when star points are replicated while the variability is unaffected by replication of center points (with the
exception of C-path “A” for both cases).
5. The change in A decreases as q decreases. When two models differ by one xi xj
term:
(a) For the n0 = 1 case, the change in A is larger for the model having one
less xi xj term when q decreases from 4 to 3. Otherwise, the change in A
is similar (except when the models in Q-paths “E” and “F” and Q-paths
“F” and “G” are compared).
(b) For the n0 = 3 case, the change in A is larger for the model having one
less xi xj term when q decreases from 4 to 3 (except when the models in Qpaths “A” and “B” are compared). Otherwise, the change in A is similar
(except when the models in Q-paths “E” and “F” and Q-paths “F” and
“G” are compared).
6. The change in A increases as c decreases from 6 → 5 to 5 → 4 to 4 → 3, and
then decreases as c decreases from 4 → 3 to 3 → 2, and then increases. When
two models differ by one x2i term, the change in A is similar (except when the
models in C-paths “B” and “C” and C-paths “C” and “D” are compared).
210
For G, paths with dv = 4:
1. G tends to decrease with removal of an x2i term (except for Q-paths “D”, “E”,
“F”, “G”, “M”, and “U”).
2. G increases as c decreases from 6 to 5 to 4. Otherwise, removal of an xi xj term
can either increase or decrease G.
3. The variability within a Q-path is unaffected by replication of star or center
points (with the exception of Q-paths “D”, “E”, “F” and “G” when n0 = 1).
4. Within a C-path, there is slightly more variability when star points are replicated while the variability is unaffected by replication of center points (with the
exception of C-path “A”).
5. When n0 = 1, the change in G drops as q decreases from 4 → 3 to 3 → 2,
and then looks fairly constant as q decreases from 3 → 2 to 2 → 1, and then
increases. When n0 = 3, the change in G looks fairly constant as q decreases
from 4 → 3 to 3 → 2 to 2 → 1, and then increases.
6. When two models differ by one xi xj term, the change in G when an x2i term
is removed is similar (except for n0 = 1, when the models in Q-paths “E” and
“F” and for all cases, when the models in Q-paths “F” and “G” are compared).
211
7. The change in G drops as c decreases from 6 → 5 to 5 → 4, and then increases
as c decreases from 5 → 4 to 4 → 3, and then decreases as c decreases from
4 → 3 to 3 → 2 to 2 → 1. It then increases except when rs = 1, n0 = 1.
For IV , paths with dv = 4:
1. IV decreases as q or c decreases. However, the decrease in IV when an xi xj
term is removed is smaller.
2. IV tends to be higher when star points are replicated and lower when center
points are replicated.
3. Within a Q-path, there is more variability when star points are replicated and
less variability when center points are replicated.
4. The variability within a C-path is unaffected by replication of star or center
points.
5. The change in IV approaches to zero as q decreases. When two models differ
by one xi xj term, the change in IV when an x2i term is removed is similar.
6. The change in IV decreases as c decreases from 6 → 5 to 5 → 4, and then looks
fairly constant as c decreases from 5 → 4 to 4 → 3, and then decreases as c
decreases from 4 → 3 to 3 → 2, and then increases as c decreases from 3 → 2 to
2 → 1, and then decreases. When two models differ by one x2i term, the change
in IV when an xi xj term is removed is similar.
212
The Uniform Shell Designs (UNFSDs)
The UNFSDs with n0 = 1, 3 center points are examined for K = 4 design variables. For a summary of the number of Q-paths and C-paths that increase (“↑”) or
decrease (“↓”) or indicate no change (“=”), see Table 43. For UNFSDs, plots of the
D, A, G, and IV criteria and plots of the change in the D, A, G, and IV criteria are
given in Appendix C. The results based on D, A, G, and IV criteria for the 4-factor
UNFSDs are summarized as follows:
For D, paths with dv = 4:
1. When n0 = 1, D increases as q decreases from 4 to 3. For all other cases,
removal of an x2i term tends to decrease D.
2. Removal of an xi xj term increases D.
3. D tends to be lower when center points are replicated.
4. There is more variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D decreases as q decreases. When two models differ by one xi xj
term and n0 = 1, the change in D is slightly larger for the model having one less
xi xj term when q decreases from 4 to 3. Otherwise, the change in D is similar.
7. The change in D increases as c decreases. When two models differ by one x2i
term and c > 0, the change in D is similar when an xi xj terms is removed.
213
Table 43. The Optimality Criteria Across the Reduced Models for the UNFSD (K =
4).
Q
Criterion n0
1
dv
4
3
2
1
4→3
↑
3→2
2→1
1→0
dv
6→5
5→4
4→3
↓
↓
↓
4
↑
↑
↑
↑(3) ↓(1) ↑(1) ↓(11) ↑(1) ↓(10)
4→3
↑
↑
↑
↑
↑
4→3→2
↑
↑
↑
4→3→2→1
3
D
3
4
↓
↓
↓
↓
4
↑
↑
↑
3
↑(1) ↓(3) ↑(1) ↓(11) ↑(1) ↓(10)
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4
↑
↑
↑
↑(27) ↓(1)
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2)
3
↑
↑
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
A
3
4
↑
↑
↑
↑(27) ↓(1)
4
↑
↑
↑
3
↑
↑
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4 ↑(1) ↓(10)
↓
↑(4) ↓(39) ↑(20) ↓(8)
4
↓
↑(1) ↓(9)
↓
3
↓
↑(2) ↓(10) ↑(7) ↓(4)
4→3
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
G
3
4
↓
↓
↑(4) ↓(39) ↑(19) ↓(9)
4
↓
↑(1) ↓(9)
↓
3
↓
↑(2) ↓(10) ↑(7) ↓(4)
4→3
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
1
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
IV
3
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
C
3→2
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑(16) ↓(2)
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑(3) ↓(15)
↓
↓
↓
↑(1) ↓(2)
↑(3) ↓(15)
↑(2) ↓(6)
↓
↓
↑(1) ↓(2)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
2→1
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑(10) ↓(1)
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑(1) ↓(10)
↑(1) ↓(7)
↓
↓
↓
↑(2) ↓(9)
↑(1) ↓(7)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
1→0
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
↓
↓
For A, paths with dv = 4:
1. Removal of an x2i term increases A (except for Q-path “U”).
2. Removal of an xi xj term increases A (except for C-paths “A” and “a” when
n0 = 1).
3. There is less variability within a Q-path when center points are replicated.
214
4. The variability within a C-path is unaffected by replication of center points
(with the exception of C-path “A”).
5. The change in A decreases as q decreases. When two models differ by one xi xj
term, the change in A when an x2i term is removed is slightly larger for the
model having one less xi xj term.
6. The change in A increases as c decreases. When two models differ by one x2i
term, the change in A is slightly larger for the model having one less x2i term
when c decreases from 1 to 0. Otherwise, the change in A is similar.
For G, paths with dv = 4:
1. G tends to decrease as q decreases from 4 to 3 to 2 (except for Q-path “G”
when n0 = 1). Otherwise, removal of an x2i term can either increase or decrease
G.
2. G tends to decrease as c decreases from 6 to 5 to 4 to 3 (except for C-path “E”).
Otherwise, G can either increase or decrease with removal of an xi xj term.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points
(with the exception of C-path “A”).
5. When n0 = 1, the change in G drops as q decreases from 4 → 3 to 3 → 2, and
then increases. When n0 = 3, the change in G increases as q decreases. When
215
two models differ by one xi xj term, the change in G when an x2i term is removed
is similar (except when the models in Q-paths “E”and “F” and Q-paths “F”
and “G” are compared).
6. The mean change in G is fairly constant as c decreases from 6 → 5 to 5 → 4,
and then decreases as c decreases from 4 → 3 to 3 → 2, and then increases
slightly.
For IV , paths with dv = 4:
1. IV decreases as q or c decreases. The decrease in IV when an xi xj term is
removed, however, is smaller.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is similar.
6. The change in IV decreases as c decreases from 6 → 5 to 5 → 4, and then
increases. All of these changes, however, are very small. When two models
differ by one x2i term, the change in IV when an xi xj term is removed is similar.
216
The Hybrid 416A Designs (416As)
The 416A designs with n0 = 1, 3 center points are examined for K = 4 design
variables. For a summary of the number of Q-paths and C-paths that increase (“↑”)
or decrease (“↓”) or indicate no change (“=”), see Table 44. For 416A designs, plots
of the D, A, G, and IV criteria and plots of the change in the D, A, G, and IV
criteria are given in Appendix C. The results based on D, A, G, and IV criteria for
the 4-factor 416A designs are summarized as follows:
Table 44. The Optimality Criteria Across the Reduced Models for the 416A (K = 4).
Q
Criterion n0
1
dv 4→3
3→2
2→1
1→0
dv
6→5
5→4
4→3
4
↑ ↑(11) ↓(18) ↑(4) ↓(39) ↑(26) ↓(2)
4
↑
↑
↑
3
↑
↑(2) ↓(10)
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
D
3
4
↓
↑(3) ↓(26) ↑(4) ↓(39) ↑(12) ↓(16)
4
↑
↑
↑
3
↑(3) ↓(1) ↑(4) ↓(8)
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4
↑
↑
↑
↑
4
↑(3) ↓(2) ↑(7) ↓(3) ↑(14) ↓(5)
3
↑
↑
↑
4→3
↑(3) ↓(1) ↑(3) ↓(1) ↑(3) ↓(1)
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
A
3
4
↑
↑
↑(41) ↓(2)
↑
4
↑
↑
↑
3
↑
↑
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4
↓
↑(3) ↓(26) ↑(4) ↓(39) ↑(27) ↓(1)
4
↓
↓
↑(2) ↓(17)
3
↓
↓
↑
4→3
↓
↓
↑(1) ↓(3)
2
↑(1) ↓(1)
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
G
3
4
↓
↑(3) ↓(26) ↑(4) ↓(39) ↑(27) ↓(1)
4
↓
↓
↑(2) ↓(17)
3
↓
↓
↑
4→3
↓
↓
↑(1) ↓(3)
2
↑(1) ↓(1)
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
1
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
IV
3
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
C
3→2
↑(17) ↓(1)
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑(12) ↓(6)
↑(5) ↓(3)
↓
↓
↑
↑(17) ↓(1)
↑(6) ↓(2)
↓
↓
↑
↑(2) ↓(16)
↑(1) ↓(7)
↓
↓
↓
↑(2) ↓(16)
↑(1) ↓(7)
↓
↓
↓
↑(1) ↓(17)
↓
↓
↓
↓
↑(1) ↓(17)
↓
↓
↓
↓
2→1
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑(6) ↓(5)
↑(2) ↓(6)
↓
↓
↑
↑(10) ↓(1)
↑(2) ↓(6)
↓
↓
↑
↓
↓
↑(1) ↓(2)
↓
↓
↓
↓
↑(1) ↓(2)
↓
↓
↓
↑(1) ↓(7)
↓
↓
↓
↓
↑(1) ↓(7)
↓
↓
↓
1→0
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑(2) ↓(2)
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
↓
↓
217
For D, paths with dv = 4:
1. When n0 = 1, D increases as q decreases from 4 to 3. For all other cases,
removal of an x2i term can either increase or decrease D.
2. Removal of an xi xj term tends to increase D.
3. D tends to be lower when center points are replicated.
4. There is less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D decreases as q decreases from 4 → 3 to 3 → 2 to 2 → 1, and
then increases. When two models differ by one xi xj term, the change in D when
an x2i term is removed is similar.
7. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D is slightly larger for the model having one less x2i term
when c decreases from 1 to 0. Otherwise, the change in D is similar.
For A, paths with dv = 4:
1. Removal of an x2i term increases A (except for Q-path “D” when n0 = 3).
2. Removal of an xi xj term tends to increase A when n0 = 3 (except for C-paths
“C” and “h1”). When n0 = 1, A can either increase or decrease as c decreases.
3. There is less variability within a Q-path when center points are replicated.
218
4. The variability within a C-path is unaffected by replication of center points.
5. The change in A decreases as q decreases from 4 → 3 to 3 → 2 to 2 → 1,
and then increases. When two models differ by one xi xj term and n0 = 1, the
change in A when an x2i term is removed is slightly larger for the model having
one less xi xj term. When n0 = 3, the change in A is similar (except when the
models in Q-paths “E” and “F” and Q-paths “F” and “G” are compared).
6. When n0 = 1, the change in A increases as c decreases from 6 → 5 to 5 → 4
to 4 → 3, and then decreases slightly. However, when n0 = 3, the change in A
increases as c decreases. When two models differ by one x2i term, the change in
A when an xi xj term is removed is similar.
For G, paths with dv = 4:
1. For most Q-paths, G tends to decrease as q decreases from 4 to 3 to 2 to 1.
When q decreases from 1 to 0, G tends to increase.
2. For most C-paths, G tends to decrease as c decreases.
3. G tends to be lower when center points are replicated.
4. There is slightly less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points
(with the exception of C-paths “C” and “c”).
219
6. The change in G increases slightly as q decreases from 4 → 3 to 3 → 2, and
then slightly decreases as q decreases from 3 → 2 to 2 → 1, and then increases.
When two models differ by one xi xj term, there is no pattern to the change in
G when q decreases from 1 to 0. Otherwise, the change in G is similar.
7. The mean change in G decreases as c decreases from 6 → 5 to 5 → 4, and
then increases as c decreases from 5 → 4 to 4 → 3, and then decreases as c
decreases from 4 → 3 to 3 → 2 to 2 → 1, and then increases. When two models
differ by one x2i term, the change in G when an xi xj term is removed is similar
(except when the models in C-paths “B” and “C” and C-paths “C” and “D”
are compared).
For IV , paths with dv = 4:
1. IV decreases as q or c decreases (except for C-path “C”) However, the decrease
in IV when an xi xj term is removed is smaller.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. When n0 = 1, the change in IV decreases as q decreases. When n0 = 3, the
change in IV decreases as q decreases from 4 → 3 to 3 → 2 to 2 → 1, and then
220
slightly increases. When two models differ by one xi xj term, the change in IV
when an x2i term is removed is similar.
6. The change in IV is constant as c decreases from 6 → 5 to 5 → 4, and then
increases as c decreases from 5 → 4 to 4 → 3, and then decreases as c decreases
from 4 → 3 to 3 → 2 to 2 → 1, and then increases. When two models differ by
one x2i term, the change in IV when an xi xj term is removed is similar.
The Hybrid 416B Designs (416Bs)
The 416B designs with n0 = 1, 3 center points are examined for K = 4 design
variables. For a summary of the number of Q-paths and C-paths that increase (“↑”)
or decrease (“↓”) or indicate no change (“=”), see Table 45. For 416B designs, plots
of the D, A, G, and IV criteria and plots of the change in the D, A, G, and IV
criteria are given in Appendix C. The results based on D, A, G, and IV criteria for
the 4-factor 416B designs are summarized as follows:
For D, paths with dv = 4:
1. When n0 = 1, D increases as q decreases from 4 to 3. For all other cases,
removal of an x2i term can either increase or decrease D.
2. Removal of an xi xj term tends to increase D.
3. D tends to be lower when center points are replicated.
221
Table 45. The Optimality Criteria Across the Reduced Models for the 416B (K = 4).
Q
Criterion n0
1
dv
4
3
2
1
4→3
↑
3→2
↑
↑
2→1
↑(39) ↓(4)
↑
↑
C
1→0
↑
↑
↑
↑
dv
6→5
5→4
4→3
4
↑
↑
↑
4→3
↑
↑
↑
4→3→2
↑
↑
4→3→2→1
3
D
3
4 ↑(2) ↓(9) ↑(7) ↓(22) ↑(8) ↓(35) ↑(25) ↓(3)
4
↑
↑
↑
3
↑
↑
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4
↑
↑
↑
↑
4
↑(2) ↓(3) ↑(3) ↓(7) ↑(10) ↓(9)
3
↑
↑
↑
4→3
↑(2) ↓(2) ↑(2) ↓(2) ↑(2) ↓(2)
2
↑
↑
4→3→2
↑(2) ↓(1) ↑(2) ↓(1)
1
↑
4→3→2→1
3
A
3
4
↑
↑
↑
↑
4
↑(4) ↓(1) ↑(9) ↓(1) ↑(17) ↓(2)
3
↑
↑
↑
4→3
↑
↑
↑
2
↑
↑
4→3→2
↑
↑
1
↑
4→3→2→1
3
1
4
↓
↑(3) ↓(26) ↑(1) ↓(42)
↑
4
↓
↓
↑(2) ↓(17)
3
↓
↓
↑
4→3
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
G
3
4
↓
↑(3) ↓(26) ↑(1) ↓(42)
↑
4
↓
↓
↓
3
↓
↓
↑
4→3
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
1
↑
4→3→2→1
3
1
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
IV
3
4
↓
↓
↓
↓
4
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
1
↓
4→3→2→1
3
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
3→2
↑
↑(6) ↓(2)
↓
↓
↑
↑
↑(6) ↓(2)
↓
↓
↑
↑(7) ↓(11)
↑(4) ↓(4)
↓
↓
↑
↑(15) ↓(3)
↑(6) ↓(2)
↓
↓
↑
↑(1) ↓(17)
↓
↓
↓
↓
↑(1) ↓(17)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↑(1) ↓(7)
↓
↓
↓
2→1
↑
↑(2) ↓(6)
↓
↓
↑
↑
↑(2) ↓(6)
↓
↓
↑
↑(3) ↓(8)
↑(2) ↓(6)
↓
↓
↑
↑(9) ↓(2)
↑(2) ↓(6)
↓
↓
↑
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↑(2) ↓(1)
↓
↓
1→0
↑
↑
↑
↑(1) ↓(1)
↑
↑
↑
↑(1) ↓(1)
↑(1) ↓(3)
↑
↑
↑(1) ↓(1)
↑(3) ↓(1)
↑
↑
↑(1) ↓(1)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↓
↑(1) ↓(3)
↑(1) ↓(2)
↑(1) ↓(1)
↓
↓
↓
↓
↓
↓
↓
↓
4. There is slightly less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D decreases as q decreases from 4 → 3 to 3 → 2 to 2 → 1, and
then increases. When two models differ by one xi xj term, the change in D when
an x2i term is removed is similar.
222
7. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D is slightly larger for the model having one less x2i term
when c decreases from 1 to 0. Otherwise, the change in D is similar.
For A, paths with dv = 4:
1. Removal of an x2i term increases A.
2. Removal of an xi xj term can either increase or decrease A.
3. There is less variability within a Q-path when center points are replicated.
4. There is slightly less variability within a C-path when center points are replicated.
5. When n0 = 1, the change in A decreases as q decreases from 4 → 3 to 3 → 2
to 2 → 1, and then increases. When n0 = 3, the change in A increases as q
decreases. When two models differ by one xi xj term and n0 = 1, the change in
A is slightly larger for the model having one less xi xj term. When n0 = 3, the
change in A is slightly larger for the model having one less xi xj term when q
decreases from 1 to 0. Otherwise, the change in A is similar.
6. When n0 = 1, the change in A slightly decreases as c decreases from 6 → 5
to 5 → 4, and then increases as c decreases from 5 → 4 to 4 → 3, and then
decreases. However, when n0 = 3, the change in A increases as c decreases from
6 → 5 to 5 → 4 to 4 → 3 to 3 → 2, and then slightly decreases. When two
223
models differ by one x2i term, the change in A when an xi xj term is removed is
similar.
For G, paths with dv = 4:
1. For most Q-paths, G tends to decrease as q decreases from 4 to 3 to 2 to 1.
When q decreases from 1 to 0, G tends to increase.
2. For most C-paths, G tends to decrease as c decreases.
3. G tends to be lower when center points are replicated.
4. There is slightly less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in G is fairly constant as q decreases from 4 → 3 to 3 → 2 to
2 → 1, and then increases. When two models differ by one xi xj term, there is
no pattern to the change in G when q decreases from 1 to 0. Otherwise, the
change in G is similar.
7. The mean change in G decreases as c decreases from 6 → 5 to 5 → 4, and then
increases as c decreases from 5 → 4 to 4 → 3 to 3 → 2, and then decreases as
c decreases from 3 → 2 to 2 → 1, and then looks fairly constant. When two
models differ by one x2i term, the change in G when an xi xj term is removed is
similar.
224
For IV , paths with dv = 4:
1. IV decreases as q or c decreases. However, the decrease in IV when an xi xj
term is removed is smaller.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. When n0 = 1, the change in IV decreases as q decreases. When n0 = 3, the
change in IV decreases slightly as q decreases from 4 → 3 to 3 → 2 to 2 → 1,
and then increases. When two models differ by one xi xj term, the change in
IV when an x2i term is removed is similar.
6. The change in IV is constant as c decreases from 6 → 5 to 5 → 4, and then
increases as c decreases from 5 → 4 to 4 → 3, and then decreases as c decreases
from 4 → 3 to 3 → 2 to 2 → 1, and then increases. When two models differ by
one x2i term, the change in IV when an xi xj term is removed is similar.
The Hybrid 416C Designs (416Cs)
The 416C designs with n0 = 1, 2 center points are examined for K = 4 design
variables. For a summary of the number of Q-paths and C-paths that increase (“↑”)
or decrease (“↓”) or indicate no change (“=”), see Table 46. For 416C designs, plots
of the D, A, G, and IV criteria and plots of the change in the D, A, G, and IV
225
criteria are given in Appendix C. The results based on D, A, G, and IV criteria for
the 4-factor 416C designs are summarized as follows:
Table 46. The Optimality Criteria Across the Reduced Models for the 416C (K = 4).
Q
Criterion
n0
1
dv
4
3
2
1
4→3 3→2 2→1
↑
↑
↑
↑
↑
↑
C
1→0
↑
↑
↑
↑
dv
6→5
5→4
4→3
3→2
2→1
1→0
4
↑
↑
↑
↑
↑
↑
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
4→3→2
↑
↑
↓
↓
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
D
2
4
↑
↑
↑
↑(25) ↓(3)
4
↑
↑
↑
↑
↑
↑
3
↑
↑
↑
4→3
↑
↑
↑
↑(6) ↓(2) ↑(2) ↓(6)
↑
2
↑
↑
4→3→2
↑
↑
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
1
4
↑
↑
↑
↑
4
↑(1) ↓(4) ↑(2) ↓(8) ↑(6) ↓(13) ↑(6) ↓(12) ↑(4) ↓(7) ↑(1) ↓(3)
3
↑
↑
↑
4→3
↑(1) ↓(3) ↑(2) ↓(2) ↑(2) ↓(2)
↑(4) ↓(4) ↑(2) ↓(6) ↑(3) ↓(1)
2
↑
↑
4→3→2
↑(2) ↓(1) ↑(1) ↓(2)
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑(2) ↓(1)
↑
A
2
4
↑
↑
↑
↑
4
↑(2) ↓(3) ↑(4) ↓(6) ↑(7) ↓(12) ↑(9) ↓(9) ↑(6) ↓(5) ↑(2) ↓(2)
3
↑
↑
↑
4→3
↑(2) ↓(2) ↑(2) ↓(2) ↑(2) ↓(2)
↑(4) ↓(4) ↑(2) ↓(6) ↑(3) ↓(1)
2
↑
↑
4→3→2
↑(2) ↓(1) ↑(1) ↓(2)
↓
↓
↑
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑
↑
1
4
↓
↓
↓
↑
4
↓
↑(1) ↓(9)
↓
↓
↓
↓
3
↓
↓
↑
4→3
↓
↓
↑(1) ↓(3)
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
↓
↓
↑(1) ↓(2)
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑(1) ↓(2)
↓
G
2
4
↓
↓
↓
↑
4
↓
↑(1) ↓(9)
↓
↓
↓
↓
3
↓
↓
↑
4→3
↓
↓
↑(1) ↓(3)
↓
↓
↑(1) ↓(3)
2
↓
↑
4→3→2
↓
↓
↓
↓
↑(1) ↓(2)
1
↑
4→3→2→1
↓
↓
↑(1) ↓(1)
3
↑(1) ↓(2)
↓
1
4
↓
↓
↓
↓
4
↓
↓
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
↓
↓
1
↓
4→3→2→1
↓
↓
↓
3
↓
↓
IV
2
4
↓
↓
↓
↓
4
↓
↓
↓
↓
↓
↓
3
↓
↓
↓
4→3
↓
↓
↓
↓
↓
↓
2
↓
↓
4→3→2
↓
↓
↓
↓
↓
1
↓
4→3→2→1
↓
↓
↓
3
↓
↓
Notation: ’↑’ indicates all Q or C-path criterion values increase,
’↓’ indicates all Q or C-path criterion values decrease,
’=’ indicates all Q or C-path criterion values do not change,
’↑(#)’ indicates the number of Q or C-paths with criterion values that increase,
’↓(#)’ indicates the number of Q or C-paths with criterion values that decrease,
’=(#)’ indicates the number of Q or C-paths with criterion values that do not change.
For D, paths with dv = 4:
1. D increases as q decreases (except for Q-paths “J”, “P”, and “U” when n0 = 3).
2. Removal of an xi xj term increase D.
3. D tends to be lower when center points are replicated.
226
4. There is less variability within a Q-path when center points are replicated.
5. The variability within a C-path is unaffected by replication of center points.
6. The change in D decreases as q decreases. When two models differ by one xi xj
term, the change in D is similar (except when the models in Q-paths “F” and
“G” are compared).
7. The change in D increases as c decreases. When two models differ by one x2i
term, the change in D is slightly larger for the model having one less x2i term
when c decreases from 1 to 0. Otherwise, the change in D is similar.
For A, paths with dv = 4:
1. Removal of an x2i term increases A.
2. Removal of an xi xj term can either increase or decrease A.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. When n0 = 1, the mean change in A decreases as q decreases. However, when
n0 = 3, the mean change in A decreases slightly as q decreases from 4 → 3 to
3 → 2 to 2 → 1, and then increases. When two models differ by one xi xj term,
the change in A when an x2i term is removed is slightly larger for the model
having one less xi xj term.
227
6. The change in A slightly increases as c decreases from 6 → 5 to 5 → 4, and
then slightly decreases as c decreases from 5 → 4 to 4 → 3, and then increases
as c decreases from 4 → 3 to 3 → 2, and then decreases. When two models
differ by one x2i term, the change in A when an xi xj term is removed is similar.
For G, paths with dv = 4:
1. G tends to decrease as q decreases from 4 to 3 to 2 to 1 and increases as q
decreases from 1 to 0.
2. G tends to decrease as c decreases (except for C-path “E”).
3. G tends to be lower when center points are replicated.
4. The variability within a Q-path or a C-path is unaffected by replication of center
points.
5. The change in G decreases slightly as q decreases from 4 → 3 to 3 → 2 to
2 → 1, and then increases. When two models differ by one xi xj term, there is
no pattern to the change in G when q decreases from 1 to 0. Otherwise, the
change in G is similar.
6. There is no pattern to the mean change in G as c decreases. When two models
differ by one x2i term, the change in G when an xi xj term is removed is similar.
For IV , paths with dv = 4:
228
1. IV decreases as q or c terms decreases. However, the decrease in IV when an
xi xj term is removed is smaller.
2. IV tends to be lower when center points are replicated.
3. There is less variability within a Q-path when center points are replicated.
4. The variability within a C-path is unaffected by replication of center points.
5. The change in IV decreases as q decreases. When two models differ by one xi xj
term, the change in IV when an x2i term is removed is similar.
6. There is no pattern to the change in IV as c decreases. When two models differ
by one x2i term, the change in IV when an xi xj term is removed is similar.
229
General Results for the Reduced Models
To study the robustness of 3 and 4 factor spherical response surface designs across
the set of reduced models, the 7 response surface designs for 3 design variables: CCD,
BBD, SCD, UNFSD, hybrid 310, 311A, and 311B designs and 8 response surface
designs for 4 design variables: CCD, BBD, SCD, PBCD, UNFSD, hybrid 416A,
416B, and 416C designs are considered. Summaries based on computed values for the
four criteria (D, A, G, and IV ) for the set of reduced models will now be presented.
Removing an x2i term from a model:
1. For 3 design variables with dv = 3:
(a) For D: D tends to increase for BBDs and hybrid 310 designs. For other
designs, the effects on D can vary.
(b) For A: A tends to increase for BBDs, UNFSDs, hybrid 310, and 311A
designs while the effects on A vary for the other designs.
(c) For G: removing an x2i term has varying effects on G.
(d) For IV : removing an x2i term improves the IV criterion.
2. For 4 design variables with dv = 4:
(a) For D: D tends to decrease for CCDs and BBDs, increase for the hybrid
416C designs, while the effects on D vary for the other designs.
230
(b) For A: A tends to increase for CCDs, BBDs, UNFSDs, and the hybrid
416A, 416B, and 416C designs. The effects on A vary for SCDs and
PBCDs.
(c) For G: G tends to decrease for SCDs and PBCDs. The effects on G vary
for the other designs.
(d) For IV : The IV criterion improves for all designs.
Removing an xi xj term from a model:
1. For 3 design variables with dv = 3:
(a) For D: D tends to increase except for the hybrid 310 designs.
(b) For A: A tends to increase for SCDs, UNFSDs, and the hybrid 311B
designs. The effects on A vary for the other designs.
(c) For G: The effects on G vary for all designs.
(d) For IV : The IV criterion improves for all designs.
2. For 4 design variables with dv = 4:
(a) For D: D tends to increase for all designs.
(b) For A: A tends to increase for PBCDs and UNFSDs. The effects on A
vary for the other designs.
(c) For G: G tends to decrease for BBDs and the hybrid 416C designs. The
effects on G vary for the other designs.
231
(d) For IV : The IV criterion improves for all designs.
In addition, for K = 3 design variables, of the 44 reduced models considered,
there are 34 models with dv = 3 and 10 models with dv = 1 or 2. For K = 4 design
variables, of the 224 models considered, there are 170 models with dv = 4 and 54
models with dv = 1, 2 or 3. Tables 47 and 48 indicate the following results when
K = 3 and K = 4, respectively.
1. For D: D-efficiency for the full second-order model tends to be smaller relative
to the set of reduced models when dv = K and larger when dv < K.
2. For A: A-efficiency for the full second-order model tends to be smaller relative
to the set of reduced models when dv = K and larger when dv < K. (For K = 3,
exceptions include the CCD, BBD, UNFSD, and 311B design having n0 = 1,
and the SCD having rs = 1. For K = 4, exceptions include the CCD, BBD,
UNFSD, PBCD, 416A, and 416C designs having n0 = 1, the PBCD having
rs = 2, n0 = 3, and all SCDs).
3. For G: For K = 3, G-efficiency for the full second-order model tends to be
larger relative to the set of reduced models when dv = 3 (except for the n0 = 1
CCD and all SCDs). When K = 4 and dv = 4, there is no consistent increasing
or decreasing pattern to G-efficiencies across the designs considered. For K = 3
or 4, G-efficiency also tends to be larger when dv < K (except for all SCDs).
232
4. For IV : The IV -criterion for the full second-order model will always be the
largest relative to the set of reduced models (as indicated in the IV criterion
plots). Thus, the IV criterion for the full second-order model is the worst
relative to the set of reduced models.
Table 47. The Number of Models the D, A, and G-Criteria Values are Greater Than
(for dv = 3), or Smaller Than (for dv = 1, 2) the Full Second-Order Model Criteria
Values when K = 3.
Design
CCD
BBD
SCD
UNFSD
310
311A
311B
rs
1
2
1
2
1
2
1
2
-
n0
1
1
3
3
1
3
1
1
3
3
1
3
0
1
3
1
3
1
3
dv = 3
dv = 1, 2
(maximum = 33)
(maximum = 10)
D-Eff
33
28
24
23
33
33
26
26
24
23
30
22
32
31
32
30
23
28
22
A-Eff
30
30
33
33
30
30
33
33
30
31
30
33
30
27
26
30
33
30
33
G-Eff
15
28
0
3
4
4
23
19
23
19
1
1
8
8
8
1
1
1
1
D-Eff
9
8
10
9
9
10
7
7
7
7
9
10
9
9
9
9
10
9
10
A-Eff
1
1
9
6
4
9
1
10
4
6
2
8
6
9
9
5
9
2
9
G-Eff
9
4
10
10
10
10
0
3
0
3
10
10
8
8
8
10
10
10
10
233
Table 48. The Number of Models the D, A, and G-Criteria Values are Greater Than
(for dv = 4), or Smaller Than (for dv = 1, 2, and 3) the Full Second-Order Model
Criteria Values when K = 4 .
Design
CCD
BBD
SCD
PBCD
UNFSD
416A
416B
416C
rs
1
2
1
2
1
2
1
2
1
2
1
2
-
n0
1
1
3
3
1
3
1
1
3
3
1
1
3
3
1
3
1
3
1
3
1
2
dv = 4
dv = 1, 2, 3
(maximum = 169)
(maximum = 54)
D-Eff
164
142
130
111
164
130
146
130
123
117
152
137
124
118
134
101
150
113
163
139
169
163
A-Eff
159
159
159
159
159
159
167
167
167
168
159
159
169
169
159
169
159
169
159
159
159
159
G-Eff
98
156
0
1
98
0
134
113
138
114
99
100
101
104
1
1
1
1
5
5
3
4
D-Eff
53
50
54
50
53
54
44
42
49
46
51
46
51
50
51
53
53
53
54
54
53
54
A-Eff
6
1
46
28
6
46
7
4
13
22
6
2
30
21
7
45
19
53
45
53
20
48
G-Eff
50
30
54
54
50
54
2
17
2
16
34
31
34
30
53
53
54
54
53
53
54
54
D-Efficiencies Greater Than 100%
It was previously mentioned that there are cases for which D > 100%. To see how
this can happen, we need to examine the following definition of D-efficiency given by
Mitchell [41],
234
D − efficiency
=
100
|X0 X|1/p
,
N
where X is the expanded design matrix, p is the number of model parameters, and
N is the design size.
For a hypercube design region X , −1 ≤ xi ≤ 1, where xi is the coded level of the
ith design variable. Therefore, max |X0 X| ≤ N p for any design X having all design
points in X . Thus, the D-efficiency value for any design in the hypercube is less than
or equal to 100%.
However, for a spherical design region X , the coded levels of the K design variables
K
√
√
P
x1 , x2 , . . . , xK in the model satisfy − K ≤ xi ≤ K subject to
x2i ≤ K. Hence,
i=1
it is possible that max |X0 X| > N p for certain designs for a given model. Therefore,
the D-efficiency value can be greater than 100% for a spherical design region.
Comparison of Design Optimality Criteria of Reduced Models
In this section, the results of research related to the comparison of design optimality criteria based on D, A, G, and IV criteria of the spherical response surface
designs for the set of reduced models for 3 and 4 design variables will be presented.
For the set of reduced models for 3 and 4 factor spherical response surface designs,
three comparisons are performed: (i) across the full set of 44 reduced models for K = 3
and 224 reduced models for K = 4, (ii) across the set of 32 reduced models for K = 3
235
and 181 reduced models for K = 4 having at least one squared term (i.e., q ≥ 1),
and (iii) across the set of 15 reduced models for K = 3 and 109 reduced models for
K = 4 having at least two squared terms (q ≥ 2).
For each pairwise design comparison, the percentage of models for which Design 1
is superior to Design 2 is determined for the four criteria (D, A, G, and IV criteria).
Superior implies the criterion difference is greater than or equal to zero for D, A,
and G and is less than or equal to zero for the IV criteria. Ties will be considered
superior for both designs. Thus, when two designs’ criteria are equal, each design is
considered superior to the other. Therefore, the sum of percentages for Design 1 vs
Design 2 and for Design 2 vs Design 1 may exceed 100%. These percentages for 3
and 4 factor designs are given in Tables 49 to 51, 53 to 55, 57 to 59, 61 to 63, 65 to
67, 69 to 74, 76 to 78, and 80 to 82.
In addition, these percentages are then ranked for each of the four criteria for
designs of equal size N . The ranks based on the corresponding percentage design
comparisons are given in Tables 52, 56, 60, 64, 68, 75, 79, and 83. The summaries in
Tables 49 to 83 are analogous to the comparisons of response surface designs in the
hypercube given in Borkowski and Valeroso [11, 12].
For the comparison ranking tables, each row/column entry contains 3 ranks
(r0 , r1 , r2 ). Each rank ranges from 1 (’best’) to the number of designs to be compared (’worst’). Ranks r0 , r1 , and r2 represent a design’s rank relative to the other
designs across the full set of reduced models, across the set of reduced models with
236
q ≥ 1, and across the set of reduced models with q ≥ 2, respectively. In case of ties,
average ranks are shown. In case the order of the ranks across designs and criteria is
uncertain, comparison ranking and comparison plots for the D, A, G, or IV criteria
will be jointly considered. Each comparison plot contains a reference line indicating
when optimality criterion values are equal. The plotting symbol is the number of
squared terms (q) in model. The “?” notation indicates lack of transitivity which
means that Design A is superior to Design B and Design B is superior to Design C
but it does not imply that Design A is superior to Design C. In this case, the “?”
notation is used to indicate an indeterminate ranking.
Based on the results in Tables 49 to 52, the 311B design is recommended over
any of the other three 11-point designs because it is the superior design for the D, A,
G, and IV criteria. The 310 design performs very poorly for the D and A criteria.
Note the lack of transitivity in Table 52 for the SCD, 310, and 311A designs for the
IV criterion across models having q ≥ 2 (e.g., when models having q ≥ 2 and the
IV -criterion are considered, although the 311A design is superior to the 310 design
for 73.3% of all models, and the 310 design is superior to the SCD for 53.3% of all
models, it does not imply that the 311A is superior to the SCD because Table 51
indicates the SCD is superior to the 311A design for 53.3% of all models).
237
Table 49. Comparisons of D, A, G, and IV Criteria for K = 3, N = 11.
DESIGN 1
SCD
(rs = 1, n0 = 1)
310
(n0 = 1)
311A
(n0 = 1)
311B
(n0 = 1)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
36.4
56.8
45.5
50.0
56.8
70.5
70.5
61.4
84.1
86.4
88.6
84.1
DESIGN 2
310
311A
63.6
43.2
43.2
29.6
54.6
29.6
50.0
38.6
–
2.3
–
15.9
–
2.3
–
9.1
97.7
–
84.1
–
97.7
–
90.9
–
88.6
79.6
72.7
79.6
79.6
79.6
86.4
81.8
311B
15.9
13.6
13.6
18.2
11.4
27.3
20.5
13.6
20.5
20.5
20.5
18.2
–
–
–
–
Note: Values are % of the 44 models for which DESIGN 1 is superior to DESIGN 2.
Table 50. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 3, N = 11.
DESIGN 1
SCD
(rs = 1, n0 = 1)
310
(n0 = 1)
311A
(n0 = 1)
311B
(n0 = 1)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
21.9
50.0
34.4
40.6
50.0
68.8
68.8
56.3
78.1
81.3
84.4
78.1
DESIGN 2
310
311A
78.1
50.0
50.0
31.3
65.6
31.3
59.4
43.8
–
0
–
12.5
–
0
–
12.5
100.0
–
87.5
–
100.0
–
87.5
–
100.0
100.0
87.5
100.0
100.0
100.0
87.5
100.0
311B
21.9
18.8
18.8
21.9
0
12.5
0
12.5
0
0
0
0
–
–
–
–
Note: Values are % of the 32 models (q ≥ 1) for which DESIGN 1 is superior to DESIGN 2.
238
Table 51. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 3, N = 11.
DESIGN 1
SCD
(rs = 1, n0 = 1)
310
(n0 = 1)
311A
(n0 = 1)
311B
(n0 = 1)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
DESIGN 2
310
311A
80.0
53.3
40.0
26.7
60.0
26.7
46.7
53.3
–
0
–
26.7
–
0
–
26.7
100.0
–
73.3
–
100.0
–
73.3
–
100.0
100.0
73.3
100.0
100.0
100.0
73.3
100.0
SCD
–
–
–
–
20.0
60.0
40.0
53.3
46.7
73.3
73.3
46.7
73.3
80.0
86.7
73.3
311B
26.7
20.0
20.0
26.7
0
26.7
0
26.7
0
0
0
0
–
–
–
–
Note: Values are % of the 15 models (q ≥ 2) for which DESIGN 1 is superior to DESIGN 2.
Table 52. Design Criteria Comparison Ranking for K = 3, N = 11.
Design Criterion
D
A
G
IV
SCD
3, 2.5, 2
4, 3.5, 4
3, 3, 3
3.5, 3, ?
DESIGN
310
311A
4, 4, 4
2, 2.5, 3
3, 3.5, 3
2, 2, 2
4, 4, 4
2, 2, 2
3.5, 4, ?
2, 2, ?
311B
1, 1, 1
1, 1, 1
1, 1, 1
1, 1, 1
For r0 , r1 , r2 : r0 = Rank across 44 models,
r1 = Rank across 32 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 15 models with at least 2 squared terms (q ≥ 2),
? indicates lack of transitivity.
Based on the results in Tables 53 to 56, the UNFSD is the superior design for D,
and it also fares very well based on the A, G, and IV criteria (for r0 and r1 ). Overall,
239
the 311B design is the superior design for the G and IV and it fares very well based
on the A and D criteria. The BBD performs very well for only the D criterion. The
SCD and 310 design, however, perform very poorly across all 4 criteria. Thus, when
running a 13-point design, it is recommended that the experimenter should choose
either the UNFSD or the 311B design depending on the criterion. The comparison
plots in Figure 64 correspond to the ranking results in Tables 53 to 56.
Table 53. Comparisons of D, A, G, and IV Criteria for K = 3, N = 13.
DESIGN 1
SCD
(rs = 1, n0 = 3)
310
(n0 = 3)
311A
(n0 = 3)
311B
(n0 = 3)
BBD
(n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
34.1
70.5
45.5
52.3
63.6
72.7
70.5
70.5
84.1
86.4
86.4
70.5
81.8
77.3
81.8
54.6
93.2
79.6
97.7
70.5
310
65.9
29.6
54.6
47.7
–
–
–
–
97.7
93.2
97.3
100.0
88.6
81.8
79.6
100.0
100.0
86.4
100.0
68.2
100.0
90.9
95.5
84.1
DESIGN 2
311A
311B
36.4
15.9
27.3
13.6
29.6
13.6
29.6
29.6
2.3
11.4
6.8
18.2
2.3
20.5
0
0
–
20.5
–
20.5
–
20.5
–
77.3
79.6
–
79.6
–
79.6
–
97.7
–
93.2
72.7
61.4
54.6
97.7
31.8
43.2
40.9
100.0
95.5
75.0
63.6
95.5
61.4
68.2
63.6
BBD
18.2
22.7
18.2
45.5
0
13.6
0
31.8
6.8
38.6
2.3
56.8
27.3
45.5
68.2
59.1
–
–
–
–
59.1
65.9
68.2
79.6
UNFSD
6.8
20.5
2.3
29.6
0
9.1
4.6
15.9
0
25.0
4.6
31.8
4.6
36.4
38.6
36.4
40.9
34.1
31.8
25.0
–
–
–
–
Note: Values are % of the 44 models for which DESIGN 1 is superior to DESIGN 2.
240
Table 54. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 3, N = 13.
DESIGN 1
SCD
(rs = 1, n0 = 3)
310
(n0 = 3)
311A
(n0 = 3)
311B
(n0 = 3)
BBD
(n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
18.8
68.8
34.4
43.8
59.4
71.9
68.8
68.8
78.1
81.3
81.3
68.8
75.0
68.8
75.0
37.5
90.6
71.9
96.9
59.4
310
81.3
31.3
65.6
56.3
–
–
–
–
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
81.3
100.0
56.3
100.0
87.5
100.0
78.1
DESIGN 2
311A
311B
40.6
21.9
28.1
18.8
31.3
18.8
31.3
31.3
0
0
0
0
0
0
0
0
–
0
–
0
–
0
–
78.1
100.0
–
100.0
–
100.0
–
96.9
–
90.6
62.5
46.9
37.5
96.9
6.3
21.9
18.8
100.0
93.8
65.6
50.0
100.0
46.9
56.3
50.0
BBD
25.0
31.3
25.0
62.5
0
18.8
0
43.8
9.4
53.1
3.1
78.1
37.5
62.5
93.8
81.3
–
–
–
–
71.9
81.3
84.4
100.0
UNFSD
9.4
28.1
3.1
40.6
0
12.5
0
21.9
0
34.4
0
43.8
6.3
50.0
53.1
50.0
28.1
18.8
15.6
3.1
–
–
–
–
Note: Values are % of the 32 models (q ≥ 1) for which DESIGN 1 is superior to DESIGN 2.
241
Table 55. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 3, N = 13.
DESIGN 1
SCD
(rs = 1, n0 = 3)
310
(n0 = 3)
311A
(n0 = 3)
311B
(n0 = 3)
BBD
(n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
13.3
73.3
40.0
46.7
46.7
73.3
73.3
73.3
73.3
80.0
80.0
73.3
66.7
53.3
80.0
6.7
86.7
60.0
93.3
26.7
310
86.7
26.7
60.0
53.3
–
–
–
–
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
60.0
100.0
6.7
100.0
73.3
100.0
53.3
DESIGN 2
311A
311B
53.3
26.7
26.7
20.0
26.7
20.0
26.7
26.7
0
0
0
0
0
0
0
0
–
0
–
0
–
0
–
86.7
100.0
–
100.0
–
100.0
–
100.0
–
80.0
40.0
0
0
100.0
6.7
0
0
100.0
86.7
26.7
0
100.0
13.3
6.7
6.7
BBD
33.3
46.7
20.0
93.3
0
40.0
0
93.3
20.0
100.0
0
100.0
60.0
100.0
93.3
100.0
–
–
–
–
86.7
86.7
73.3
100.0
UNFSD
13.3
40.0
6.7
73.3
0
26.7
0
46.7
0
73.3
0
93.3
13.3
100.0
86.7
93.3
13.3
13.3
26.7
6.7
–
–
–
–
Note: Values are % of the 15 models (q ≥ 2) for which DESIGN 1 is superior to DESIGN 2.
Table 56. Design Criteria Comparison Ranking for K = 3, N = 13.
Design Criterion
D
A
G
IV
SCD
5, 5, 4
6, 6, 6
5, 5, 5
6, 4, 3
310
6, 6, 6
5, 5, 5
6, 6, 6
5, 6, 5
DESIGN
311A
311B
4, 4, 5
3, 3, 2
4, 3, 2
3, 1.5, 1
4, 4, 4
2, 1, 1
3, 3, 2
2, 1.5, 1
For r0 , r1 , r2 : r0 = Rank across 44 models,
r1 = Rank across 32 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 15 models with at least 2 squared terms (q ≥ 2).
BBD
2, 2, 3
2, 4, 4
3, 3, 3
4, 5, 6
UNFSD
1, 1, 1
1, 1.5, 3
1, 2, 2
1, 1.5, 4
242
Figure 64. The D, A, G, and IV -Criteria Comparison Plots for 13-Point 3-Factor
Designs (Plotting Symbol = q).
243
Figure 64. cont’d
Based on the results in Tables 57 to 60, the CCD is the overall superior design
across all 4 criteria. Thus, the CCD is recommended over the other two 15-point
designs. Note that the UNFSD improves as q increases (i.e., r0 ≥ r1 ≥ r2 ). The
comparison plots in Figure 65 correspond to the ranking results in Table 60.
244
Table 57. Comparisons of D, A, G, and IV Criteria for K = 3, N ≥ 13.
DESIGN 1
CCD
(N = 15)
CCD
(N = 17)
BBD
(N = 13)
BBD
(N = 15)
UNFSD
(N = 13)
UNFSD
(N = 15)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
DESIGN 2
CCD
CCD
BBD
BBD
UNFSD UNFSD
(N = 15) (N = 17) (N = 13) (N = 15) (N = 13) (N = 15)
–
–
63.6
95.5
–
97.7
–
–
70.5
72.7
–
79.6
–
–
70.5
88.6
–
90.9
–
–
84.1
70.5
–
65.9
–
–
29.6
68.2
–
97.7
–
–
59.1
70.5
–
88.6
–
–
72.7
79.6
–
100.0
–
–
79.6
75.0
–
90.9
36.4
70.5
–
–
40.9
–
29.6
40.9
–
–
34.1
–
29.6
27.3
–
–
31.8
–
15.9
20.5
–
–
25.0
–
4.6
31.8
–
–
–
56.8
27.3
29.6
–
–
–
69.5
11.4
20.5
–
–
–
34.1
29.6
25.0
–
–
–
52.3
–
–
59.1
–
–
–
–
–
65.9
–
–
–
–
–
68.2
–
–
–
–
–
79.6
–
–
–
2.3
2.3
–
43.2
–
–
20.5
11.4
–
34.1
–
–
9.1
0
–
65.9
–
–
34.1
9.1
–
50.0
–
–
Note: Values are % of the 44 models for which DESIGN 1 is superior to DESIGN 2.
245
Table 58. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 3, N ≥ 13.
DESIGN 1
CCD
(N = 15)
CCD
(N = 17)
BBD
(N = 13)
BBD
(N = 15)
UNFSD
(N = 13)
UNFSD
(N = 15)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
CCD
(N = 15)
–
–
–
–
–
–
–
–
21.9
12.5
12.5
12.5
0
21.9
0
40.6
–
–
–
–
3.1
28.1
12.5
46.9
CCD
(N = 17)
–
–
–
–
–
–
–
–
59.4
18.8
0
0
15.6
12.5
0
9.4
–
–
–
–
3.1
15.6
0
12.5
DESIGN 2
BBD
BBD
(N = 13) (N = 15)
78.1
100.0
87.5
78.1
87.5
100.0
87.5
59.4
40.6
84.4
81.3
87.5
100.0
100.0
100.0
90.6
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
71.9
–
81.3
–
84.4
–
100.0
–
–
50.0
–
37.5
–
81.3
–
59.4
UNFSD
(N = 13)
–
–
–
–
–
–
–
–
28.1
18.8
15.6
3.1
–
–
–
–
–
–
–
–
–
–
–
–
UNFSD
(N = 15)
96.9
71.9
87.5
53.1
96.9
84.4
100.0
87.5
–
–
–
–
50.0
62.5
18.8
40.6
–
–
–
–
–
–
–
–
Note: Values are % of the 32 models (q ≥ 1) for which DESIGN 1 is superior to DESIGN 2.
246
Table 59. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 3, N ≥ 13.
DESIGN 1
CCD
(N = 15)
CCD
(N = 17)
BBD
(N = 13)
BBD
(N = 15)
UNFSD
(N = 13)
UNFSD
(N = 15)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
CCD
(N = 15)
–
–
–
–
–
–
–
–
0
26.7
26.7
26.7
0
46.7
0
86.7
–
–
–
–
6.7
60.0
26.7
100.0
CCD
(N = 17)
–
–
–
–
–
–
–
–
33.3
0
0
0
0
13.3
0
20.0
–
–
–
–
6.7
26.7
0
26.7
DESIGN 2
BBD
BBD
(N = 13) (N = 15)
100.0
100.0
73.3
53.3
73.3
100.0
73.3
13.3
66.7
100.0
100.0
86.7
100.0
100.0
100.0
80.0
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
86.7
–
86.7
–
73.3
–
100.0
–
–
80.0
–
53.3
–
66.7
–
73.3
UNFSD
(N = 13)
–
–
–
–
–
–
–
–
13.3
13.3
26.7
6.7
–
–
–
–
–
–
–
–
–
–
–
–
UNFSD
(N = 15)
93.3
40.0
73.3
0
93.3
73.3
100.0
73.3
–
–
–
–
20.0
46.7
33.3
26.7
–
–
–
–
–
–
–
–
Note: Values are % of the 15 models (q ≥ 2) for which DESIGN 1 is superior to DESIGN 2.
Table 60. Design Criteria Comparison Ranking for K = 3, N = 15.
Design Criterion
D
A
G
IV
CCD
1, 1, 1
1, 1, 2
1, 1, 1
1, 1, 2
DESIGN
BBD
2, 2.5, 3
2, 2, 3
3, 3, 3
3, 3, 3
UNFSD
3, 2.5, 2
3, 3, 1
2, 2, 2
2, 2, 1
For r0 , r1 , r2 : r0 = Rank across 44 models,
r1 = Rank across 32 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 15 models with at least 2 squared terms (q ≥ 2).
247
Figure 65. The A and IV -Criteria Comparison Plots for 15-Point 3-Factor Designs
(Plotting Symbol = q).
248
Based on the results in Tables 61 to 64, the 416C design is the superior design for
the A, G, IV , and D criteria, and therefore, it is the recommended 17-point design.
The 416B design performs very well based on the A and IV criteria. The SCD
performs very well for the D criterion (except across the full set of reduced models)
but performs very poorly based on the A and G criteria. The 416A fares very well
based on the G criteria but performs very poorly for the D and IV criteria. Although
Tables 61 to 63 indicate that the 416C design is superior to the 416B design based on
the D criterion, the comparison plot (416C vs 416B) in Figure 66 indicates that all
differences in D criterion values are very close to zero. Thus, for all practical purposes,
if the D criterion is considered, either the 416B or 416C design is recommended.
Table 61. Comparisons of D, A, G, and IV Criteria for K = 4, N = 17.
DESIGN 1
SCD
(rs = 1, n0 = 1)
416A
(n0 = 1)
416B
(n0 = 1)
416C
(n0 = 2)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
42.0
65.6
62.5
50.0
54.0
71.9
63.4
61.6
55.8
74.1
63.8
63.8
DESIGN 2
416A
416B
58.0
46.0
34.4
28.1
37.5
36.6
50.5
38.4
–
13.0
–
5.8
–
82.6
–
9.8
87.1
–
94.2
–
17.4
–
90.2
–
84.8
67.0
94.6
77.2
73.7
80.4
92.9
81.3
416C
44.2
25.9
36.2
36.2
15.2
5.4
26.3
7.1
33.0
22.8
19.6
18.8
–
–
–
–
Note: Values are % of the 224 models for which DESIGN 1 is superior to DESIGN 2.
249
Table 62. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 4, N = 17.
DESIGN 1
SCD
(rs = 1, n0 = 1)
416A
(n0 = 1)
416B
(n0 = 1)
416C
(n0 = 2)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
32.0
59.7
57.5
41.4
45.3
67.4
57.5
55.8
48.1
70.2
57.5
58.6
DESIGN 2
416A
416B
68.0
54.7
40.3
32.6
42.5
42.5
58.6
44.2
–
13.8
–
5.0
–
100.0
–
9.9
86.2
–
95.0
–
0
–
90.1
–
83.4
76.8
95.6
89.5
69.6
93.9
93.4
93.9
416C
51.9
29.8
42.5
41.4
16.6
4.4
30.4
6.6
23.2
10.5
6.1
6.1
–
–
–
–
Note: Values are % of the 181 models (q ≥ 1) for which DESIGN 1 is superior to DESIGN 2.
250
Table 63. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 4, N = 17.
DESIGN 1
SCD
(rs = 1, n0 = 1)
416A
(n0 = 1)
416B
(n0 = 1)
416C
(n0 = 2)
D
A
G
IV
D
A
G
IV
D
A
G
IV
D
A
G
IV
DESIGN 2
416A
416B
74.3
60.6
33.0
31.2
33.9
33.9
60.6
36.7
–
18.4
–
3.7
–
100.0
–
5.5
81.7
–
96.3
–
0
–
94.5
–
75.2
72.5
97.3
89.9
56.9
97.3
96.3
97.3
SCD
–
–
–
–
25.7
67.0
66.1
39.5
39.5
68.8
66.1
63.3
37.6
69.7
66.1
67.9
416C
62.4
30.3
33.9
32.1
24.8
2.8
43.1
3.7
27.5
10.1
2.8
2.8
–
–
–
–
Note: Values are % of the 109 models (q ≥ 2) for which DESIGN 1 is superior to DESIGN 2.
Table 64. Design Criteria Comparison Ranking for K = 4, N = 17.
Design Criterion
D
A
G
IV
SCD
3, 1, 1
4, 4, 4
4, 4, 4
3, 3, 3
DESIGN
416A
416B
4, 4, 4
2, 3, 3
3, 3, 3
2, 2, 2
2, 2, 2
3, 3, 3
4, 4, 4
2, 2, 2
For r0 , r1 , r2 : r0 = Rank across 224 models,
r1 = Rank across 181 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 109 models with at least 2 squared terms (q ≥ 2).
416C
1, 2, 2
1, 1, 1
1, 1, 1
1, 1, 1
251
Figure 66. The D-Criterion Comparison Plots for 17-Point 4-Factor Designs (Plotting
Symbol = q).
252
Based on the results in Tables 65 to 68, the 416B design is the superior design for
the A and IV criteria and it also performs very well for the D and IV criteria. The
416A design is the superior design for the G criterion and it is second best based on
the A and IV criteria. The 416A design, however, performs poorly based on the D
criterion. The SCD performs very poorly for the A, G, and IV criteria but performs
very well based on the D criterion. Thus, to run a 19-point design, if the D criterion
is considered, either 416B or SCD is recommended over the 416A design. If the A or
IV criteria are considered, the 416B design is the recommended design, and if the G
criterion is considered, the 416A design is the recommended design. The comparison
plot in Figure 67 corresponds to the results in Table 68.
Table 65. Comparisons of D, A, G, and IV Criteria for K = 4, N = 19.
DESIGN 1
SCD
(rs = 1, n0 = 3)
416A
(n0 = 3)
416B
(n0 = 3)
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
46.9
67.9
62.5
62.5
56.7
77.2
63.4
64.7
DESIGN 2
416A
53.1
32.1
37.5
37.5
–
–
–
–
80.8
94.2
17.4
86.6
416B
43.3
22.8
36.6
35.3
19.2
5.8
82.6
13.4
–
–
–
–
Note: Values are % of the 224 models for which DESIGN 1 is superior to DESIGN 2.
253
Table 66. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 4, N = 19.
DESIGN 1
SCD
(rs = 1, n0 = 3)
416A
(n0 = 3)
416B
(n0 = 3)
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
38.1
62.4
57.5
57.5
48.6
74.0
57.5
59.7
DESIGN 2
416A
61.9
37.6
42.5
42.5
–
–
–
–
78.5
95.0
0
85.6
416B
51.4
26.0
42.5
40.3
21.6
5.0
100.0
14.4
–
–
–
–
Note: Values are % of the 181 models (q ≥ 1) for which DESIGN 1 is superior to DESIGN 2.
Table 67. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 4, N = 19.
DESIGN 1
SCD
(rs = 1, n0 = 3)
416A
(n0 = 3)
416B
(n0 = 3)
D
A
G
IV
D
A
G
IV
D
A
G
IV
SCD
–
–
–
–
32.1
66.1
66.1
66.1
38.5
71.6
66.1
67.9
DESIGN 2
416A
67.9
33.9
33.9
33.9
–
–
–
–
68.8
95.4
0
85.3
416B
61.5
28.4
33.9
32.1
31.2
4.6
100.0
14.7
–
–
–
–
Note: Values are % of the 109 models (q ≥ 2) for which DESIGN 1 is superior to DESIGN 2.
254
Table 68. Design Criteria Comparison Ranking for K = 4, N = 19.
Design Criterion
D
A
G
IV
SCD
2, 1, 1
3, 3, 3
3, 3, 3
3, 3, 3
DESIGN
416A
3, 3, 3
2, 2, 2
1, 1, 1
2, 2, 2
416B
1, 2, 2
1, 1, 1
2, 2, 2
1, 1, 1
For r0 , r1 , r2 : r0 = Rank across 224 models,
r1 = Rank across 181 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 109 models with at least 2 squared terms (q ≥ 2).
Figure 67. The D-Criterion Comparison Plot for 19-Point 4-Factor Designs (Plotting
Symbol = q).
255
Based on the results in Tables 69 to 75, the PBCD is the superior design for the
D, A, and G criteria while the UNFSD is the superior design for the IV criterion.
Hence, when N = 21 or 23, if the D, A, and G criteria are considered, the PBCD
is recommended over the UNFSD. However, if the IV criterion is considered, the
UNFSD is recommended.
Table 69. Comparisons of D, A, G, and IV Criteria for K = 4, N = 21.
DESIGN 1
PBCD
(rs = 1, n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
PBCD
–
–
–
–
27.7
37.5
47.3
54.5
DESIGN 2
UNFSD
73.2
63.4
58.9
47.8
–
–
–
–
Note: Values are % of the 224 models for which DESIGN 1 is superior to DESIGN 2.
Table 70. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 4, N = 21.
DESIGN 1
PBCD
(rs = 1, n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
PBCD
–
–
–
–
31.9
37.4
47.3
52.5
DESIGN 2
UNFSD
69.2
63.7
60.4
49.2
–
–
–
–
Note: Values are % of the 181 models (q ≥ 1 ) for which DESIGN 1 is superior to DESIGN 2.
256
Table 71. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 4, N = 21.
DESIGN 1
PBCD
(rs = 1, n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
PBCD
–
–
–
–
40.7
42.6
51.9
56.0
DESIGN 2
UNFSD
59.3
57.4
55.6
44.0
–
–
–
–
Note: Values are % of the 109 models (q ≥ 2 ) for which DESIGN 1 is superior to DESIGN 2.
Table 72. Comparisons of D, A, G, and IV Criteria for K = 4, N = 23.
DESIGN 1
PBCD
(rs = 1, n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
PBCD
–
–
–
–
27.7
39.3
44.6
62.1
DESIGN 2
UNFSD
73.2
61.6
58.0
41.1
–
–
–
–
Note: Values are % of the 224 models for which DESIGN 1 is superior to DESIGN 2.
Table 73. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 4, N = 23.
DESIGN 1
PBCD
(rs = 1, n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
PBCD
–
–
–
–
31.9
39.6
44.0
60.8
DESIGN 2
UNFSD
69.2
61.5
59.3
40.9
–
–
–
–
Note: Values are % of the 181 models (q ≥ 1 ) for which DESIGN 1 is superior to DESIGN 2.
257
Table 74. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 4, N = 23.
DESIGN 1
PBCD
(rs = 1, n0 = 1)
UNFSD
(n0 = 1)
D
A
G
IV
D
A
G
IV
PBCD
–
–
–
–
38.9
46.3
46.3
66.1
DESIGN 2
UNFSD
61.1
53.7
53.7
33.9
–
–
–
–
Note: Values are % of the 109 models (q ≥ 2 ) for which DESIGN 1 is superior to DESIGN 2.
Table 75. Design Criteria Comparison Ranking for K = 4, N = 21 and 23.
Design Criterion
D
A
G
IV
PBCD
1, 1, 1
1, 1, 1
1, 1, 1
2, 2, 2
DESIGN
UNFSD
2, 2, 2
2, 2, 2
2, 2, 2
1, 1, 1
For r0 , r1 , r2 : r0 = Rank across 224 models,
r1 = Rank across 181 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 109 models with at least 2 squared terms (q ≥ 2).
Based on the results in Tables 76 to 79, the CCD and the BBD are equally
superior for all 4 criteria (except on the IV criterion across models having q ≥ 2).
Hence, when N = 25, either the CCD or BBD is recommended over the SCD based
on the D, A, G and IV criteria. The comparison plot in Figure 68 corresponds to
the results in Table 79.
258
Table 76. Comparisons of D, A, G, and IV Criteria for K = 4, N = 25.
DESIGN 1
BBD (n0 = 1)
or
CCD
(rs = 1, n0 = 1)
SCD
(rs = 2, n0 = 1)
D
A
G
IV
D
A
G
IV
DESIGN 2
BBD or CCD
–
–
–
–
17.0
12.1
7.6
36.2
SCD
83.9
88.8
95.1
65.2
–
–
–
–
Note: Values are % of the 224 models for which DESIGN 1 is superior to DESIGN 2.
Table 77. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One Squared Term) for K = 4, N = 25.
DESIGN 1
BBD (n0 = 1)
or
CCD
(rs = 1, n0 = 1)
SCD
(rs = 2, n0 = 1)
D
A
G
IV
D
A
G
IV
DESIGN 2
BBD or CCD
–
–
–
–
18.8
12.7
7.2
42.5
SCD
81.2
87.3
95.0
57.5
–
–
–
–
Note: Values are % of the 181 models (q ≥ 1) for which DESIGN 1 is superior to DESIGN 2.
Table 78. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two Squared Terms) for K = 4, N = 25.
DESIGN 1
BBD (n0 = 1)
or
CCD
(rs = 1, n0 = 1
SCD
(rs = 2, n0 = 1)
D
A
G
IV
D
A
G
IV
DESIGN 2
BBD or CCD
–
–
–
–
27.5
17.4
8.3
59.6
SCD
72.5
82.6
95.4
40.4
–
–
–
–
Note: Values are % of the 109 models (q ≥ 2) for which DESIGN 1 is superior to DESIGN 2.
259
Table 79. Design Criteria Comparison Ranking for K = 4, N = 25.
Design Criterion
D
A
G
IV
DESIGN
BBD or CCD
1, 1, 1
1, 1, 1
1, 1, 1
1, 1, 2
SCD
2, 2, 2
2, 2, 2
2, 2, 2
2, 2, 1
For r0 , r1 , r2 : r0 = Rank across 224 models,
r1 = Rank across 181 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 109 models with at least 2 squared terms (q ≥ 2).
Figure 68. The IV -Criterion Comparison Plot for 25-Point 4-Factor Designs (Plotting
Symbol = q).
260
Based on the results in Tables 80 to 83, the 27-point CCD or the BBD is recommended over the SCD because they are superior designs for all 4 criteria.
Table 80. Comparisons of D, A, G, and IV Criteria for K = 4, N = 27.
DESIGN 1
BBD (n0 = 3)
or
CCD
(rs = 1, n0 = 3)
SCD
(rs = 2, n0 = 3)
D
A
G
IV
D
A
G
IV
DESIGN 2
BBD or CCD
–
–
–
–
17.0
10.3
6.3
18.3
SCD
83.9
90.6
94.6
82.2
–
–
–
–
Note: Values are % of the 224 models for which DESIGN 1 is superior to DESIGN 2.
Table 81. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
One squared Term) for K = 4, N = 27.
DESIGN 1
BBD (n0 = 3)
or
CCD
(rs = 1, n0 = 3)
SCD
(rs = 2, n0 = 3)
D
A
G
IV
D
A
G
IV
DESIGN 2
BBD or CCD
–
–
–
–
18.8
10.5
5.5
20.4
SCD
81.2
89.5
94.5
79.6
–
–
–
–
Note: Values are % of the 181 models (q ≥ 1) for which DESIGN 1 is superior to DESIGN 2.
261
Table 82. Comparisons of D, A, G, and IV Criteria (Across Models with at Least
Two squared Terms) for K = 4, N = 27.
DESIGN 1
BBD (n0 = 3)
or
CCD
(rs = 1, n0 = 3)
SCD
(rs = 2, n0 = 3)
D
A
G
IV
D
A
G
IV
DESIGN 2
BBD or CCD
–
–
–
–
27.5
13.8
5.5
26.6
SCD
72.5
86.2
94.5
73.4
–
–
–
–
Note: Values are % of the 109 models (q ≥ 2) for which DESIGN 1 is superior to DESIGN 2.
Table 83. Design Criteria Comparison Ranking for K = 4, N = 27.
Design Criterion
D
A
G
IV
DESIGN
BBD or CCD
1, 1, 1
1, 1, 1
1, 1, 1
1, 1, 1
SCD
2, 2, 2
2, 2, 2
2, 2, 2
2, 2, 2
For r0 , r1 , r2 : r0 = Rank across 224 models,
r1 = Rank across 181 models with at least 1 squared term (q ≥ 1),
r2 = Rank across 109 models with at least 2 squared terms (q ≥ 2).
The results of the research related to weighted design optimality criteria of the
response surface designs assuming a spherical design region for 3 and 4 design variables
will be presented in Chapter 5.
262
CHAPTER 5
WEIGHTED DESIGN OPTIMALITY CRITERIA FOR SPHERICAL RESPONSE
SURFACE DESIGNS
In this chapter, new types of D, A, G, and IV optimality criteria for response
surface designs in a spherical design region are developed by using prior probability assignments to model effects in a method analogous to the method adopted by
Borkowski [8]. The four new D, A, G, and IV criteria that use prior probability
assignments to the model effects will be referred to as weighted design optimality
criteria.
In this chapter of the dissertation, the terminology and notation of Chipman [18]
and the set of reduced models for weak heredity and strong heredity of Borkowski [8]
are adopted.
Inheritance Principles for Reduced Models
Because design selection based on an optimality criterion is highly dependent upon
the approximating response surface model, we will get different design optimality
criterion values for different models. In practice, this means the experimenter selects
a design that is based on a model proposed prior to data collection. When data are
collected and the model’s parameters are fitted, it is often determined that many
parameter estimates are not statistically significant. Thus, a reduced model retaining
263
only significant terms is adopted. Therefore, a robust design should be considered
over the set of potential reduced models and not over a single model.
Chipman [18] and Chipman and Hamada [19] studied classes of reduced models.
Two specific classes of reduced models are formed by removing terms based on hierarchical structures. These models are based on the following two heredity concepts.
1. Weak heredity (WH) requires that (i) if a model contains an x2i term, then it
must contain the corresponding xi term and (ii) if a model contains an xi xj
term, then it must contain either the xi or xj term (or both).
2. Strong heredity (SH) requires that (i) if a model contains an x2i term, then it
must contain the corresponding xi term and (ii) if a model contains an xi xj
term, then it must contain both of the xi and xj terms.
By definition, xi and xj are the two parents of xi xj , and xi is the one parent of
x2i . Or, equivalently, xi xj is a child of parents xi and xj and x2i is a child of parent
xi . A term T1 inherits from a term T2 if the parents of T2 are also parents of T1 .
A term T1 inherits immediately from another term T2 if T1 inherits from T2 , and
T2 is of the next lower order (Chipman [18]). For example, if an xi x2j term is in a
model, it inherits immediately from xi xj and x2j , and it inherits (but not immediately)
from xi and xj . Weak and strong inheritance possess the immediate inheritance
principle. The immediate inheritance principle is defined as the assumption that
given the importance of its parents, the importance of a child term is independent
264
of all other terms. An effect is active if, for a given model, its corresponding term is
in the model. Otherwise, it is inactive. Weak heredity is a more liberal assumption
than strong heredity because weak heredity requires only one parent of a term to be
active while strong heredity requires both parents to be active.
A model can be defined by a vector δ where each element of δ is either “1” or
“0”. The “1” indicates an active effect and the “0” indicates an inactive effect. Let
δi , δii , and δij represent the indicator function values of the ith first-order effect, the
ith second-order effect, and the ij th interaction effect, respectively. Then,
δ = (δ1 , δ2 , δ3 , δ12 , δ13 , δ23 , δ11 , δ22 , δ33 ) for K = 3, and
δ = (δ1 , δ2 , δ3 , δ4 , δ12 , δ13 , δ14 , δ23 , δ24 , δ34 , δ11 , δ22 , δ33 , δ44 ) for K = 4
will represent the corresponding δ-vectors. For example, δ = (1, 0, 1, 1, 0, 1, 0, 0, 1)
corresponds to the model y = β0 + β1 x1 + β3 x3 + β12 x1 x2 + β23 x2 x3 + β33 x23 + .
K X
K
2Ki−i(i−1)/2 reduced models of the full second-order
For WH, there are
i
i=0
model. Thus, for K = 3 and 4 design variables, there are 185 and 3905 WH models,
K X
K
respectively. For SH, there are
2i(i+1)/2 reduced models of the full secondi
i=0
order model. Thus, for K = 3 and 4 design variables, there are 95 and 1337 SH
models, respectively (Borkowski [8]).
The number of reduced models may seem impractically large especially when
K = 4. However, by exploiting the symmetry of the designs, the number of models
that actually need to be considered is much smaller and can be easily handled. For
example, X0X for the 3 factor 15-point CCD (rs = 1, n0 = 1) is:
265

X0XCCD







=







15
0
0
0
0
0
0
14
14
14
0
14
0
0
0
0
0
0
0
0
0
0
14
0
0
0
0
0
0
0
0
0
0
14
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
8
0
0
0
14
0
0
0
0
0
0
26
8
8
14
0
0
0
0
0
0
8
26
8
14
0
0
0
0
0
0
8
8
26
















Recall from Chapter 2 that any permutation of the labels x1 , x2 , and x3 of the 3 factor
CCD will yield the same X0X matrix, or, in other words, the CCD is a symmetric
design. Next, suppose WH holds and consider the reduced model
y = β0 + β1 x1 + β2 x2 + β12 x1 x2 + β13 x1 x3 + β23 x2 x3 + β11 x21 + .
(5.1)
Then X0X associated with ( 5.1) would be formed from rows and columns 1, 2, 3, 5,
6, 7, and 8 of X0XCCD . Because of the symmetry of the CCD, this model is equivalent
to 5 other WH models with respect to the D, A, G and IV criteria. Specifically, it is
equivalent to the following models:
y = β0 + β1 x1 + β2 x2 + β12 x1 x2 + β13 x1 x3 + β23 x2 x3 + β22 x22 + y = β0 + β1 x1 + β3 x3 + β12 x1 x2 + β13 x1 x3 + β23 x2 x3 + β11 x21 + y = β0 + β1 x1 + β3 x3 + β12 x1 x2 + β13 x1 x3 + β23 x2 x3 + β33 x23 + y = β0 + β2 x2 + β3 x3 + β12 x1 x2 + β13 x1 x3 + β23 x2 x3 + β22 x22 + y = β0 + β2 x2 + β3 x3 + β12 x1 x2 + β13 x1 x3 + β23 x2 x3 + β33 x23 + .
266
For symmetric designs, a defining set of nonequivalent models was determined by
first generating all WH or SH models and then applying an algorithm developed by
Borkowski [8] that determined the set of nonequivalent models and the frequency
(m(i)) of each model i in the defining set. This defining set of models will be used to
study weighted design optimality criteria for the 3-factor CCDs, BBDs, SCDs, and
hybrid 311B designs and the 4-factor CCDs, BBDs, SCDs, and PBCDs in a spherical
design region. Specifically, the 185 and 3905 WH models for K = 3 and 4 design
variables, respectively, can be reduced to 41 and 138 nonequivalent WH models, and
the 95 and 1337 SH models for K = 3 and 4 design variables, respectively, can be
reduced to 25 and 60 nonequivalent SH models.
Moreover, it was previously mentioned in Chapter 3 that the hybrid 310 and
311A designs, and the UNFSDs for K = 3 and the hybrid 416A, 416B, and 416C
designs, and the UNFSDs for K = 4 are nonsymmetric designs with respect to an
optimality criterion. That is, the value of the criterion is not necessarily unique over
the set of permutations of the design variables for any particular reduced model, or,
equivalently, relabeling the design variables may yield multiple optimality criterion
values for certain reduced models. Therefore, for K = 3, all 4 optimality criteria (D,
A, G, and IV ) will be calculated for all relevant permutations of the design variables
for these nonsymmetric designs across the set of 185 WH and 95 SH models. Then,
the minimum values of D, A, and G and the maximum value of IV are chosen from
the set of permutations of the design variables. These minimum D, A, and G, or
267
maximum IV optimality criteria values will represent these nonsymmetric designs.
Because it is computationally demanding to calculate the G criterion, the computer
time greatly increases as the number of reduced models and the number of model
terms increase when the number of design variables increases from K = 3 to K = 4.
Therefore, for K = 4, only 3 optimality criteria (D, A, and IV ) will be calculated
for all relevant permutations of the design variables for nonsymmetric designs across
the set of 3905 WH and 1337 SH models. Then the minimum values of D and A
and the maximum value of IV are chosen from the set of permutations of the design
variables. However, for the 4-factor UNFSDs, only one permutation (x 1 , x2 , x3 , and
x4 ) of 24 permutations based on 3 optimality criteria (D, A, and IV ) across the set
of 3905 WH and 1337 SH models will be studied.
After calculating the set of optimality criterion values for all WH and SH models,
weights selected by the experimenter are assigned to each of the reduced models, or,
in other words, a prior specification of probabilities to the WH or SH models will be
applied to each optimality criterion value.
Model Probabilities
In this research, when prior specification of probabilities are applied to the weak
and strong heredity models, the independence-of-effects principle is assumed as in
Chipman [18]. That is, we assume (i) the linear effects (δi ’s) are independent, (ii) the
interaction effects (δij ’s) are independent of each other and each δij only depends on
268
its parents (δi and δj ), and (iii) the quadratic effects (δii ’s) are independent of each
other and each δii only depends on the parent δi . Therefore, if either weak or strong
heredity can be reasonably assumed, the joint density of δ can be written as:
Pr(δ) =
K
Y
i=1
Pr(δi )
!
K
Y
i<j
Pr(δij |δi , δj )
!
K
Y
i=1
Pr(δii |δi )
!
(5.2)
where Pr(δi ), Pr(δij |δi , δj ), and Pr(δii |δi ) are experimenter-assigned prior probabilities
that the corresponding xi , xi xj , and x2i terms, respectively, are in the model.
If each variable is treated as equally important so that prior probabilities do not
depend on particular variables, but only on the type of model term, prior probabilities
are equal for linear effects (5.3), for interaction effects (5.4), and for quadratic effects
(5.5). That is,
Pr(δi = 1) = pl
Pr(δij = 1|δi , δj ) = p0
for all i,
(5.3)
if (δi , δj ) = (0, 0),
= p1
if (δi , δj ) = (0, 1) or (1, 0),
= p2
if (δi , δj ) = (1, 1),
Pr(δii = 1|δi ) = pq
if δi = 1,
= p3
if δi = 0.
(5.4)
(5.5)
Hence, for WH, (p0 , p1 , p2 ) = (0, p1 , p2 ) for some specified values of p1 and p2 . For
SH, (p0 , p1 , p2 ) = (0, 0, p2 ) for some specified value of p2 . For both WH and SH, p3
= 0.
269
In this research, WH and SH models with pl ∈ {.6, .7, .8, .9}, p1 ∈ {.4, .6, .8},
p2 ∈ {.5, .7, .9}, and pq ∈ {.5, .7, .9} for the response surface designs in a spherical
design region are studied.
Weighted Design Optimality Criteria
To study how robust a design is to model misspecification, the assumption of
either WH or SH and the experimenter-assignment of prior probabilities are used to
calculate a weighted average of the criterion values across all WH or SH models.
1. For WH and prior pl , p1 , p2 , and pq probabilities: the weighted D-optimality
criterion under WH will be defined as
Dw =
M
X
D(i) Pr(δ = ∆i )
(5.6)
i=1
where M
= number of reduced WH models
= 185 and 3905 for K = 3 and 4 design variables
∆i
= δ-vector for model i, i = 1, 2, . . . , M
D(i)
= D-criterion for model i, i = 1, 2, . . . , M
Pr(δ = ∆i ) = Pr(δ) in 5.2 evaluated for ∆i .
2. For SH and prior pl , p2 , and pq probabilities: the weighted D-optimality criterion
under SH will be defined as
Ds =
N
X
i=1
D(i) Pr(δ = ∆i )
(5.7)
270
where the summation is now over the N = 95 and 1337 reduced SH models for
K = 3 and 4 design variables, respectively.
The weighted A, G, and IV -optimality criteria under WH, denoted Aw , Gw , and IVw
are defined by replacing D(i) with A(i), G(i), and IV (i) in ( 5.6). The weighted
A, G, and IV -optimality criteria under SH, denoted As , Gs , and IVs are defined by
replacing D(i) with A(i), G(i), and IV (i) in ( 5.7).
For symmetric designs, an alternate form to exploit the symmetry can be used.
That is, the weighted D-optimaltity criterion under WH can be written as
∗
Dw =
M
X
m(i) D(i) Pr(δ = ∆∗i )
(5.8)
i=1
where M ∗
= number of reduced nonequivalent WH models
= 41 and 138 for K = 3 and 4 design variables
m(i)
= the number of models equivalent to model i
∆∗i
= δ-vector for model i, i = 1, 2, . . . , M ∗
D(i)
= D-criterion for model i, i = 1, 2, . . . , M ∗
Pr(δ = ∆∗i ) = Pr(δ) in 5.2 evaluated at ∆∗i .
Similarly, the weighted D-optimaltity criterion under SH can be written as
∗
Ds =
N
X
m(i) D(i) Pr(δ = ∆∗i )
(5.9)
i=1
where the summation is now over the N ∗ = 25 and 60 reduced nonequivalent SH
models for K = 3 and 4 design variables, respectively. Alternate forms for A w , Gw ,
271
and IVw across WH models and As , Gs , and IVs across SH models are defined by
replacing D(i) with A(i), G(i), and IV (i) in ( 5.8) and ( 5.9), respectively.
An Example
Suppose a 3-factor 15-point CCD is considered. The D, A, G, and IV criteria
for the set of 41 nonequivalent WH models and for the set of 25 nonequivalent SH
models are shown in Tables 84 and 85, respectively. In Tables 84 and 85, each row
represents one of the 41 nonequivalent WH models and 25 nonequivalent SH models,
respectively. The m(i) column indicates the number of models equivalent to the
model i. For the model in row i, the D, A, G, and IV criteria are, respectively, in
the D(i), A(i), G(i), and IV (i) columns. These tables show how optimality measures
can significantly vary across models and an experimenter should not rely only on the
criteria associated with the full second-order model (which is model i = 41 in Table
84 and model i = 25 in Table 85).
Table 86 shows how the weighted Dw , Gw , Ds , and Gs optimality criteria for the
3-factor 15-point CCD are calculated. Two sets of prior probabilities for 41 nonequivalent WH and 25 nonequivalent SH models are considered. That is, (i) for WH models
with (pl , p1 , p2 , pq ) = (.9, .4, .5, .7) and (ii) for SH models with (pl , p2 , pq ) = (.9, .5,
.7). For the WH and SH models, the Pr(δ = ∆∗i ) columns, respectively, correspond to
the WH and SH prior probabilities defined in Equation 5.2. Then, these probabilities
are multiplied by the corresponding m(i)D(i) or m(i)G(i) as it is defined in Equation
5.8 or 5.9 and the ith model’s contributions to the weighted optimality criteria Dw ,
272
Gw , Ds , and Gs are found. These products are given in their respective columns and
rows. To calculate the Dw , Gw , Ds , and Gs , the summation of these corresponding
columns are computed and given at the bottom of the table.
Table 84. Optimality Criteria of a 15-Point CCD for WH Models, K = 3.
Model
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
δ1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
δ2
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
δ3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
δ12
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
δ13
0
0
0
0
0
1
1
0
0
0
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
δ23
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
δ11
0
0
1
0
1
0
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
δ22
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
δ33
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
m(i)
1
3
3
6
6
3
3
3
6
3
6
12
6
3
6
3
3
6
3
6
12
6
3
6
3
1
3
3
1
3
9
9
3
3
9
9
3
1
3
3
1
D(i)
100.0000
55.7766
31.0047
52.8344
48.7581
71.7804
74.4609
72.8825
62.0621
54.5101
82.5587
83.2787
81.4683
75.6502
77.3175
76.6839
55.0391
51.1946
47.4461
75.6502
77.3175
76.6839
71.3687
73.3227
73.2809
94.9553
93.1407
89.4320
80.4718
84.6087
84.8755
83.0658
76.4387
78.3452
79.4248
78.5904
73.4421
74.1572
75.5678
75.2769
71.1296
A(i)
100.0000
47.4567
19.1536
43.9500
38.2930
68.7113
62.5660
71.1851
51.0581
36.4919
79.7144
69.5602
54.1513
72.5383
66.2031
54.0330
47.4320
41.4836
33.4814
72.5383
66.2031
54.0330
68.4314
63.9970
53.9445
94.9134
78.3150
58.3856
27.7358
82.1103
72.6439
57.6060
29.5059
75.3356
69.0712
57.0348
31.0471
71.1428
66.6141
56.5984
32.4011
G(i)
100.0000
47.4567
32.7226
43.3543
43.6301
57.8058
54.5377
71.1851
43.6301
54.4132
57.8058
54.5377
65.2959
71.7403
65.4452
76.1785
49.8116
53.4909
62.4159
62.2645
64.1890
72.8186
67.1286
74.8872
83.2212
94.9134
54.5377
65.2959
46.6667
62.2645
64.1890
72.8186
53.3333
74.7174
74.8872
82.9391
60.0000
72.4269
82.3331
91.0192
66.6667
IV (i)
1.0000
1.0357
1.5022
2.7685
3.7735
3.1301
4.3532
2.9295
3.9346
5.5512
3.3590
4.5821
6.9346
3.7487
4.9718
7.3243
3.5261
4.5312
6.1478
3.7487
4.9718
7.3243
4.1384
5.3615
7.7141
3.5879
4.8110
7.1635
16.7762
3.9776
5.2007
7.5532
17.1659
4.3673
5.5904
7.9429
17.5556
4.7570
5.9801
8.3327
17.9453
273
Table 85. Optimality Criteria of a 15-Point CCD for SH Models, K = 3.
Model
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
δ1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
δ2
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
δ3
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
δ12
0
0
0
0
0
0
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
δ13
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
δ23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
δ11
0
0
1
0
1
1
0
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
δ22
0
0
0
0
0
1
0
0
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
δ33 m(i) D(i)
A(i)
G(i)
IV (i)
0
1 100.0000 100.0000 100.0000 1.0000
0
3 55.7766 47.4567 47.4567 1.0357
0
3 31.0047 19.1536 32.7226 1.5022
0
3 72.8825 71.1851 71.1851 2.9295
0
6 62.0621 51.0581 43.6301 3.9346
0
3 54.5101 36.4919 54.4132 5.5512
0
3 55.0391 47.4320 49.8116 3.5261
0
6 51.1946 41.4836 53.4909 4.5312
0
3 47.4461 33.4814 62.4159 6.1478
0
1 94.9553 94.9134 94.9134 3.5879
0
3 93.1407 78.3150 54.5377 4.8110
0
3 89.4320 58.3856 65.2959 7.1635
1
1 80.4718 27.7358 46.6667 16.7762
0
3 84.6087 82.1103 62.2645 3.9776
0
9 84.8755 72.6439 64.1890 5.2007
0
9 83.0658 57.6060 72.8186 7.5532
1
3 76.4387 29.5059 53.3333 17.1659
0
3 78.3452 75.3356 74.7174 4.3673
0
9 79.4248 69.0712 74.8872 5.5904
0
9 78.5904 57.0348 82.9391 7.9429
1
3 73.4421 31.0471 60.0000 17.5556
0
1 74.1572 71.1428 72.4269 4.7570
0
3 75.5678 66.6141 82.3331 5.9801
0
3 75.2769 56.5984 91.0192 8.3327
1
1 71.1296 32.4011 66.6667 17.9453
274
Table 86. The WH and SH Model Probabilities for a 3 Factor 15-Point CCD with
pl = .9, p1 = .4, p2 = .5, and pq = .7.
Weak Heredity
Model
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
Pr(δ = ∆∗i )
0.001000000
0.000972000
0.002268000
0.000648000
0.001512000
0.000432000
0.001008000
0.001312200
0.003061800
0.007144200
0.000874800
0.002041200
0.004762800
0.000583200
0.001360800
0.003175200
0.001312200
0.003061800
0.007144200
0.000874800
0.002041200
0.004762800
0.000583200
0.001360800
0.003175200
0.002460375
0.005740875
0.013395375
0.031255875
0.002460375
0.005740875
0.013395375
0.031255875
0.002460375
0.005740875
0.013395375
0.031255875
0.002460375
0.005740875
0.013395375
0.031255875
Strong Heredity
m(i)D(i)
Pr(δ = ∆∗i )
0.1000
0.1626
0.2110
0.2054
0.4423
0.0930
0.2252
0.2869
1.1401
1.1683
0.4333
2.0399
2.3281
0.1324
0.6313
0.7305
0.2167
0.9405
1.0169
0.3971
1.8938
2.1914
0.1249
0.5987
0.6980
0.2336
1.6041
3.5939
2.5152
0.6245
4.3853
10.0143
7.1675
0.5783
4.1037
9.4747
6.8865
0.1825
1.3015
3.0251
2.2232
m(i)G(i)
Pr(δ = ∆∗i )
0.1000
0.1384
0.2226
0.1686
0.3958
0.0749
0.1649
0.2802
0.8015
1.1662
0.3034
1.3359
1.8659
0.1255
0.5343
0.7256
0.1961
0.9827
1.3377
0.3268
1.5723
2.0809
0.1174
0.6114
0.7927
0.2335
0.9393
2.6240
1.4586
0.4596
3.3165
8.7789
5.0009
0.5515
3.8693
9.9990
5.6261
0.1782
1.4180
3.6577
2.0837
Dw =
76.3221
Gw =
66.6168
Pr(δ = ∆∗i )
0.001000000
0.002700000
0.006300000
–
–
–
–
0.003645000
0.008505000
0.019845000
–
–
–
–
–
–
0.003645000
0.008505000
0.019845000
–
–
–
–
–
–
0.002460375
0.005740875
0.013395375
0.031255875
0.002460375
0.005740875
0.013395375
0.031255875
0.002460375
0.005740875
0.013395375
0.031255875
0.002460375
0.005740875
0.013395375
0.031255875
m(i)D(i)
Pr(δ = ∆∗i )
0.1000
0.4518
0.5860
–
–
–
–
0.7970
3.1670
3.2453
–
–
–
–
–
–
0.6019
2.6125
2.8247
–
–
–
–
–
–
0.2336
1.6041
3.5939
2.5152
0.6245
4.3853
10.0143
7.1675
0.5783
4.1037
9.4747
6.8865
0.1825
1.3015
3.0251
2.2232
m(i)G(i)
Pr(δ = ∆∗i )
0.1000
0.3844
0.6185
–
–
–
–
0.7784
2.2264
3.2395
–
–
–
–
–
–
0.5447
2.7296
3.7159
–
–
–
–
–
–
0.2335
0.9393
2.6240
1.4586
0.4596
3.3165
8.7789
5.0009
0.5515
3.8693
9.9990
5.6261
0.1782
1.4180
3.6577
2.0837
Ds =
72.3000
Gs =
64.5322
275
Table 87. Weighted Optimality Criteria for the 3-Factor 15-Point CCD Across WH
Models.
pl
.60
.60
.60
..
.
pq
.50
.50
.50
..
.
p1
.40
.40
.40
..
.
p2
.50
.70
.90
..
.
Dw
60.0636
58.9372
57.8879
..
.
Aw
52.0843
51.1790
50.3374
..
.
Gw
48.2801
49.4657
50.6761
..
.
IVw
4.4597
4.5687
4.6778
..
.
.60
.60
.60
.70
.70
.70
..
.
.90
.90
.90
.50
.50
.50
..
.
.80
.80
.80
.40
.40
.40
..
.
.50
.70
.90
.50
.70
.90
..
.
64.8389
64.0555
63.3258
61.2073
59.7481
58.4112
..
.
54.6559
54.3052
53.9764
52.8283
51.6875
50.6481
..
.
52.4496
53.9933
55.5364
48.5234
50.1989
51.9137
..
.
5.7141
5.8163
5.9185
5.0038
5.1496
5.2954
..
.
.80
.80
.80
.90
.90
.90
..
.
.90
.90
.90
.50
.50
.50
..
.
.80
.80
.80
.40
.40
.40
..
.
.50
.70
.90
.50
.70
.90
..
.
65.8738
64.5011
63.2560
66.0423
63.8745
61.9668
..
.
53.3396
52.7653
52.2428
56.9573
55.3741
54.0064
..
.
54.1440
56.8593
59.5734
52.4217
55.3954
58.4527
..
.
6.9497
7.1306
7.3115
6.0164
6.2487
6.4811
..
.
.90
.90
.90
.90
.90
.90
.80
.80
.80
.50
.70
.90
67.1907
65.4661
63.9231
53.4643
52.7685
52.1464
55.9717
59.3899
62.8065
7.5226
7.7510
7.9795
To study the behavior of a design with respect to choices of pl , p1 , p2 , and pq
for WH models and pl , p2 , and pq for SH models, the weighted design optimality
criteria with respect to various choices of these prior probabilities are evaluated. In
this dissertation, WH and SH models with pl ∈ {.6, .7, .8, .9}, p1 ∈ {.4, .6, .8},
p2 ∈ {.5, .7, .9}, and pq ∈ {.5, .7, .9} for the response surface designs in a spherical
design region are studied. Then, a full second-order response surface model is fitted.
That is, for WH, Dw , Aw , Gw , and IVw are evaluated at the 108 points of the 4 × 33
factorial design in pl , p1 , p2 , and pq . Table 87 shows a subset of the results from the
276
108 points for the 3-factor 15-point CCD. For WH, the 15-parameter full second-order
model to be fit is
Eff w = β0 + βl pl + βq pq + β1 p1 + β2 p2 + βll p2l + βqq p2q + β11 p21 + β22 p22
+ βlq pl pq + βl1 pl p1 + βl2 pl p2 + βq1 pq p1 + βq2 pq p2 + β12 p1 p2 + (5.10)
where Eff w is Dw , Aw , Gw or IVw . Tables 88 to 91 and Tables 96 to 98 contain the
estimated β-coefficients for Dw , Aw , Gw or IVw of the 3 and 4 factor response surface
designs, respectively.
For SH, Ds , As , Gs , and IVs are evaluated at the 36 points of the 4 × 32 factorial
design having pl ∈ {.6, .7, .8, .9}, p2 ∈ {.5, .7, .9}, and pq ∈ {.5, .7, .9}. For SH,
the 10-parameter full second-order model to be fit is
Eff s = β0 + βl pl + βq pq + β2 p2 + βll p2l + βqq p2q + β22 p22
+ βlq pl pq + βl2 pl p2 + βq2 pq p2 + (5.11)
where Eff s is Ds , As , Gs or IVs . Tables 92 to 95, respectively, contain the estimated
β-coefficients for Ds , As , Gs , and IVs of the 3-factor response surface designs. Tables
99 to 101, respectively, contain the Ds , As , and IVs of the 4-factor designs.
For 19 3-factor designs across all 4 criteria and 22 4-factor designs across 3 criteria,
a total of 284 models were fit based on Equations 5.10 and 5.11. The minimum
R2 is .9780. In summary, 282 of 284 models have R2 > .98 and 279 of 284 models
have R2 > .99. These models will provide reliable interpolation estimates of weighted
optimality criterion values for other choices of pl , p1 , p2 , and pq .
277
Weighted Design Optimality Criteria Comparisons
Examples of comparisons of weighted design optimality criteria for 3-factor small
spherical response surface designs by ranking within a design optimality criterion and
a design size are given in Tables 102, 103, and 104. Table 102 contains the D, A, G,
and IV criteria for the full second-order model for the small response surface designs
of design sizes N = 11 and 13. The weighted optimality criteria under WH with
(pl , p1 , p2 , pq ) = (.9, .4, .5, .7) are given in Table 103. Table 104 contains the
weighted optimality criteria under SH with (pl , p2 , pq ) = (.8, .5, .5).
Tables 102 to 104 indicate the following results:
1. D, A, and IV criterion values are conservative while the G criterion value is
not: Every weighted D, A, and IV optimality criterion value in Tables 103 and
104 is better than its associated D, A, or IV value in Table 102. However, for
G, most of the weighted optimality criterion values in Tables 103 and 104 are
smaller than the associated G criterion values in Table 102 with the exception
being the SCDs. Thus, if only the full model optimality values are considered,
the experimenter is being conservative for the D, A, and IV criteria because
they are underestimates relative to the weighted design optimality criteria values. Conversely, the experimenter is often using a liberal criterion value when
the G-criterion is considered.
278
2. Large differences can occur between full-model and weighted criterion values:
The 13-point UNFSD is tied for being the worst design based on IV = 16.36.
However, in Table 104, its IVs = 5.58 is the third best.
3. Weighted criteria rankings can dramatically change: The 13-point UNFSD has
A-rank of 5 in Table 102 but As -rank of 1 in Table 104.
4. Designs that were obviously strongest by D, A, G, or IV , now have competitors:
For the 11-point designs and the IV criterion, the 310 design is 3.75 units better
than the 311B design. However, the IVw values are nearly identical in Table
103. Similarly, the A-efficiencies for the BBD and 311A designs are 15% apart
but the As values are less than 1% apart in Table 104.
For experimenters considering larger designs and designs with more factors, the results on small designs serve as a reminder that the chosen design may not be as
efficient as you believe it to be. Therefore, if we assume that the full second-order
response surface model is reduced after an experiment has been performed, then the
experimenter should exercise caution when choosing a design. When a researcher
must decide which response surface design is ‘best’ based on one or more design optimality criteria, it is important that the optimality criteria are determined over a
subset of possible reduced models.
279
Table 88. Coefficients of WH Dw Models for 3-Factor Designs.
Dsgn
rs
310
–
SCD
1
310
–
311A
–
311B
–
BBD
–
SCD
1
UNFSD –
310
–
311A
–
311B
–
CCD
1
BBD
–
UNFSD –
CCD
1
SCD
2
SCD
2
CCD
2
CCD
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
Dsgn
rs
310
–
SCD
1
310
–
311A
–
311B
–
BBD
–
SCD
1
UNFSD –
310
–
311A
–
311B
–
CCD
1
BBD
–
UNFSD –
CCD
1
SCD
2
SCD
2
CCD
2
CCD
2
βb0
55.16
51.48
55.71
48.76
45.20
49.60
56.45
45.20
56.25
52.87
51.06
43.66
52.11
50.59
49.49
49.78
55.85
40.75
48.27
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
βbl
0.03
6.27
-8.91
12.86
21.20
18.79
-16.86
23.47
-22.26
-8.50
-3.68
27.80
1.60
0.97
5.97
10.05
-11.23
31.80
9.56
N
βblq
10 -4.62
11 1.30
11 -0.29
11 -4.43
11 -3.63
13 -7.22
13 12.24
13 -6.58
13 1.47
13 5.57
13 8.47
15 -9.12
15 1.92
15 6.16
17 4.71
17 -5.77
19 9.10
21 -14.23
23 3.03
Linear Terms
Quadratic
βbq
βb1
βb2
βbll
βbqq
-20.34 52.57 -4.48 26.83 2.57
4.44 22.39 4.16 31.08 -6.63
-19.08 48.50 -3.43 30.04 2.38
-6.91 46.88 -2.71 25.00 -2.40
1.77 41.30 -1.24 24.32 -5.43
-15.64 59.21 -4.52 20.73 0.70
-2.55 17.93 6.26 38.80 -2.78
0.42 43.74 -0.76 23.18 -5.08
-15.27 41.68 -2.58 34.57 1.32
-10.36 39.97 -0.69 32.28 -0.36
-5.09 34.78 1.08 32.66 -1.95
0.74 48.31 -2.83 20.64 -5.40
-16.38 52.38 -2.64 26.82 1.50
-6.39 37.97 1.61 30.87 -1.67
-7.12 42.85 -0.38 28.29 -1.67
16.40 9.19 1.23 32.88 -10.77
5.19
7.10 4.57 40.32 -5.33
15.62 33.41 -2.32 22.29 -10.97
2.80 30.20 0.82 30.28 -5.22
Interaction
βbl2
βbl1
-46.34 -2.45
-23.25 -46.42
-42.80 -4.37
-41.49 -13.23
-36.67 -22.66
-51.99 -2.79
-18.96 -45.51
-38.41 -21.51
-36.93 -6.06
-35.43 -15.39
-30.94 -24.14
-42.69 -16.01
-46.04 -5.97
-33.35 -23.39
-37.87 -18.47
-11.21 -53.17
-9.15 -53.80
-30.07 -29.76
-27.19 -32.02
Terms
βbq1
5.29
8.19
5.33
6.77
7.54
5.70
7.28
7.26
5.18
6.35
6.89
7.27
5.69
6.79
6.89
7.96
7.28
8.04
7.59
βbq2
4.98
4.75
4.15
4.98
4.68
4.99
1.45
5.17
3.49
2.93
2.09
5.52
3.46
2.54
3.05
5.84
1.71
6.70
3.34
βb12
1.25
3.75
1.26
1.82
2.42
1.36
3.57
2.03
1.24
1.80
2.35
2.06
1.37
2.02
2.05
4.90
4.72
3.07
3.02
Terms
βb11
βb22
-7.88 0.33
-7.87 6.20
-7.59 0.50
-8.78 1.46
-9.31 2.62
-8.88 0.36
-7.01 6.19
-9.52 1.94
-6.92 0.66
-8.00 1.70
-8.38 2.82
-9.45 1.79
-8.36 0.64
-8.76 2.20
-8.83 2.05
-7.44 7.90
-6.87 8.05
-9.29 3.71
-8.79 3.98
R2
.9989
.9975
.9986
.9983
.9983
.9989
.9982
.9983
.9981
.9982
.9986
.9982
.9986
.9986
.9985
.9977
.9986
.9974
.9986
280
Table 89. Coefficients of WH Aw Models for 3-Factor Designs.
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
βb0
56.50
56.34
56.56
37.77
31.40
46.97
62.38
33.86
52.74
48.06
46.30
26.51
52.10
47.31
43.57
53.86
64.24
19.64
41.06
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
βbl
-11.16
-13.65
-20.59
24.63
38.84
16.18
-40.52
38.28
-27.84
-9.85
-4.41
51.18
-7.48
-1.10
8.47
-7.68
-37.90
61.34
14.70
βblq
-25.97
-31.62
-9.05
-50.24
-59.67
-45.04
-1.71
-63.24
-1.89
-9.37
-12.03
-73.48
-9.43
-16.35
-22.83
-48.02
-12.97
-90.79
-38.68
Linear Terms
βbq
βb1
-25.83 52.52
14.25 11.37
-27.96 51.43
11.71 47.86
29.23 40.27
-10.79 57.66
-1.68
8.07
25.35 41.01
-20.83 47.41
-8.99 43.42
0.01 35.45
33.37 47.81
-22.99 55.36
-2.35 37.31
0.55 44.43
30.08 -4.42
6.46
-5.86
60.34 29.24
20.12 26.87
βb2
-7.06
-12.97
-5.86
-10.12
-12.97
-7.68
-4.60
-11.96
-4.11
-5.40
-6.28
-11.90
-5.70
-5.26
-7.79
-22.88
-14.20
-17.86
-13.07
Interaction
βbl2
βbl1
-45.85 5.58
-19.80 -29.77
-45.04 2.59
-43.21 -0.44
-37.60 -6.36
-50.37 6.53
-15.92 -39.97
-37.51 -6.96
-41.63 -1.43
-39.02 -7.88
-32.92 -15.32
-43.58 -0.99
-48.51 1.54
-33.80 -15.58
-40.30 -7.52
-6.10 -28.59
-4.28 -38.23
-29.36 -9.98
-26.91 -16.15
Terms
βbq1
0.80
7.06
2.25
3.66
4.96
0.52
5.27
4.69
3.28
4.29
4.75
4.00
2.68
4.70
4.48
6.22
4.96
5.79
5.40
Quadratic Terms
βbll
βbqq
βb11
βb22
29.49 7.33 -5.75 -0.17
33.34 -13.05 -2.59 6.79
34.39 7.36 -6.36 -0.04
19.41 -8.50 -6.50 0.43
15.97 -16.08 -6.52 1.31
19.54 1.83 -6.15 -0.21
45.85 -3.52 -2.24 9.53
15.64 -14.78 -6.84 0.58
38.43 4.46 -6.59 0.25
33.80 -0.37 -6.87 1.09
33.62 -3.87 -6.60 2.35
10.07 -16.89 -6.62 0.59
30.22 5.28 -7.08 0.02
32.02 -3.37 -7.11 1.30
27.18 -4.40 -7.02 1.14
32.78 -19.13 -1.16 8.19
46.24 -7.38 -0.95 10.46
8.57 -28.74 -5.63 2.31
26.88 -12.88 -5.72 3.01
βbq2
5.92
18.13
5.44
10.61
14.23
6.25
7.05
15.27
4.41
6.55
7.66
12.50
5.40
9.14
8.97
23.02
12.31
19.19
14.40
βb12
0.95
5.06
1.00
1.75
2.80
1.05
5.40
2.06
1.05
1.97
3.10
2.07
1.17
2.34
2.31
7.71
7.98
3.93
4.21
R2
.9997
.9940
.9993
.9993
.9981
.9998
.9958
.9986
.9988
.9985
.9976
.9990
.9993
.9980
.9985
.9916
.9941
.9970
.9958
281
Table 90. Coefficients of WH Gw Models for 3-Factor Designs.
Dsgn
rs
310
–
SCD
1
310
–
311A
–
311B
–
BBD
–
SCD
1
UNFSD –
310
–
311A
–
311B
–
CCD
1
BBD
–
UNFSD –
CCD
1
SCD
2
SCD
2
CCD
2
CCD
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
Dsgn
rs
310
–
SCD
1
310
–
311A
–
311B
–
BBD
–
SCD
1
UNFSD –
310
–
311A
–
311B
–
CCD
1
BBD
–
UNFSD –
CCD
1
SCD
2
SCD
2
CCD
2
CCD
2
βb0
87.34
72.95
83.25
74.40
69.69
93.31
71.47
78.97
77.80
73.00
69.15
23.96
85.84
74.15
74.05
78.78
75.98
15.90
68.66
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
βbl
-83.89
-61.79
-84.48
-66.95
-52.26
-88.00
-72.50
-57.88
-87.00
-76.32
-68.46
18.46
-86.52
-68.38
-70.72
-69.43
-74.96
36.81
-62.13
N
βblq
10 10.77
11 9.35
11 8.18
11 18.68
11 21.48
13 37.30
13 10.81
13 12.47
13 3.57
13 17.50
13 26.47
15 -44.08
15 26.35
15 20.25
17 33.82
17 4.95
19 11.42
21 -72.33
23 26.24
Linear Terms
βbq
βb1
-50.49 23.49
1.66 -13.43
-42.77 22.07
-23.52 26.17
-12.58 25.64
-60.82 28.54
0.68 -11.98
-17.07 12.90
-32.54 19.40
-21.08 22.51
-16.24 21.36
58.10 29.57
-46.95 25.65
-17.18 10.71
-25.76 26.18
2.32 -26.16
2.32 -23.79
85.65 14.81
-10.68 13.27
βb2
-9.73
9.02
-9.26
-17.10
-21.73
-16.44
7.74
-21.88
-7.97
-15.35
-11.55
-13.22
-14.12
-8.59
-16.65
-3.16
-1.62
-14.77
-7.59
Interaction
βbl2
βbl1
-20.71 23.74
2.33 -47.34
-19.59 22.62
-25.72 35.81
-26.80 34.15
-25.95 34.84
2.30 -44.39
-15.54 19.05
-17.33 19.86
-22.30 30.98
-22.41 22.75
-29.70 34.34
-23.39 30.77
-12.99 4.80
-26.41 34.07
11.68 -36.71
10.64 -44.02
-19.02 17.48
-17.20 10.32
Terms
βbq1
-2.99
0.44
-2.57
1.15
3.81
-1.86
0.58
2.85
-2.12
1.22
3.19
2.73
-1.62
2.66
2.61
-0.07
0.15
3.44
3.21
βbll
49.21
44.78
50.31
46.18
39.67
50.67
50.09
42.87
52.46
50.05
47.91
16.41
51.42
50.00
47.85
48.18
53.24
10.07
47.58
βbq2
0.34
-0.16
0.31
4.23
9.85
3.80
-3.89
15.35
0.06
4.42
2.91
1.10
2.81
2.69
5.86
6.21
-5.24
6.41
0.35
Quadratic Terms
βbqq
βb11
βb22
20.68 0.71 0.30
0.61
2.93 8.98
17.42 0.67 0.28
6.22
0.88 0.29
0.84
1.07 -0.66
22.05 0.76 0.23
1.55
2.54 10.83
1.55 -0.09 1.24
13.31 0.60 0.26
5.30
0.79 0.23
4.25
0.95 -0.68
-23.90 0.99 -0.47
17.02 0.70 0.22
5.08 -0.21 2.98
6.04
0.88 -0.66
-1.30 4.93 7.32
1.40
4.43 14.91
-33.78 2.25 0.22
3.10
2.06 0.58
βb12
R2
0.22 .9988
2.91 .9969
0.21 .9988
0.36 .9991
0.00 .9987
0.26 .9989
2.54 .9969
1.90 .9959
0.20 .9988
0.32 .9989
0.04 .9991
0.23 .9926
0.25 .9987
1.59 .9988
0.15 .9993
7.42 .9955
6.68 .9964
1.56 .9780
1.49 .9993
282
Table 91. Coefficients of WH IVw Models for 3-Factor Designs.
Dsgn
rs
310
–
SCD
1
310
–
311A
–
311B
–
BBD
–
SCD
1
UNFSD –
310
–
311A
–
311B
–
CCD
1
BBD
–
UNFSD –
CCD
1
SCD
2
SCD
2
CCD
2
CCD
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
βb0
10 2.70
11 8.38
11 0.12
11 6.23
11 7.63
13 6.42
13 1.32
13 9.45
13 -0.10
13 0.56
13 0.98
15 11.71
15 0.24
15 1.30
17 1.71
17 16.63
19 3.43
21 21.00
23 4.43
Dsgn
rs
310
–
SCD
1
310
–
311A
–
311B
–
BBD
–
SCD
1
UNFSD –
310
–
311A
–
311B
–
CCD
1
BBD
–
UNFSD –
CCD
1
SCD
2
SCD
2
CCD
2
CCD
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
βbl
-0.02
-9.39
5.03
-5.92
-8.22
-6.72
3.23
-11.46
6.22
4.54
3.77
-15.34
4.66
3.02
2.15
-22.88
-0.23
-30.65
-2.44
Linear Terms
βbq
βb1
-6.13 3.10
-15.44 4.71
-0.76 2.71
-12.03 3.02
-14.35 3.09
-12.92 3.14
-2.58 4.96
-17.69 3.19
0.29 2.38
-1.28 2.76
-2.11 2.98
-21.73 3.06
-0.95 2.65
-2.75 3.02
-3.58 2.86
-30.16 5.11
-6.81 5.33
-37.85 3.38
-8.54 3.34
Quadratic Terms
βb2
βbll
βbqq
βb11
βb22
-0.49 0.75
1.96 -0.45
0
-2.85 3.86
5.48 -0.34
0
-0.54 -1.20 0.15 -0.29
0
-0.58 2.74
4.18 -0.39
0
-0.67 3.47
5.06 -0.36
0
-0.44 3.20
4.38 -0.50
0
-3.36 -0.78 0.87 -0.24
0
-0.85 4.65
6.21 -0.38 0.04
-0.64 -1.73 -0.08 -0.13
0
-0.69 -1.15 0.39 -0.23
0
-0.80 -0.95 0.69 -0.24
0
-0.57 6.15
7.71 -0.41
0
-0.51 -1.07 0.22 -0.29
0
-0.97 -0.68 0.87 -0.25 0.05
-0.65 -0.27 1.18 -0.29
0
-2.54 8.43 10.88 -0.34
0
-2.84 0.13
2.39 -0.28
0
-0.80 11.53 13.65 -0.38
0
-0.88 1.23
2.99 -0.31
0
βblq
13.59
19.08
6.08
17.27
18.54
20.21
5.01
22.59
3.74
4.59
4.83
26.94
5.35
5.94
7.06
33.53
9.31
42.12
11.74
Interaction Terms
βbq2
βbq1
βbl2
βbl1
-2.81 1.49 0.49
0
-3.99 6.66 0.31
0
-2.45 1.63 0.20
0
-2.71 -1.76 0.41
0
-2.76 2.03 0.35
0
-2.84 1.33 0.60
0
-4.17 7.87 0.13
0
-2.91 2.23 0.46 0.04
-2.15 1.93 -0.06
0
-2.48 2.08 0.13
0
-2.66 2.40 0.14
0
-2.74 1.73 0.45
0
-2.39 1.54 0.21
0
-2.75 2.52 0.22 0.05
-2.56 1.96 0.22
0
-4.40 6.48 0.30
0
-4.58 7.24 0.19
0
-3.01 2.42 0.38
0
-2.97 2.65 0.25
0
βb12
R2
-0.05 .9999
-0.09 .9991
-0.06 1.0000
-0.06 .9995
-0.07 .9991
-0.05 .9996
-0.11 .9996
-0.04 .9991
-0.07 1.0000
-0.07 1.0000
-0.09 .9999
-0.06 .9989
-0.06 1.0000
-0.05 .9999
-0.07 .9999
-0.14 .9982
-0.16 .9995
-0.09 .9980
-0.10 .9994
283
Table 92. Coefficients of SH Ds Models for 3-Factor Designs.
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
βb0
98.46
81.34
96.91
93.02
157.56
97.45
82.91
88.61
93.36
92.63
153.28
90.08
96.30
90.15
92.32
78.75
82.55
82.86
87.83
Linear Terms
βbl
βbq
βb2
-100.09 -36.53 -7.65
-44.94 -20.23 -1.70
-103.31 -35.28 -6.46
-85.58 -27.07 -6.37
-275.74 -41.11 -69.43
-92.53 -33.02 -7.90
-61.11 -25.18
0.89
-71.30 -21.34 -4.53
-106.09 -30.98 -5.38
-95.73 -29.48 -4.06
-271.53 -46.42 -60.88
-74.96 -20.87 -6.76
-99.93 -33.62 -5.80
-84.36 -27.04 -1.92
-87.90 -27.84 -4.07
-36.28
-9.30
-5.59
-53.15 -18.96 -1.86
-55.25
-8.76
-7.13
-71.57 -20.58 -3.77
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
Quadratic Terms
βbll
βbqq
βb22
86.28
3.74
0.33
52.61
-6.00
6.20
85.61
3.41
0.50
81.44
-1.38
1.46
225.92 -2.30 29.86
87.19
1.89
0.36
56.57
-2.28
6.19
76.63
-4.17
1.94
83.22
2.21
0.66
81.59
0.45
1.70
213.68
0.65
27.90
79.31
-4.40
1.79
86.73
2.47
0.64
78.39
-0.94
2.20
81.33
-0.86
2.05
50.04 -10.44 7.90
55.22
-5.05
8.05
68.75 -10.23 3.71
73.18
-4.59
3.98
Interaction Terms
βbq2
βbl2
βblq
11.77 -0.56
6.93
26.83 -41.99 7.80
16.28 -2.53
6.01
16.38 -10.89 7.24
33.83 -2.10 18.91
10.44 -0.72
6.99
35.74 -41.34 4.15
16.02 -19.06 7.58
17.69 -4.33
5.21
25.48 -13.17 1.70
44.95 -5.52 14.80
13.25 -13.43 7.92
19.69 -3.99
5.33
27.74 -21.03 4.75
26.30 -15.98 5.26
20.82 -47.76 9.34
34.13 -48.59 4.90
11.11 -26.33 9.58
27.41 -28.67 6.03
R2
.9999
.9995
.9998
.9999
.9997
.9999
.9996
.9998
.9998
.9998
.9998
.9998
.9999
.9999
.9999
.9993
.9996
.9995
.9998
284
Table 93. Coefficients of SH As Models for 3-Factor Designs.
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
βb0
88.41
76.71
90.95
74.55
117.06
80.53
78.67
68.39
87.04
84.40
129.79
64.52
89.23
80.96
81.51
80.19
87.20
57.68
77.55
Linear Terms
βbl
βbq
βb2
-87.35 -34.83 -10.44
-30.10
-9.47 -24.32
-101.52 -39.08 -9.13
-56.27
-4.24 -14.96
-196.39
4.53
-65.67
-64.16 -20.20 -11.35
-50.32 -21.54 -15.19
-34.52
7.81
-17.32
-107.28 -33.30 -7.15
-89.35 -25.15 -9.93
-228.69 -27.80 -55.31
-31.12
16.26
17.42
-94.59 -35.33 -9.21
-71.85 -19.49 -10.32
-73.49 -16.66 -13.07
-34.27
6.67
-38.34
-59.23 -13.72 -28.84
-11.78
39.16 -26.49
-55.45
0.09
-21.38
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
Quadratic Terms
βbll
βbqq
βb22
76.86
8.88
-0.17
29.02 -12.72 6.79
83.81
8.55
-0.04
66.11
-6.77
0.43
177.48 -12.62 22.56
69.65
3.69
-0.21
38.49
-3.26
9.53
55.94 -13.32 0.58
86.14
5.35
0.25
79.18
0.64
1.09
186.26 -1.54 23.13
56.92 -15.09 0.59
83.18
6.43
0.02
70.90
-2.47
1.30
73.46
-3.26
1.14
32.70 -19.29 8.19
43.78
-7.48 10.46
45.35 -27.67 2.31
61.98 -12.12 3.01
Interaction Terms
βbq2
βbl2
βblq
-18.75
7.23
8.39
-8.89 -22.24 24.16
1.10
4.25
7.77
-35.60
2.20
13.93
-38.43
9.06
25.47
-37.73
8.38
8.84
17.62 -32.01 11.47
-46.45 -3.99 18.96
10.23
0.17
6.56
6.46
-5.12
9.39
12.30
-1.68 17.99
-57.74
2.10
16.21
2.19
3.45
7.74
0.68
-12.41 12.26
-6.10
-4.26 12.19
-26.14 -17.49 30.29
6.26
-26.77 17.95
-70.71 -4.59 24.48
-19.28 -10.50 18.98
R2
.9998
.9971
.9997
.9994
.9983
.9998
.9986
.9989
.9997
.9997
.9995
.9990
.9998
.9995
.9995
.9962
.9980
.9975
.9982
285
Table 94. Coefficients of SH Gs Models for 3-Factor Designs.
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
βb0
102.14
61.26
97.74
101.48
181.33
113.48
62.45
94.84
91.07
97.06
164.05
57.08
104.42
88.28
103.86
80.74
78.59
47.30
97.89
Linear Terms
βbl
βbq
βb2
-117.87 -51.76 -12.56
0.57
-14.46
0.78
-117.27 -44.69 -11.93
-122.44 -35.78 -20.87
-330.47 -58.43 -94.16
-135.04 -63.41 -19.42
-20.38 -14.20
0.41
-83.82 -32.75 -24.19
-116.41 -35.03 -10.37
-125.15 -32.41 -18.73
-305.34 -55.00 -73.30
-49.07
42.02 -17.46
-129.06 -50.19 -16.85
-90.89 -31.49 -10.83
-131.07 -40.57 -20.50
-37.74
-8.66 -18.10
-47.53
-8.61 -15.25
-17.56
68.42 -22.94
-112.51 -27.07 -15.17
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
Quadratic Terms
βbll
βbqq
βb22
71.20
20.32
0.30
-7.24
3.99
8.98
71.28
17.29
0.28
77.83
8.09
0.29
224.54
2.44
25.24
80.48
21.56
0.23
6.04
4.72
10.83
54.43
4.55
1.24
71.02
13.46
0.26
77.64
7.02
0.23
205.23
5.34
21.49
54.41 -21.59 -0.47
78.13
16.92
0.22
59.78
7.62
2.98
81.57
8.12
-0.66
12.29
1.54
7.32
21.29
4.13
14.91
35.14 -30.75 0.22
70.76
6.00
0.58
Interaction Terms
βbq2
βbl2
βblq
9.56
24.58
2.95
20.08 -41.42 2.69
7.61
23.42
2.78
26.81
36.94
7.75
59.74
58.51 24.03
37.86
35.73
6.56
20.56 -39.26 -1.15
23.52
19.60 19.22
3.59
20.57
2.27
25.16
31.98
7.58
59.27
43.50 14.90
-32.49 35.39
5.32
27.35
31.59
5.34
30.46
5.27
6.35
44.65
35.00
9.70
12.35 -22.80 7.07
18.81 -31.50 -4.21
-58.95 22.74
9.26
38.80
15.11
3.20
R2
.9990
.9990
.9990
.9996
.9996
.9995
.9986
.9979
.9988
.9995
.9997
.9955
.9994
.9991
.9997
.9986
.9988
.9795
.9994
286
Table 95. Coefficients of SH IVs Models for 3-Factor Designs.
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
Dsgn
310
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
CCD
BBD
UNFSD
CCD
SCD
SCD
CCD
CCD
βb0
3.70
11.16
0.23
7.10
16.50
7.80
3.93
10.62
-0.84
0.56
9.13
12.76
0.44
1.76
2.07
18.95
5.58
22.01
5.05
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
rs
–
1
–
–
–
–
1
–
–
–
–
1
–
–
1
2
2
2
2
n0
0
1
1
1
1
1
3
1
3
3
3
1
3
3
3
1
3
1
3
Linear Terms
βbl
βbq
βb2
-2.39
-7.10 -0.40
-17.15 -15.93 -2.69
4.64
-1.30 -0.44
-8.19 -12.72 -0.48
-30.76 -17.85 -6.30
-9.98 -13.98 -0.36
-4.37
-2.83 -3.18
-14.50 -18.39 -0.75
7.70
0.18
-0.52
4.21
-1.58 -0.56
-17.40 -5.14 -6.67
-18.05 -22.46 -0.47
4.06
-1.47 -0.42
1.58
-3.14 -0.86
0.99
-4.01 -0.53
-29.56 -30.61 -2.31
-6.68
-7.11 -2.58
-33.48 -38.43 -0.66
-4.38
-8.96 -0.72
N
10
11
11
11
11
13
13
13
13
13
13
15
15
15
17
17
19
21
23
Interaction Terms
βbq2
βbl2
βblq
14.88 1.40
0
19.79 6.51
0
6.75 1.54
0
18.25 1.65
0
21.04 3.53 1.33
21.67 1.25
0
5.36 7.69
0
23.60 2.14 0.03
3.84 1.81
0
4.99 1.96
0
6.64 4.06 1.42
27.98 1.62
0
5.99 1.44
0
6.48 2.41 0.05
7.64 1.84
0
34.21 6.24
0
9.76 6.97
0
42.98 2.27
0
12.33 2.49
0
Quadratic Terms
βbll
βbqq
βb22
2.18
1.84
0
9.13
5.37
0
-0.85
0.13
0
4.22
4.04
0
19.12 5.06 2.07
5.15
4.21
0
4.55
0.82
0
6.61
6.07 0.04
-2.40 -0.05
0
-0.72
0.36
0
14.07 0.89 2.07
7.89
7.55
0
-0.58
0.20
0
0.41
0.82 0.05
0.62
1.12
0
13.08 10.77
0
4.74
2.32
0
13.45 13.51
0
2.68
2.90
0
R2
.9999
.9993
1.0000
.9995
.9992
.9996
.9999
.9991
1.0000
1.0000
.9998
.9989
1.0000
.9999
.9999
.9985
.9997
.9981
.9995
287
Table 96. Coefficients of WH Dw Models for 4-Factor Designs.
Dsgn
r s n0
416C
– 1
SCD
1 1
416A
– 1
416B
– 1
416C
– 2
SCD
1 3
416A
– 3
416B
– 3
PBCD
1 1
UNFSD – 1
PBCD
1 3
UNFSD – 3
CCD or 1 1
BBD
– 1
SCD
2 1
CCD or 1 3
BBD
– 3
SCD
2 3
PBCD
2 1
PBCD
2 3
CCD
2 1
CCD
2 3
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βb0
βbl
16.24 81.40
9.68 84.96
17.18 72.22
18.64 70.94
18.50 71.48
15.71 64.01
21.08 55.29
20.47 57.88
9.01 90.76
12.44 87.89
14.81 71.66
18.18 68.56
5.25 100.29
25 10.63
27 11.37
81.47
81.89
βbq
-5.28
14.48
1.46
-7.53
-7.59
6.27
-2.78
-6.97
10.36
14.92
2.21
6.17
8.09
βb2
βbll
-4.71 -11.22
4.33 -4.49
-4.08 -6.47
-3.59 -7.35
-3.57 -7.57
6.14 2.70
-2.37 -0.44
-2.70 -2.78
2.12 -6.24
-2.06 -9.24
4.17 0.42
0.25 -2.25
-4.81 -11.41
27.85 26.39 -0.21
-0.51 66.35 -2.61
27 17.35 62.13 16.39
29 7.21 91.79 25.75
31 14.25 72.57 13.92
33 2.34 103.19 24.90
35 10.07 83.58 12.26
Dsgn
r s n0
416C
– 1
SCD
1 1
416A
– 1
416B
– 1
416C
– 2
SCD
1 3
416A
– 3
416B
– 3
PBCD
1 1
UNFSD – 1
PBCD
1 3
UNFSD – 3
CCD or 1 1
BBD
– 1
SCD
2 1
CCD or 1 3
BBD
– 3
SCD
2 3
PBCD
2 1
PBCD
2 3
CCD
2 1
CCD
2 3
βb1
78.76
47.75
64.61
74.03
74.55
42.32
58.03
66.69
56.39
57.65
51.37
52.60
71.43
23.82 2.50
37.70 0.55
34.88 3.24
52.93 -4.49
49.76 -1.82
βbl2
-6.27
-46.05
-14.60
-8.20
-8.06
-45.98
-16.46
-9.78
-34.24
-29.68
-35.58
-31.22
-14.02
βbqq
-1.47
-7.57
-3.44
-0.21
-0.19
-3.57
-1.17
0.03
-7.06
-7.46
-3.13
-3.40
-6.34
βb11
-18.50
-9.59
-17.36
-17.81
-17.88
-8.81
-16.13
-16.55
-10.39
-18.80
-9.76
-17.66
-10.28
βb22
1.28
7.39
2.78
1.56
1.54
7.52
3.07
1.81
4.07
4.97
4.41
5.24
2.70
-0.01
-4.84
-11.19
-2.44
-8.15
-9.82
11.26
3.05
6.90
-3.90
3.07
-9.74
-2.53
-6.04
-11.49
-6.19
-11.69
-6.21
-7.65
-9.57
-9.10
-9.92
-9.52
11.37
7.80
8.06
6.06
6.35
βbq1 βbq2 βb12
R2
4.33 3.89 2.23 .9989
5.09 0.74 5.75 .9976
4.55 3.30 3.02 .9984
4.19 2.78 2.30 .9989
4.18 2.75 2.29 .9989
4.47 -1.85 5.58 .9981
4.15 1.46 3.05 .9986
3.94 1.82 2.34 .9987
5.48 3.20 3.77 .9984
4.38 2.01 4.36 .9981
4.98 0.61 3.78 .9988
3.98 -0.51 4.33 .9987
5.02 3.78 3.16 .9991
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βblq
-4.90
1.58
-3.73
0.68
0.77
11.64
4.40
3.87
-3.46
-6.58
7.65
5.08
-8.97
βbl1
-62.62
-48.97
-50.93
-58.71
-59.12
-43.63
-45.54
-52.74
-54.44
-44.09
-49.68
-40.02
-67.21
25
27
-2.17
3.14
-30.16 -55.23 4.16 -0.61 8.08 .9980
-62.48 -16.52 4.72 1.59 3.20 .9993
27 10.16 -27.51
29 -6.22 -38.39
31 7.05 -35.64
33 -10.88 -51.49
35 3.42 -48.47
-55.82
-47.34
-48.68
-30.19
-32.28
3.71 -3.61 7.85 .9985
4.92 1.61 6.05 .9981
4.49 -1.38 5.97 .9988
5.15 2.64 5.14 .9983
4.81 -0.04 5.11 .9991
288
Table 97. Coefficients of WH Aw Models for 4-Factor Designs.
Dsgn
r s n0
416C
– 1
SCD
1 1
416A
– 1
416B
– 1
416C
– 2
SCD
1 3
416A
– 3
416B
– 3
PBCD
1 1
UNFSD – 1
PBCD
1 3
UNFSD – 3
CCD or 1 1
BBD
– 1
SCD
2 1
CCD or 1 3
BBD
– 3
SCD
2 3
PBCD
2 1
PBCD
2 3
CCD
2 1
CCD
2 3
N
βb0
βbl
βbq
βb1
βb2
βbll
16 5.64
94.40 13.76 76.49 -10.23 -19.00
17 3.54
83.78 37.54 43.97 -9.92 -12.19
17 -0.57 97.19 30.09 65.42 -12.34 -18.51
17 14.81 71.72 -4.60 74.41 -8.47 -9.82
17 15.05 71.61 -5.29 74.82 -8.38 -9.78
19 16.82 49.04
9.37 39.81 -1.62
3.00
19 14.39 59.87
0.76 61.18 -7.48 -3.33
19 17.34 56.15 -11.14 70.04 -5.45 -2.97
21 -5.11 110.20 46.20 49.41 -11.29 -18.66
21 -6.77 117.48 54.06 52.78 -17.50 -24.60
23 12.54 68.64
9.73 46.14 -2.32 -1.32
23 13.97 70.66 13.83 49.56 -10.38 -5.47
25 -23.38 143.73 61.15 70.61 -13.80 -31.41
25
27
-3.67
1.50
βbqq
-7.58
-19.83
-14.56
-0.92
-0.66
-6.16
-2.76
1.91
-23.95
-24.64
-7.92
-8.09
-25.35
βb11
βb22
-13.79 -0.45
-5.06 5.07
-13.69 0.53
-14.45 -0.13
-14.52 -0.14
-4.88 7.33
-14.06 1.48
-14.81 0.57
-7.54 0.97
-13.96 2.02
-7.75 2.20
-14.10 3.14
-6.92 0.30
90.48
93.39
58.26
15.41
24.30 -19.31 -12.05 -26.30
67.69 -10.71 -11.94 -8.38
-3.33
-7.51
7.27
0.81
27 14.77 50.37
29 -13.90 120.57
31 8.24
74.29
33 -36.95 162.59
35 -7.26 106.47
22.89
70.64
28.55
94.88
42.66
22.18
31.76
29.74
52.51
50.19
-3.21
-6.05
-6.03
-6.25
-6.45
8.90
3.87
4.93
2.58
3.24
Dsgn
r s n0
416C
– 1
SCD
1 1
416A
– 1
416B
– 1
416C
– 2
SCD
1 3
416A
– 3
416B
– 3
PBCD
1 1
UNFSD – 1
PBCD
1 3
UNFSD – 3
CCD or 1 1
BBD
– 1
SCD
2 1
CCD or 1 3
BBD
– 3
SCD
2 3
PBCD
2 1
PBCD
2 3
CCD
2 1
CCD
2 3
-11.51 4.77 -10.88
-17.93 -21.05 -32.44
-9.73 -2.11 -14.65
-19.42 -36.68 -39.27
-15.72 -15.05 -19.25
βbl2
8.31
-24.81
2.19
4.41
4.43
-35.06
-5.49
-1.50
-17.17
-7.59
-27.83
-16.32
5.13
βbq1
0.28
3.76
2.02
1.15
1.14
2.55
2.25
1.82
4.68
2.75
3.70
2.39
1.08
βbq2
8.45
15.27
10.95
7.19
7.11
5.86
6.36
4.53
19.60
16.84
11.02
9.79
11.75
βb12
1.30
5.16
2.15
1.50
1.49
5.56
2.55
1.78
2.66
3.74
3.17
4.21
2.19
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βblq
-55.09
-46.05
-58.39
-25.82
-25.04
-9.00
-15.12
-5.79
-68.72
-78.59
-20.68
-26.92
-94.15
βbl1
-62.47
-49.83
-53.58
-60.37
-60.68
-45.31
-49.44
-56.32
-49.86
-43.17
-46.19
-39.93
-66.68
25
27
-61.47
-37.25
-31.79 -23.30 2.76 16.85 7.65 .9827
-63.81 -0.44 1.73 9.10 2.51 .9989
R2
.9995
.9906
.9984
.9994
.9994
.9917
.9984
.9990
.9945
.9959
.9950
.9964
.9983
27 -21.23 -29.38 -31.57 1.92 8.46 7.87 .9897
29 -82.01 -35.33 -22.88 4.03 21.69 5.33 .9881
31 -34.04 -33.00 -31.36 3.08 13.57 5.71 .9913
33 -110.53 -52.62 -4.50 2.88 16.41 4.36 .9939
35 -53.87 -50.19 -9.55 2.62 12.73 4.68 .9949
289
Table 98. Coefficients of WH IVw Models for 4-Factor Designs.
Dsgn
r s n0
416C
– 1
SCD
1 1
416A
– 1
416B
– 1
416C
– 2
SCD
1 3
416A
– 3
416B
– 3
PBCD
1 1
UNFSD – 1
PBCD
1 3
UNFSD – 3
CCD or 1 1
BBD
– 1
SCD
2 1
CCD or 1 3
BBD
– 3
SCD
2 3
PBCD
2 1
PBCD
2 3
CCD
2 1
CCD
2 3
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βb0
13.62
22.97
15.04
4.25
4.06
6.35
2.94
1.36
27.19
26.78
6.10
5.61
31.85
βbl
-16.14
-30.88
-17.87
0.02
0.35
-2.84
3.11
5.78
-37.96
-37.29
-2.62
-1.77
-45.94
βbq
-23.34
-35.55
-25.12
-6.57
-6.23
-6.78
-3.58
-0.54
-44.58
-44.77
-8.46
-8.67
-53.60
βb1
3.42
4.69
3.50
3.07
3.06
4.69
3.14
2.66
4.56
4.18
4.43
4.01
3.57
βb2
-1.10
-5.93
-1.36
-1.18
-1.17
-6.63
-1.53
-1.32
-3.74
-2.20
-4.09
-2.31
-1.16
βbll
6.37
11.72
6.87
0.34
0.23
1.21
-0.96
-1.82
14.12
13.83
0.87
0.56
17.38
βbqq
8.26
12.81
8.99
2.25
2.13
2.38
1.26
0.19
16.06
16.05
2.95
2.96
19.30
βb11
-0.40
-0.03
-0.24
-0.16
-0.16
0.16
0.07
0.12
-0.21
-0.23
-0.01
0
-0.44
βb22
0
0.36
0
0
0
0.41
0
0
0.36
0.09
0.39
0.11
0
25 38.74 -55.76 -62.30 5.74 -4.93 20.45 22.66 -0.13 0.22
27 6.36 -3.41 -10.13 3.28 -1.25 1.47 3.52 -0.24
0
27
29
31
33
35
10.56
44.60
11.59
50.35
12.51
Dsgn
r s n0
416C
– 1
SCD
1 1
416A
– 1
416B
– 1
416C
– 2
SCD
1 3
416A
– 3
416B
– 3
PBCD
1 1
UNFSD – 1
PBCD
1 3
UNFSD – 3
CCD or 1 1
BBD
– 1
SCD
2 1
CCD or 1 3
BBD
– 3
SCD
2 3
PBCD
2 1
PBCD
2 3
CCD
2 1
CCD
2 3
N
16
17
17
17
17
19
19
19
21
21
23
23
25
-9.24
-65.63
-11.28
-75.46
-13.28
-15.24
-72.84
-17.83
-83.38
-20.42
5.87
5.27
5.31
4.05
3.97
-5.32 2.95 5.46 -0.03 0.24
-4.14 24.12 26.50 -0.14 0.41
-4.42 3.68 6.38 -0.03 0.43
-1.53 28.21 30.33 -0.37
0
-1.63 4.86 7.30 -0.26
0
βblq
31.58
38.23
31.26
13.80
13.43
10.42
8.91
6.11
47.99
48.16
13.09
13.29
57.76
βbq1 βbq2 βb12
βbl2
βbl1
R2
-3.00 3.39 0.57
0
-0.14 .9991
-3.80 12.01 0.41
0
-0.22 .9975
-3.14 4.19 0.44
0
-0.17 .9986
-2.77 3.62 0.30
0
-0.15 .9998
-2.76 3.60 0.30
0
-0.15 .9999
-3.77 13.42 0.19
0
-0.25 .9993
-2.93 4.69 0.12
0
-0.20 .9999
-2.51 4.05 0.03
0
-0.17 1.0000
-4.12 7.94 0.53
0
0.02 .9970
-3.88 5.73 0.67 0.17 -0.15 .9969
-4.03 8.69 0.26
0
0.02 .9996
-3.77 6.08 0.34 0.14 -0.16 .9996
-3.12 3.64 0.64
0
-0.16 .9967
25 61.17 -4.75 11.52 0.37
27 15.77 -2.87 3.94 0.33
0
0
-0.33
-0.18
.9949
.9996
27
29
31
33
35
0
0
0
0
0
-0.35
-0.03
-0.03
-0.22
-0.23
.9988
.9944
.9986
.9940
.9984
17.83
71.54
20.91
81.91
23.98
-4.85 12.44 0.24
-4.75 9.17 0.44
-4.80 9.81 0.29
-3.49 4.81 0.51
-3.43 5.10 0.34
290
Table 99. Coefficients of SH Ds Models for 4-Factor Designs.
Dsgn
416C
SCD
416A
416B
416C
SCD
416A
416B
PBCD
UNFSD
PBCD
UNFSD
CCD or
BBD
SCD
CCD or
BBD
SCD
PBCD
PBCD
CCD
CCD
rs
–
1
–
–
–
1
–
–
1
–
1
–
1
–
2
1
–
2
2
2
2
2
n0
1
1
1
1
2
3
3
3
1
1
3
3
1
1
1
3
3
3
1
3
1
3
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βb0
71.91
53.50
68.77
72.14
72.21
56.14
69.01
70.12
59.40
62.36
62.13
65.20
66.74
βbl
-53.73
3.58
-48.55
-58.05
-58.02
-10.20
-55.87
-60.73
-17.36
-25.95
-29.09
-37.83
-39.54
βbq
-26.56
-12.57
-21.96
-28.68
-28.65
-19.10
-25.29
-27.53
-15.35
-11.59
-22.40
-19.27
-15.80
βb2
-13.13
-8.92
-13.04
-11.79
-11.94
-6.38
-10.85
-10.46
-8.32
-12.16
-5.84
-9.40
-14.18
βbll
74.18
34.42
67.91
73.78
73.87
37.60
67.51
71.28
52.49
59.72
54.69
61.77
69.00
βbqq
0.05
-6.90
-2.13
1.13
1.05
-3.03
-0.11
1.18
-5.98
-6.46
-2.25
-2.60
-5.00
βb22
1.91
9.14
3.60
2.20
2.03
9.23
3.89
2.45
5.48
6.15
5.80
6.41
3.56
25
27
53.93
69.83
5.04
-50.32
0.21
-23.74
-16.74
-11.66
33.61
70.79
-11.09
-1.32
13.86
3.92
27
29
31
33
35
58.30
54.91
59.73
59.91
65.30
-9.60
-3.68
-17.92
-20.59
-34.68
-9.93
-1.85
-12.63
-2.12
-13.91
-13.35
-12.84
-9.70
-16.30
-13.30
38.05
44.27
48.40
57.77
61.68
-5.97
-10.99
-5.78
-10.86
-5.50
13.90
9.88
10.10
7.50
7.79
Dsgn
416C
SCD
416A
416B
416C
SCD
416A
416B
PBCD
UNFSD
PBCD
UNFSD
CCD or
BBD
SCD
CCD or
BBD
SCD
PBCD
PBCD
CCD
CCD
rs
–
1
–
–
–
1
–
–
1
–
1
–
1
–
2
1
–
2
2
2
2
2
n0
1
1
1
1
2
3
3
3
1
1
3
3
1
1
1
3
3
3
1
3
1
3
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βblq
14.83
27.15
18.21
20.44
20.27
35.75
25.66
23.24
21.06
18.42
31.22
29.20
13.55
βbl2
-0.86
-37.42
-8.71
-2.89
-2.45
-37.71
-10.80
-4.68
-27.17
-22.82
-28.70
-24.57
-7.82
βbq2
7.32
5.15
6.96
6.09
6.34
2.11
4.84
4.92
7.14
6.00
4.27
6.41
7.52
R2
.9999
.9997
.9999
.9999
.9999
.9997
.9999
.9999
.9998
.9998
.9999
.9999
.9998
25
27
24.26
25.19
-43.64
-10.45
4.00
5.11
.9994
.9999
27
29
31
33
35
35.36
20.26
32.54
14.92
28.49
-44.60
-37.80
-39.37
-21.92
-24.17
0.61
5.98
2.71
6.83
7.79
.9996
.9996
.9998
.9996
.9998
291
Table 100. Coefficients of SH As Models for 4-Factor Designs.
Dsgn
416C
SCD
416A
416B
416C
SCD
416A
416B
PBCD
UNFSD
PBCD
UNFSD
CCD or
BBD
SCD
CCD or
BBD
SCD
PBCD
PBCD
CCD
CCD
rs
–
1
–
–
–
1
–
–
1
–
1
–
1
–
2
1
–
2
2
2
2
2
n0
1
1
1
1
2
3
3
3
1
1
3
3
1
1
1
3
3
3
1
3
1
3
Dsgn
416C
SCD
416A
416B
416C
SCD
416A
416B
PBCD
UNFSD
PBCD
UNFSD
CCD or
BBD
SCD
CCD or
BBD
SCD
PBCD
PBCD
CCD
CCD
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βb0
38.93
30.48
36.06
51.29
36.30
41.33
52.70
55.83
25.46
28.51
43.01
49.97
18.97
βbl
7.37
57.59
9.43
-21.54
6.55
27.26
-30.72
-39.98
51.25
39.05
10.36
-9.12
47.95
βbq
5.58
17.46
16.51
-14.82
10.93
-8.67
-13.86
-23.55
28.47
38.58
-7.69
-1.55
46.79
βb2
-19.02
-32.16
-23.12
-17.18
-19.29
-23.21
-17.94
-13.88
-25.19
-31.97
-16.18
-24.50
-25.07
βbll
40.15
-9.89
37.63
52.28
40.46
2.74
53.73
59.80
10.21
22.95
27.01
42.50
24.51
βbqq
-4.86
-19.47
-12.04
0.94
-8.01
-5.90
-1.40
3.03
-22.05
-22.61
-6.76
-6.98
-22.26
βb22
-0.01
7.64
1.34
0.40
-0.01
10.32
2.46
1.21
2.82
3.44
4.31
4.81
1.03
25
27
38.78
46.18
33.54
-7.57
39.75
0.48
-51.60
-21.85
5.09
46.60
-26.53
-6.50
12.17
1.73
27
29
31
33
35
54.84
24.33
45.28
11.50
41.01
-2.55
54.88
10.99
64.18
8.18
6.28
52.14
11.09
76.59
25.08
-43.06
-40.53
-32.06
-37.62
-33.69
19.92
6.91
24.61
15.21
36.73
-11.08
-31.76
-14.23
-37.28
-17.91
14.18
7.28
8.61
4.48
5.34
rs
–
1
–
–
–
1
–
–
1
–
1
–
1
–
2
1
–
2
2
2
2
2
n0
1
1
1
1
2
3
3
3
1
1
3
3
1
1
1
3
3
3
1
3
1
3
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βblq
-50.15
-29.70
-47.81
-17.89
-49.03
6.45
-2.39
5.06
-52.91
-65.68
-4.78
-13.22
-83.54
βbl2
13.56
-11.83
8.81
9.71
13.23
-21.95
1.19
3.74
-8.15
1.83
-18.61
-6.78
12.06
βbq2
12.40
22.97
15.51
10.90
12.90
11.52
10.29
7.91
24.60
22.20
15.42
14.22
16.60
R2
.9992
.9950
.9987
.9997
.9991
.9986
.9997
.9998
.9964
.9967
.9991
.9992
.9978
25
27
-45.66
-24.75
-2.32
6.65
24.31
13.36
.9917
.9994
27
29
31
33
35
-6.61
-65.24
-18.08
-95.27
-38.59
-10.71
-7.33
-15.74
7.62
2.70
14.22
27.87
18.85
22.49
18.02
.9974
.9923
.9973
.9943
.9977
292
Table 101. Coefficients of SH IVs Models for 4-Factor Designs.
Dsgn
416C
SCD
416A
416B
416C
SCD
416A
416B
PBCD
UNFSD
PBCD
UNFSD
CCD or
BBD
SCD
CCD or
BBD
SCD
PBCD
PBCD
CCD
CCD
rs
–
1
–
–
–
1
–
–
1
–
1
–
1
–
2
1
–
2
2
2
2
2
n0
1
1
1
1
2
3
3
3
1
1
3
3
1
1
1
3
3
3
1
3
1
3
Dsgn
416C
SCD
416A
416B
416C
SCD
416A
416B
PBCD
UNFSD
PBCD
UNFSD
CCD or
BBD
SCD
CCD or
BBD
SCD
PBCD
PBCD
CCD
CCD
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βb0
14.66
26.81
15.50
4.19
14.85
9.69
2.04
0.06
29.50
27.42
7.51
5.87
32.96
βbl
-19.20
-42.94
-19.78
-0.38
-19.23
-13.95
4.41
8.18
-44.80
-40.36
-7.34
-3.31
-49.45
βbq
-24.09
-35.86
-25.53
-6.98
-24.42
-6.88
-3.55
-0.39
-45.00
-44.86
-8.67
-9.03
-54.14
βb2
-0.64
-5.30
-0.79
-0.68
-0.56
-5.92
-0.89
-0.77
-3.20
-1.22
-3.51
-1.71
-0.69
βbll
8.45
20.25
8.38
0.88
8.42
9.36
-1.30
-2.89
18.84
16.01
4.40
1.93
19.84
βbqq
7.75
12.40
8.53
2.02
7.93
2.21
1.12
0.16
15.54
15.34
2.73
2.74
18.68
βb22
0
0.36
0
0
0
0.41
0
0
0.30
0
0.32
0.13
0
25
27
41.06
6.28
-64.13
-4.01
-62.55
-10.45
-3.99
-0.75
26.85
2.25
22.27
3.26
0.22
0
27
29
31
33
35
12.50
46.50
12.98
50.99
12.49
-16.86
-71.84
-16.34
-78.17
-14.41
-15.40
-73.15
-18.04
-83.74
-20.69
-4.32
-3.40
-3.64
-0.92
-0.97
9.00
28.66
7.59
30.40
6.15
5.22
26.03
6.10
29.80
6.98
0.24
0.33
0.35
0
0
rs
–
1
–
–
–
1
–
–
1
–
1
–
1
–
2
1
–
2
2
2
2
2
n0
1
1
1
1
2
3
3
3
1
1
3
3
1
1
1
3
3
3
1
3
1
3
N
16
17
17
17
17
19
19
19
21
21
23
23
25
βblq
32.91
39.00
32.16
14.47
32.91
10.71
9.01
5.97
49.01
49.35
13.55
13.92
59.01
βbl2
2.95
11.41
3.65
3.16
2.96
12.75
4.08
3.53
7.51
5.17
8.23
5.50
3.20
βbq2
0
0
0
0
-0.09
0
0
0
0
-0.04
0
0.16
0
R2
.9990
.9977
.9986
.9998
.9989
.9997
.9999
1.0000
.9971
.9970
.9997
.9997
.9966
25
27
61,86
16.39
10.63
3.46
0
0
.9954
.9996
27
29
31
33
35
18.25
72.36
21.43
82.86
24.61
11.48
8.57
9.16
4.23
4.48
0
0
0
0
0
.9992
.9947
.9989
.9941
.9985
293
Table 102. Full Model Optimality Criteria for Small Response Surface Designs.
Designs
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
rs
1
–
–
–
–
1
–
–
–
–
n0
1
1
1
1
1
3
1
3
3
3
N
11
11
11
11
13
13
13
13
13
13
D
59.0785
60.6397
67.6003
70.9973
69.5854
55.7945
69.5913
55.0194
63.8425
67.0507
A
28.1641
45.7457
37.4090
37.8798
35.5007
32.8879
34.0475
47.1490
50.6899
50.9072
4
3
2
1
2
5
1
6
4
3
G
32.7923
45.0198
78.6243
90.9091
76.9140
27.7473
76.9231
38.9577
69.0153
77.4084
4
1
3
2
4
6
5
3
2
1
IV
17.0840
10.6710
14.4549
14.4290
16.3622
12.1843
16.3622
9.6415
9.2126
9.2126
4
3
2
1
3
6
2
5
4
1
4
1
3
2
5.5
4
5.5
3
1.5
1.5
Note: Italicized values indicates rank within column and design size.
Table 103. Weighted Optimality Criteria for Small Response Surface Designs Across
WH Models with pl = .9, p1 = .4, p2 = .5, and pq = .7.
Designs
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
rs
1
–
–
–
–
1
–
–
–
–
n0
1
1
1
1
1
3
1
3
3
3
N
11
11
11
11
13
13
13
13
13
13
Dw
69.9093
62.0937
71.3373
76.3152
71.9468
64.0253
75.7160
56.4586
65.7684
70.0299
3
4
2
1
2
5
1
6
4
3
Aw
43.0961
48.4534
50.6298
53.9327
48.8510
43.3953
51.7501
48.1721
54.3365
56.6074
4
3
2
1
5
6
3
4
2
1
Gw
43.0227
38.5379
60.6570
69.5248
57.3411
37.1165
62.3268
33.9802
53.0084
60.3640
3
4
2
1
3
5
1
6
4
2
IVw
8.9139
7.7151
8.3490
7.9384
9.4461
7.6786
8.6074
7.2912
6.6877
6.4029
Note: Italicized values indicate rank within column and design size.
4
1
3
2
6
4
5
3
2
1
294
Table 104. Weighted Optimality Criteria for Small Response Surface Designs Across
SH Models with pl = .8, p2 = .5, and pq = .5.
Designs
SCD
310
311A
311B
BBD
SCD
UNFSD
310
311A
311B
rs
1
–
–
–
–
1
–
–
–
–
n0
1
1
1
1
1
3
1
3
3
3
N
11
11
11
11
13
13
13
13
13
13
Ds
64.1062
56.3093
64.0995
67.9600
65.1038
57.9465
67.8953
51.0743
58.3290
61.5502
2
4
3
1
2
5
1
6
4
3
As
46.5929
45.6463
50.8736
54.1921
49.4432
43.5300
53.2593
43.4713
49.5156
51.8310
3
4
2
1
4
5
1
6
3
2
Gs
45.7266
38.9048
54.7577
61.0947
52.3499
39.2916
57.8345
34.5094
47.8368
52.6207
3
4
2
1
3
5
1
6
4
2
IVs
5.9081
5.8809
5.6721
5.3346
6.2114
5.7934
5.5789
5.8632
5.2659
5.0286
Note: Italicized values indicate rank within column and design size.
4
3
2
1
6
4
3
5
2
1
295
CHAPTER 6
CONCLUSION AND FUTURE RESEARCH
Theoretical and computational details of evaluating optimality criteria for reduced
models for response surface designs in a spherical design region have been described in
this dissertation. For 3 design variables, robustness results were presented for CCDs,
BBDs, SCDs, UNFSDs, and hybrid 310, 311A, and 311B designs and for 4 design
variables, robustness results were presented for CCDs, BBDs, SCDs, PBCDs, UNFSDs, and hybrid 416A, 416B, and 416C designs. These results are based on the four
optimality criteria (D, A, G, and IV -criteria) of the full second-order model and sets
of reduced models. In addition, the design optimality criteria were compared across
these response surface designs. Numerous tables and figures have been used to illustrate the potential of this research. The development of weighted design optimality
criteria for the set of weak and strong heredity reduced models using specified pl , pq ,
p1 , and p2 values is a significant contribution to experimental design research.
For future research, it would be useful to expand the robustness study to 5 factor
response surface designs in a spherical design region across reduced models. However,
in a spherical region, it would currently require impractically large amounts of computational time to calculate the G and IV criteria values. The computation time for
296
the G and IV criteria dramatically increases if there are five design variables because
both the number of reduced models and the dimensions of X0X increase.
Because of the existence of cases of D-efficiencies > 100% based on D−efficiency =
0
100 |X X|
N
1/p
, it would be worthwhile to study the ratio of |X0X|1/p to the theoretical
optimal design criteria for spherical region instead of N . Then, the efficiency would
be bounded above by 100%. Finding theoretically optimum values requires finding optimal weights to assign to points in the sphere. This will be computationally
demanding given that different optimal weights have to be found for each reduced
model.
Another topic concerns the robustness of computer-generated designs in a spherical region. For example, how do designs generated by the SAS OPTEX procedure
perform across reduced models for the four optimality criteria? The impact of blocking of central composite designs or Box-Behnken designs in a spherical region on
design robustness is another potential research problem.
297
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301
APPENDICES
302
APPENDIX A
Tables of D, A, G, and IV Criteria Values for 3 and 4 Factor Response Surface
Designs
303
Table of Criteria Values for CCDs (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
D
71.1296
75.2769
73.4421
75.5678
78.5904
78.5904
76.4387
73.2809
74.1572
79.4248
79.4248
83.0658
83.0658
80.4718
73.3227
76.6839
78.3452
84.8755
84.8755
84.8755
89.4320
71.3687
77.3175
77.3175
47.4461
81.4683
84.6087
93.1407
75.6502
51.1946
83.2787
83.2787
54.5101
74.4609
94.9553
55.0391
82.5587
62.0621
71.7804
48.7581
72.8825
52.8344
31.0047
55.7766
rs = 1, n0 = 1
A
G
32.4011 66.6667
56.5984 91.5819
31.0471 60.0000
66.6141 81.5622
57.0348 82.9391
57.0348 82.9391
29.5059 53.3333
53.9445 83.2212
71.1428 72.4269
69.0712 74.4003
69.0712 74.8872
57.6060 72.8186
57.6060 72.5717
27.7358 46.6667
63.9970 74.8872
54.0330 72.8186
75.3356 71.8859
72.6439 64.1890
72.6439 64.1890
72.6439 65.4452
58.3856 65.2959
68.4314 70.1395
66.2031 64.1890
66.2031 64.1890
33.4814 62.4159
54.1513 65.2959
82.1103 62.2645
78.3150 54.5377
72.5383 62.2645
41.4836 53.4909
69.5602 54.5377
69.5602 54.5377
36.4919 54.4132
62.5660 54.5377
94.9134 94.9134
47.4320 49.8116
79.7144 57.8058
51.0581 43.6301
68.7113 57.3162
38.2930 43.6301
71.1851 71.1851
43.9500 43.3543
19.1536 32.7226
47.4567 47.4567
IV
17.5556
7.9429
17.1659
5.5904
7.5532
7.5532
17.1659
7.3243
4.3673
5.2007
5.2007
7.5532
7.5532
16.7762
4.9718
6.9346
3.9776
5.2007
5.2007
5.2007
7.1635
3.7487
4.5821
4.5821
6.1478
6.5449
3.9776
4.8110
3.3590
4.5312
4.1924
4.1924
5.5512
3.9635
3.5879
3.5261
2.9693
3.9346
2.7404
3.7735
2.9295
2.7685
1.5022
1.0357
D
67.3113
72.3330
71.7061
69.8613
78.3690
78.3690
77.6054
69.8882
64.7568
76.1833
76.1833
86.8756
86.8756
85.9088
66.8365
76.2169
70.7437
85.5116
85.5116
85.5116
99.6710
60.7249
73.4014
73.4014
49.8264
85.5555
80.0666
100.5208
66.6602
51.4472
83.6895
83.6895
61.8321
69.6764
96.4043
51.1121
76.6682
67.9250
60.9725
48.8119
73.8702
47.5478
34.7356
56.3426
rs = 2, n0 = 1
A
G
24.6659 47.6190
53.3857 75.6883
23.7362 42.8571
58.6937 67.3737
56.2057 75.1255
56.2057 75.3333
22.6682 38.0952
50.6060 72.9956
58.2115 59.4183
63.6070 67.1315
63.6070 67.6144
60.3009 65.7348
60.3009 65.9166
21.4285 33.3333
55.6437 64.6264
53.0970 65.7348
63.8290 60.3914
71.5984 57.9552
71.5984 57.9552
71.5984 58.5039
66.7895 78.1155
54.6693 56.1138
60.2710 57.9552
60.2710 57.9552
34.2726 56.3441
56.8268 62.9516
73.7995 52.6739
86.8799 65.1149
59.8797 52.3256
39.3288 48.2960
68.2125 54.5412
68.2125 53.9695
43.1018 65.0963
56.1482 53.9695
96.3830 96.3830
39.7758 42.1391
69.8681 47.5962
58.7624 52.0919
54.7942 46.6682
35.9122 43.1756
72.2872 72.2872
35.3719 35.6972
24.0233 39.0689
48.1915 48.1915
IV
23.1576
7.5637
22.6120
5.5634
7.0181
7.0181
22.6120
6.9574
4.6420
5.0178
5.0178
7.0181
7.0181
22.0664
4.9571
6.4118
4.0964
5.0178
5.0178
5.0178
6.4725
4.0357
4.4115
4.4115
5.8342
5.8662
4.0964
4.4722
3.4901
4.4839
3.8660
3.8660
4.9990
3.8053
3.5508
3.7345
2.9445
3.6486
2.8839
3.7414
2.8992
2.9920
1.3626
1.0250
Table of Criteria Values for CCDs (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
D
70.0495
69.8655
73.2151
68.6864
73.4033
73.4033
77.3741
68.4443
66.6133
72.4990
72.4990
78.2164
78.2164
83.0704
66.9289
72.2070
70.5853
77.9137
77.9137
77.9137
85.1283
64.2998
70.9756
70.9756
45.1628
77.5478
76.5472
86.1804
68.4422
47.3689
77.0554
77.0554
52.6800
68.8966
rs = 1, n0 = 3
A
G
50.3343 89.2039
57.6049 81.2868
50.7266 81.7853
61.5882 72.2636
59.2650 73.6238
59.2650 73.6238
51.2257 72.6981
55.5194 74.1049
63.5327 64.6936
64.4301 65.9554
64.4301 66.7023
61.5456 64.8418
61.5456 64.4208
51.8820 66.5153
59.4471 66.3446
56.9830 64.8418
67.4692 64.3356
68.6539 57.1734
68.6539 57.1734
68.6539 58.7775
64.8741 58.1678
61.2018 62.7508
62.1750 57.1734
62.1750 57.1734
34.9799 55.5787
59.0588 58.1678
73.8776 55.7561
75.5916 48.9812
65.1158 55.7561
38.8674 47.6445
66.4436 48.9812
66.4436 48.9812
40.4157 48.4731
59.2708 48.9812
IV
9.4271
7.0983
8.9854
5.7140
6.6567
6.6567
8.9854
6.3972
4.7187
5.2724
5.2724
6.6567
6.6567
8.5437
5.0130
5.9556
4.2770
5.2724
5.2724
5.2724
6.2150
4.0176
4.5713
4.5713
5.7477
5.5139
4.2770
4.8307
3.5759
4.6877
4.1296
4.1296
5.0715
3.8702
D
68.5948
68.3170
73.9710
65.0362
74.3318
74.3318
81.2875
66.2879
59.8992
71.1186
71.1186
82.8497
82.8497
91.7676
62.3931
72.6850
65.5789
80.1222
80.1222
80.1222
95.7442
56.2916
68.7753
68.7753
47.8634
82.1848
74.4465
94.6741
61.9811
48.4548
78.8218
78.8218
60.0023
65.6238
rs = 2, n0 = 3
A
G
41.7389 76.4730
52.7239 69.4505
42.6875 77.0282
54.8204 61.7431
56.3576 68.7429
56.3576 69.1659
43.9356 68.4695
50.2512 66.6596
53.5368 54.6549
59.7367 61.3494
59.7367 61.7523
61.8371 60.1500
61.8371 60.5202
45.6518 83.0028
52.0715 59.0581
53.6603 60.1500
58.8228 55.6269
67.8495 52.9306
67.8495 52.9306
67.8495 54.7707
71.0474 71.5394
50.3141 51.6545
56.7749 52.9306
56.7749 52.9306
34.5180 51.5572
58.9969 57.8347
68.2583 48.5382
83.7788 59.7134
55.2481 48.2143
36.9896 44.1088
64.9937 51.2150
64.9937 49.2921
45.8148 59.6161
53.0897 49.2921
IV
11.1987
7.1117
10.6012
5.7222
6.5142
6.5142
10.6012
6.4478
4.9191
5.1247
5.1247
6.5142
6.5142
10.0036
5.0582
5.8502
4.3215
5.1247
5.1247
5.1247
5.9166
4.2551
4.4607
4.4607
5.6337
5.2527
4.3215
4.5271
3.6576
4.6419
3.8631
3.8631
4.7189
3.7967
304
Table of Criteria Values for CCDs (K = 3)
Dsgn
35
36
37
38
39
40
41
42
43
44
p
4
4
4
4
4
4
3
3
3
2
dv
3
2
3
2
3
2
2
2
1
1
l
3
2
2
2
1
1
2
1
1
1
c
0
1
1
0
2
1
0
1
0
0
q
0
0
0
1
0
1
0
0
1
0
D
86.4472
50.1075
75.1613
58.1102
65.3488
45.6533
67.0478
48.6047
29.6106
52.3930
rs = 1, n0 = 3
A
G
86.1521 86.1521
42.4439 44.6049
72.0249 51.8873
49.4866 39.1850
61.8782 51.4403
36.2228 39.1850
64.6141 64.6141
39.4595 38.9154
18.7349 29.3887
43.0761 43.0761
IV
3.8353
3.8077
3.1342
4.0115
2.8748
3.8290
3.1315
2.9490
1.5954
1.1072
D
90.0462
47.7411
71.6117
64.4727
56.9512
46.3310
69.5233
44.7499
33.3995
53.8372
rs = 2, n0 = 3
A
G
89.8852 89.8852
36.6338 38.8306
64.7765 43.9118
56.7491 47.7708
50.6327 43.0468
33.9206 39.4337
67.4139 67.4139
32.6306 32.9338
23.2863 35.8281
44.9426 44.9426
IV
3.7240
3.9554
3.0600
3.7271
2.9936
3.8287
3.0406
3.1422
1.4281
1.0750
Table of Criteria Values for BBDs (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
D
69.5854
74.0927
69.6260
78.1322
74.7252
74.7252
69.6767
72.0853
82.5316
79.4955
79.4955
75.5464
75.5464
69.7419
76.2940
72.5040
84.9865
81.3504
81.3504
81.3504
76.6554
81.0070
77.5411
77.5411
42.5527
73.0660
88.5466
84.0201
83.5947
48.7619
79.3214
79.3214
45.4047
74.8854
94.1683
58.4214
87.6321
54.7125
81.5497
46.0068
72.3452
57.4193
26.3055
55.4680
n0 = 1
A
G
35.5007 76.9140
52.0740 71.7850
33.6778 69.2226
66.3230 64.7679
50.5099 63.8089
50.5099 63.8089
31.6465 61.5312
49.3836 63.8089
81.5456 82.8428
65.9286 56.6719
65.9286 56.6719
48.6319 55.8328
48.6319 55.8328
29.3691 53.8398
63.7594 56.6719
47.4413 55.8328
84.0394 80.7104
65.4100 48.5759
65.4100 48.5759
65.4100 48.5759
46.3349 47.8567
79.9921 80.3404
62.9318 48.5759
62.9318 48.5759
28.0911 47.8567
45.0774 47.8567
87.7982 69.5383
64.6976 40.4799
82.5614 69.5383
37.5200 40.4799
61.8086 40.4799
61.8086 40.4799
27.6112 39.8806
59.1666 40.4799
94.1124 94.1124
53.3286 55.6306
86.7404 65.9266
39.6991 32.3839
80.4394 65.9266
34.1848 32.3839
70.5843 70.5843
51.0590 49.4450
13.1854 24.2880
47.0562 47.0562
IV
16.3622
9.5395
16.0619
6.4501
9.2392
9.2392
15.7616
8.9139
4.5094
6.1498
6.1498
8.9389
8.9389
15.4614
5.8245
8.6137
4.2091
5.8495
5.8495
5.8495
8.6387
3.8839
5.5243
5.5243
7.2010
8.3134
3.9088
5.5493
3.5836
5.0101
5.2240
5.2240
6.7413
4.8988
3.6086
3.4061
3.2833
4.5504
2.9581
4.2440
2.9464
2.6400
1.7985
1.0417
D
67.3104
68.7859
68.1768
70.6233
69.9721
69.9721
69.2755
67.5001
73.0047
72.2887
72.2887
71.5274
71.5274
70.7142
69.3774
68.6468
75.4328
74.5704
74.5704
74.5704
73.6550
71.9006
71.0786
71.0786
40.8871
70.2061
78.9684
77.8863
74.5522
45.2021
73.5306
73.5306
44.5371
69.4185
84.5851
52.4761
78.7141
51.5787
73.2506
43.3717
65.7624
52.1946
25.5040
51.6379
n0 = 3
A
G
52.1694 66.6588
56.9523 63.6653
51.4243 59.9929
63.0927 59.7251
56.5938 56.5914
56.5938 56.5914
50.5224 53.3270
54.9730 56.5914
71.7879 72.9481
63.5620 52.2595
63.5620 52.2595
56.1396 49.5175
56.1396 49.5175
49.4083 46.6611
61.2442 52.2595
54.3237 49.5175
74.2202 71.2265
64.1988 44.7938
64.1988 44.7938
64.1988 44.7938
55.5451 42.4435
70.5812 70.8941
61.4580 44.7938
61.4580 44.7938
31.5121 42.4435
53.4815 42.4435
77.9160 61.4052
65.1119 37.3282
73.1640 61.4052
36.5978 37.3282
61.7598 37.3282
61.7598 37.3282
32.7015 35.3696
58.7359 37.3282
84.2057 84.2057
47.0545 49.1242
77.4133 58.4202
40.3571 29.8625
71.6349 58.4202
33.9360 29.8625
63.1542 63.1542
45.2787 43.8152
13.6570 22.3969
42.1028 42.1028
IV
9.2957
7.5925
8.9492
6.0336
7.2460
7.2460
8.9492
6.8707
4.5902
5.6871
5.6871
7.2460
7.2460
8.6028
5.3119
6.5243
4.2437
5.6871
5.6871
5.6871
6.8996
3.8684
4.9654
4.9654
6.3507
6.1778
4.2437
5.3407
3.5220
5.0132
4.6189
4.6189
5.8203
4.2437
3.8973
3.7125
3.1755
4.4828
2.8002
4.1292
3.1821
2.8286
1.8752
1.1250
rs = 2, n0 = 1
A
G
22.1162 33.3844
34.4606 30.1057
22.9622 39.4374
33.9907 26.7698
39.8092 35.1469
39.8092 35.2143
24.1154 47.0588
34.3594 32.6562
31.4600 23.4870
39.9659 30.7675
39.9659 30.8130
49.7340 41.9462
49.7340 42.0809
25.7799 41.1765
33.8125 28.5862
40.5507 40.3121
IV
22.4519
10.0297
21.0696
8.4494
8.6474
8.6474
19.6873
8.9173
7.7194
7.0671
7.0671
7.2651
7.2651
18.3050
7.3370
7.5350
Table of Criteria Values for SCDs (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
D
59.0785
59.3046
65.9931
55.6177
67.2006
67.2006
75.7853
59.9284
49.9903
63.5727
63.5727
78.9166
78.9166
90.5396
55.7731
69.2344
rs = 1, n0 = 1
A
G
28.1641 32.7923
36.5938 29.5968
29.8118 39.3976
36.6855 26.3142
41.4992 35.1530
41.4992 35.2780
32.1640 51.7605
38.2273 34.1527
34.5668 23.1166
42.4496 30.7681
42.4496 30.8701
50.1410 45.2985
50.1410 45.4717
35.7952 63.6364
38.5885 29.8924
44.8414 44.3789
IV
17.0840
10.2635
15.7080
8.6222
8.8874
8.8874
14.3319
8.8239
7.7656
7.2461
7.2461
7.5114
7.5114
12.9558
7.1826
7.4478
D
56.6631
58.4165
64.5050
53.5091
67.8452
67.8452
75.8506
57.0511
46.3287
62.6976
62.6976
82.2375
82.2375
93.4178
51.4334
67.4628
305
Table of Criteria Values for SCDs (K = 3)
Dsgn
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
l
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
c
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
q
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
D
57.3983
75.9768
75.9768
75.9768
97.7751
49.2695
65.2169
65.2169
48.8786
77.0781
69.6493
97.5115
57.9872
49.9070
73.2992
73.2992
60.9882
67.5905
93.0988
49.3596
65.1626
66.1577
58.8819
47.5419
71.6144
46.0959
34.0599
55.0472
rs = 1, n0 = 1
A
G
40.1773 26.5131
53.6995 39.1781
53.6995 39.1781
53.6995 39.9878
69.4140 74.6949
36.1800 25.7545
46.7900 38.0556
46.7900 38.2379
34.5744 53.8507
46.6542 40.8862
51.9915 34.0733
85.3759 62.3017
44.3777 31.9572
38.1415 46.1077
49.6364 34.8854
49.6364 34.4130
44.7772 62.2458
54.6068 51.5251
93.0207 93.0207
38.1402 40.4173
46.9653 28.7998
57.7901 49.8413
52.6313 44.7844
34.9069 41.2201
69.7656 69.7656
33.9460 34.2599
23.6717 37.3810
46.5104 46.5104
IV
6.3895
5.8700
5.8700
5.8700
6.1353
6.3260
5.8065
5.8065
5.6958
6.8763
5.0135
4.4940
4.9499
4.5588
5.2349
5.2349
4.8207
4.3669
3.6374
3.8449
4.3783
3.6837
3.5103
3.7809
2.9699
3.0671
1.3944
1.0500
D
54.4145
77.4488
77.4488
77.4488
106.2847
43.1891
61.4716
61.4716
49.1295
80.4093
68.1592
104.1082
51.6555
48.5028
74.4882
74.4882
66.9460
59.7955
95.5518
45.0441
62.8771
71.1790
47.7770
45.4798
73.2893
40.3332
37.1181
56.0100
rs = 2, n0 = 1
A
G
36.9696 26.4660
52.2013 36.5126
52.2013 36.5126
52.2013 36.7555
74.4978 84.7760
30.9008 24.5834
40.8680 34.5738
40.8680 34.6408
30.3991 43.4633
46.1101 37.8581
48.9783 31.8060
91.3573 70.6535
37.3244 28.9460
31.4712 36.9020
47.9321 31.9775
47.9321 31.7988
49.1404 70.6467
42.2015 39.6798
95.5192 95.5192
29.0906 31.1599
43.7347 26.2316
63.2409 56.5228
37.8697 32.9875
27.9844 32.1334
71.6394 71.6394
24.6785 25.5630
27.4578 42.3921
47.7596 47.7596
IV
6.3371
5.6848
5.6848
5.6848
5.8828
6.6069
5.9547
5.9547
5.9366
6.6517
4.9547
4.3025
5.2246
4.8665
5.0713
5.0713
4.5843
4.8422
3.5724
4.2692
4.3413
3.5142
4.1122
4.1152
2.9169
3.5179
1.3051
1.0313
rs = 2, n0 = 3
A
G
29.7209 29.8702
32.1741 27.0002
33.4360 35.2861
30.7991 24.0002
37.6586 31.5448
37.6586 31.6776
39.6279 42.8769
32.2504 29.3029
28.2822 21.0892
36.3722 27.6018
36.3722 27.7345
48.2287 37.5468
48.2287 37.8113
52.0118 88.4193
30.6911 25.6401
38.7240 36.2156
33.2940 23.7905
47.9380 32.6925
47.9380 32.6925
47.9380 33.4075
77.0725 75.9286
27.7987 22.0910
37.3167 31.0419
37.3167 31.2381
29.2329 38.9083
45.0226 34.0242
44.2793 28.6499
86.4026 63.3074
33.6600 26.0578
28.7858 33.0461
44.1360 29.0157
44.1360 28.4902
50.8267 63.2738
38.7244 35.5633
87.6682 87.6682
26.2292 28.1104
39.5867 23.6335
59.8605 50.6459
34.2244 29.7736
25.6602 28.7780
65.7512 65.7512
22.2737 23.0792
26.0484 37.9844
43.8341 43.8341
IV
13.3923
10.1397
11.8474
9.0537
8.5948
8.5948
10.3025
8.8964
8.4238
7.5087
7.5087
7.0498
7.0498
8.7575
7.8104
7.3515
6.8788
5.9638
5.9638
5.9638
5.5049
7.1805
6.2654
6.2654
5.9401
6.3642
5.3339
4.4189
5.6355
5.1542
5.2782
5.2782
4.4287
5.0221
3.7890
4.6051
4.6483
3.6427
4.3922
4.3144
3.0937
3.7654
1.3892
1.0938
Table of Criteria Values for SCDs (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
D
55.7945
52.8881
63.0901
48.6761
60.3247
60.3247
73.5655
53.7966
43.3211
55.9070
55.9070
71.4427
71.4427
89.6290
49.0479
62.6774
49.9390
67.2462
67.2462
67.2462
89.5174
42.8666
57.7228
57.7228
44.7506
70.5684
60.9363
87.0863
50.7331
44.5713
65.4626
65.4626
56.7246
60.3643
82.1355
43.5470
57.4890
59.8875
51.9480
43.0360
64.0669
41.2377
31.5332
50.6361
rs = 1, n0 = 3
A
G
32.8879 27.7473
33.1530 25.1055
37.3091 33.3364
31.7498 22.3171
38.3447 29.8433
38.3447 30.0430
44.8450 43.7974
35.0672 28.9914
29.4727 19.6601
37.0107 26.1146
37.0107 26.3449
48.0116 38.3354
48.0116 38.6103
60.5762 73.4255
33.5516 25.3691
42.3478 37.7310
34.3501 22.5877
47.5063 33.1831
47.5063 33.1831
47.5063 34.5379
72.3216 63.4376
30.9005 21.9372
41.1526 32.3439
41.1526 32.7577
33.2551 45.6876
45.1815 34.7740
44.7080 29.1367
78.7851 53.0693
38.0702 27.3093
33.7961 39.0310
44.1311 30.2299
44.1311 29.1487
46.5990 52.8647
48.7982 43.6191
81.6304 81.6304
32.7530 34.7392
40.4709 24.6421
53.4607 42.4554
45.4543 38.5587
31.1416 34.8953
61.2228 61.2228
29.2324 29.5075
22.0342 31.8416
40.8152 40.8152
IV
12.1843
10.5157
10.5580
9.5533
8.8894
8.8894
8.9318
8.8143
8.8626
7.9271
7.9271
7.2631
7.2631
7.3055
7.8520
7.1881
7.2363
6.3008
6.3008
6.3008
5.6369
7.1613
6.2257
6.2257
5.6904
6.5126
5.6101
4.6745
5.5350
4.9261
5.5502
5.5502
4.6562
4.5243
3.9838
4.2869
4.8595
3.8920
3.8336
4.0069
3.2528
3.3676
1.5377
1.1500
D
56.5859
54.1407
65.2082
48.9222
63.1566
63.1566
77.8564
53.1085
42.1159
57.5002
57.5002
76.9887
76.9887
97.7879
47.1698
63.1570
49.5976
71.3216
71.3216
71.3216
100.2547
39.3659
56.6084
56.6084
46.3422
75.8474
62.3563
96.4258
47.2577
44.9237
68.9915
68.9915
63.8185
55.3830
87.9044
41.4390
57.8448
66.4987
43.9532
42.4894
68.0514
37.4507
35.1805
52.9801
306
Table of Criteria Values for UNFSDs (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
D
69.5913
72.3791
71.8937
71.8986
75.9890
75.4481
75.9897
70.2120
70.5485
77.4081
75.3214
81.6000
80.9364
82.8377
69.3774
73.9063
74.2472
81.7288
81.7288
81.7288
90.4540
67.4574
75.1699
72.8112
46.5562
79.1826
81.6538
92.8772
71.0835
49.0335
79.7632
79.7635
55.3806
72.1449
94.1730
52.7922
79.1883
62.0134
66.5888
46.3268
72.3483
50.1623
31.0865
55.4695
n0 = 1
A
G
34.0475 76.9231
52.9225 69.6345
33.0835 69.2308
60.7974 64.0823
53.9112 61.8973
55.5331 61.8972
31.9526 61.5385
50.2428 64.2235
65.1141 57.7213
66.2987 60.5203
63.6846 60.5203
55.2382 56.1956
55.2382 54.1602
31.0651 53.8462
57.9704 60.5203
50.8873 56.1956
69.9011 66.4801
67.9899 51.8745
67.9899 51.8745
67.9899 53.8528
60.0966 64.8442
62.0667 54.5238
63.3249 51.8745
60.5551 51.8745
31.5230 48.1676
51.7730 50.8399
77.9199 57.7378
78.6440 54.5906
66.6647 57.7378
37.7470 43.2288
64.5869 49.9070
64.5871 45.4276
37.7904 54.0369
59.5835 45.4276
94.1172 94.1163
43.8337 46.1902
74.9975 53.0715
51.4117 43.6725
62.3355 52.3952
34.4203 36.3421
70.5877 70.5872
39.9979 39.8037
19.4023 32.7543
47.0581 47.0581
IV
16.3622
8.5522
15.9118
6.2500
8.1019
8.0118
15.4615
7.9268
4.9595
5.7095
5.7996
7.6515
7.6515
14.9211
5.6245
7.4764
4.5092
5.3493
5.2350
5.3493
6.9623
4.3341
5.0841
5.1742
6.2879
7.0261
4.0588
4.7846
3.8837
4.7246
4.7238
4.7238
5.4010
4.4586
3.6085
3.6358
3.4333
3.9184
3.2582
3.9585
2.9463
2.8697
1.5003
1.0417
D
67.3162
66.0373
70.3973
64.4728
70.1995
69.2811
75.5522
64.4730
62.4048
69.5393
67.8718
76.0738
74.9373
83.9925
62.5157
68.9011
65.9007
74.1253
74.8683
74.1253
84.6770
59.8742
67.9340
66.0373
43.9344
74.1255
72.8212
84.8696
63.3943
44.8779
73.0031
73.0034
52.6493
65.7486
84.5893
47.4197
71.1295
57.4217
59.8123
42.9826
65.7652
45.5980
29.4267
51.6393
n0 = 3
A
G
48.6477 66.6769
53.3320 60.6559
49.8454 60.0092
55.4423 56.9196
55.6512 53.9164
55.9213 53.9163
51.4283 53.3416
51.1987 56.3967
57.1410 50.5812
61.0895 52.7409
58.6792 52.7409
58.9470 50.4576
58.9470 47.1768
55.2630 65.1158
53.1131 52.7409
53.3323 50.4576
61.5368 58.4800
63.6329 45.2065
65.5223 52.5741
63.6329 50.2576
66.3975 56.5445
54.5434 47.8335
58.7740 45.2065
56.1836 45.2065
32.8202 43.2493
56.4702 44.3060
68.9635 50.8219
75.1089 48.1793
58.8218 50.8219
34.9888 37.6721
61.1315 46.6184
61.1317 40.1652
41.6711 47.1204
55.8124 40.1652
84.2101 84.2093
38.5525 40.6575
66.6644 46.8237
49.3188 38.5434
55.1705 46.2163
32.2185 32.1322
63.1574 63.1569
35.2923 35.1178
18.7978 28.9076
42.1046 42.1046
IV
9.6418
7.7395
9.1222
6.4860
7.2198
7.1239
8.6025
7.0178
5.4561
5.8624
5.9664
6.7002
6.7002
7.9790
5.7643
6.4981
4.9364
5.4467
5.3567
5.4467
6.0580
4.7344
5.1407
5.2447
5.8823
5.9785
4.4168
4.8370
4.2147
4.9258
4.7250
4.7250
4.9654
4.4190
3.8971
3.9776
3.6951
4.0289
3.4930
4.0419
3.1820
3.0937
1.6134
1.1250
Table of Criteria Values for Hybrid 310 (K = 3)
n0 = 0
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
n0 = 1
n0 = 3
p dv l c q
D
A
G
IV
D
A
G
IV
D
A
G
IV
10 3 3 3 3 62.1772 36.9127 47.3893 14.3356 60.6397 45.7457 45.0198 10.6710 55.0194 47.1490 38.9577 9.6415
9 3 3 3 2 64.4927 44.4560 42.8756 10.5009 61.7874 48.6961 42.0549 9.0377 55.6450 48.2396 37.7033 8.5669
9 3 3 2 3 62.1864 35.3211 42.6504 14.0009 61.1241 44.8018 40.5178 10.3028 55.8906 47.0815 35.0619 9.2064
8 3 3 3 1 67.0084 51.9685 38.2353 7.8790 63.2377 52.5296 38.7293 7.4755 56.3704 49.9056 36.8891 7.4585
8 3 3 2 2 64.7995 42.9319 38.1117 10.1662 62.4896 47.8744 37.3821 8.6695 56.7177 48.3000 33.5140 8.1317
8 3 3 2 2 64.7989 42.9316 38.1117 10.1662 62.4891 47.8741 37.3821 8.6695 56.7172 48.2996 33.5140 8.1317
8 3 3 1 3 62.1978 33.5148 37.9115 13.6662 61.7349 43.6753 36.0158 9.9346 56.9989 46.9974 31.1662 8.7712
8 3 2 3 2 62.3825 41.9805 38.1117 9.8149 60.1588 46.5796 37.3821 8.2830 54.6022 46.7504 33.5140 7.6750
7 3 3 3 0 75.7196 74.4993 75.7056 4.7943 69.7796 68.3883 69.5066 5.1005 60.4703 58.7501 59.7256 5.7130
7 3 3 2 1 67.7420 50.7858 33.4559 7.5442 64.2724 52.0140 33.8881 7.1073 57.7208 50.2276 32.2780 7.0234
7 3 3 2 1 67.7414 50.7854 33.4559 7.5442 64.2718 52.0136 33.8881 7.1074 57.7203 50.2271 32.2780 7.0234
7 3 3 1 2 65.1953 41.1191 33.3477 9.8315 63.4037 46.8576 32.7094 8.3013 58.1268 48.3774 29.3248 7.6966
7 3 3 1 2 65.1947 41.1188 33.3477 9.8315 63.4031 46.8572 32.7094 8.3013 58.1263 48.3769 29.3248 7.6966
7 3 3 0 3 62.2132 31.4472 33.1725 13.3314 62.5299 42.3079 31.5139 9.5664 58.4569 46.8902 27.2704 8.3360
7 3 2 3 1 64.8621 49.2760 33.4559 7.1929 61.5400 50.2785 33.8881 6.7209 55.2669 48.3240 32.2780 6.5667
7 3 2 2 2 62.4237 40.1237 33.3477 9.4802 60.7082 45.4444 32.7094 7.9149 55.6557 46.6090 29.3248 7.2399
6 3 3 2 0 78.2649 77.0652 73.9467 4.4596 72.2891 70.8870 67.9860 4.7323 62.8947 61.0917 58.5471 5.2778
6 3 3 1 1 75.5309 62.6275 43.3162 5.8761 71.1822 61.3902 41.0173 5.8293 63.6188 57.1347 36.4683 6.0192
6 3 3 1 1 68.7320 49.2896 28.6765 7.2095 65.6778 51.3416 29.0469 6.7391 59.5711 50.6629 27.6668 6.5882
6 3 3 1 1 68.7312 49.2892 28.6765 7.2095 65.6770 51.3411 29.0469 6.7392 59.5705 50.6623 27.6668 6.5883
6 3 3 0 2 65.7269 38.9274 28.5838 9.4967 64.6432 45.5671 28.0366 7.9331 60.0603 48.4809 25.1355 7.2614
6 3 2 3 0 74.3969 73.1002 73.5483 4.1083 68.7165 67.1990 67.6156 4.3459 59.7864 57.8577 58.2226 4.8212
6 3 2 2 1 65.3359 47.6374 28.6765 6.8582 62.4326 49.3793 29.0469 6.3527 56.6277 48.4190 27.6668 6.1315
6 3 2 2 1 65.3352 47.6370 28.6765 6.8582 62.4319 49.3788 29.0469 6.3527 56.6271 48.4184 27.6668 6.1316
6 2 2 1 2 36.3872 22.8090 28.5838 8.2292 35.7872 25.4920 28.0366 7.4197 33.2501 25.7789 25.1355 7.3957
6 3 2 1 2 62.4787 37.8893 28.5838 9.1455 61.4485 44.0143 28.0366 7.5467 57.0921 46.4217 25.1355 6.8048
5 3 3 1 0 81.9727 80.9693 63.7559 4.1248 75.9548 74.7083 58.6396 4.3641 66.4530 64.7020 50.5298 4.8427
5 3 3 0 1 70.1422 47.3372 23.8971 6.8748 67.6970 50.4289 24.2058 6.3709 62.2617 51.2852 23.0557 6.1531
5 3 2 2 0 77.1356 75.7867 63.7559 3.7736 71.4728 69.8596 58.6396 3.9777 62.5317 60.4105 50.5298 4.3860
5 2 2 1 1 40.5753 26.9184 23.8971 6.2967 39.1608 28.0299 24.2058 6.0326 36.0166 27.6475 23.0557 6.0969
5 3 2 1 1 66.0033 45.5174 23.8971 6.5235 63.7024 48.1720 24.2058 5.9845 58.5877 48.5512 23.0557 5.6964
5 3 2 1 1 66.0042 45.5178 23.8971 6.5235 63.7032 48.1725 24.2058 5.9845 58.5885 48.5518 23.0557 5.6964
5 2 2 0 2 38.4559 22.0440 23.8198 7.7168 38.4218 25.5742 23.3638 6.8560 36.3718 26.9334 20.9463 6.7295
5 3 1 2 1 62.1102 43.8331 23.8971 6.1722 59.9450 46.1094 24.2058 5.5981 55.1321 46.0951 23.0557 5.2397
307
Table of Criteria Values for Hybrid 310 (K = 3)
n0 = 0
Dsgn
35
36
37
38
39
40
41
42
43
44
n0 = 1
p dv l c q
D
A
G
IV
D
A
G
4 3 3 0 0 87.8678 87.6303 87.6281 3.7901 81.8061 81.2827 81.2807
4 2 2 1 0 54.2892 48.8493 51.0047 3.6071 50.5439 44.9070 46.9117
4 3 2 1 0 81.4351 80.2095 60.5067 3.4388 75.8171 74.2717 55.7731
4 2 2 0 1 44.6796 26.7538 19.1177 5.7843 43.7718 28.8762 19.3646
4 3 1 2 0 75.4735 73.9477 60.5017 3.0876 70.2668 68.3743 55.7684
4 2 1 1 1 37.4176 23.8205 19.1177 5.4565 36.6574 25.1929 19.3646
3 2 2 0 0 68.0252 65.7215 65.7211 3.0946 63.8373 60.9608 60.9606
3 2 1 1 0 53.6969 46.8304 45.3800 2.7669 50.3912 43.1860 41.8298
3 1 1 0 1 20.7068 7.7620 14.3383 2.5427 20.7982 8.5436 14.5235
2 1 1 0 0 52.9648 43.8143 43.8143 1.0941 50.5000 40.6406 40.6406
IV
3.9959
3.8264
3.6095
5.4689
3.2231
5.1084
3.2627
2.9022
2.5006
1.1535
D
72.1726
44.5919
66.8889
41.1052
61.9922
34.4242
57.1094
45.0803
20.2214
46.4532
n0 = 3
A
G
70.9973 70.9956
38.6661 40.4239
64.6933 48.2271
29.9082 18.4445
59.4177 48.2230
25.3683 18.4445
53.2469 53.2467
37.3696 36.1703
9.1051 13.8334
35.4979 35.4979
Table of Criteria Values for Hybrid 311A (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
D
67.6003
69.3012
69.5461
69.1433
71.7719
71.7722
72.0576
67.3326
71.4888
71.9443
71.9446
75.0788
75.0788
75.4205
66.8815
69.7951
75.2962
80.9627
75.8565
75.8569
79.7262
69.1509
69.6656
69.6660
42.6421
73.2195
80.9701
81.6937
73.1062
45.3429
73.7602
73.7597
48.1326
66.5962
90.2925
52.9782
79.4667
53.5709
69.9402
42.6034
69.6917
51.3493
26.0586
53.9350
n0 = 1
A
G
37.4090 78.6243
52.0496 70.8236
36.2584 70.7618
59.7690 62.9734
52.0106 62.9543
52.0108 62.9543
34.9162 62.8994
49.5500 62.9543
68.8511 70.0741
61.0014 55.1017
61.0018 55.1017
51.9608 55.0850
51.9608 55.0850
33.3298 55.0370
57.1943 55.1017
49.1725 55.0850
72.6643 69.3911
70.3875 59.3496
62.7264 47.2301
62.7268 47.2301
51.8946 47.2157
66.5111 68.1678
58.0874 47.2301
58.0877 47.2301
29.6323 47.2157
48.6783 47.2157
78.7729 60.0637
65.3119 39.3584
70.3109 60.0637
35.1224 39.3584
59.3865 39.3584
59.3861 39.3584
31.3457 39.3464
54.4461 39.3584
90.1393 90.1383
45.7951 48.0510
76.9004 56.0469
40.2459 31.4867
67.0534 55.6576
31.9563 31.4867
67.6037 67.6037
42.7604 42.0352
13.4716 23.6150
45.0691 45.0691
IV
14.4549
8.8307
14.0579
6.6155
8.4337
8.4337
13.6610
8.1691
4.9075
6.2186
6.2186
8.0368
8.0368
13.2641
5.9540
7.7722
4.5106
5.2691
5.8217
5.8217
7.6398
4.2460
5.5570
5.5570
6.6413
7.3752
4.1137
5.4247
3.8491
5.0992
5.1601
5.1601
6.0336
4.8955
3.7167
3.6424
3.4521
4.4915
3.1875
4.2889
3.0347
2.8322
1.8184
1.0729
D
63.8425
62.7452
66.4868
61.2066
65.5342
65.5345
69.9471
61.4807
61.9515
64.0979
64.0982
69.3034
69.3034
74.6618
59.5872
64.4261
65.5110
72.0467
68.1669
68.1672
74.6682
60.1643
62.6035
62.6039
39.9368
68.5743
70.8409
74.3011
63.9607
41.2398
67.0855
67.0850
46.0035
60.5698
79.6597
46.7395
70.1087
49.6106
61.7041
39.4539
62.3467
45.9375
24.8692
49.6130
n0 = 3
A
G
50.6899 69.0153
53.1710 62.9498
51.5143 62.1137
54.8642 56.5085
54.5346 55.9553
54.5349 55.9553
52.5837 55.2122
51.3732 55.9553
59.1537 60.2209
56.7973 49.4449
56.7977 49.4449
56.3945 48.9609
56.3945 48.9609
54.0256 48.3107
52.9211 49.4449
52.5708 48.9609
62.6525 59.7792
64.5792 50.8753
59.5977 42.3814
59.5981 42.3814
59.0810 41.9665
57.2551 58.7066
54.6931 42.3814
54.6935 42.3814
31.3416 41.9665
54.2576 41.9665
68.3097 51.7801
64.0165 35.3178
60.8094 51.7801
33.0874 35.3178
57.3841 35.3178
57.3836 35.3178
35.5478 34.9721
51.9962 35.3178
79.0110 79.0100
39.4444 41.4241
67.0528 48.4691
39.8676 28.2543
58.2395 48.1251
30.5806 28.2543
59.2575 59.2575
36.9931 36.3519
13.6141 21.1907
39.5050 39.5050
IV
9.2126
7.8650
8.7435
6.7570
7.3959
7.3959
8.2744
7.0831
5.4849
6.2879
6.2879
6.9268
6.9268
7.8053
5.9751
6.6140
5.0158
5.4593
5.8188
5.8188
6.4577
4.7031
5.5060
5.5060
6.2005
6.1449
4.5467
5.3497
4.2340
5.2593
5.0369
5.0369
5.4824
4.7242
4.0776
4.0475
3.7649
4.5411
3.4521
4.3017
3.3294
3.0900
1.9492
1.1771
Table of Criteria Values for Hybrid 311B (K = 3)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
p
10
9
9
8
8
8
8
8
7
7
7
7
7
7
7
7
dv
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
l
3
3
3
3
3
3
3
2
3
3
3
3
3
3
2
2
c
3
3
2
3
2
2
1
3
3
2
2
1
1
0
3
2
q
3
2
3
1
2
2
3
2
0
1
1
2
2
3
1
2
D
70.9973
73.1077
74.6030
71.7200
77.5814
77.5812
79.3683
71.1429
68.4751
76.5474
76.5467
83.7372
83.7369
85.9451
69.3313
75.8438
n0 = 1
A
G
37.8798 90.9091
56.5583 82.9752
37.1909 81.8182
62.7901 73.8205
58.3390 78.2888
58.3398 78.3737
36.3641 72.7273
54.0073 77.9571
64.2249 65.4537
66.4071 70.7230
66.4064 70.7224
60.7997 68.5027
60.8006 68.5770
35.3536 63.6364
60.1328 69.2019
55.4984 68.5027
IV
14.4290
7.8869
13.9717
6.0225
7.4296
7.4295
13.5145
7.2518
5.0089
5.5653
5.5654
6.9724
6.9723
13.0573
5.3875
6.7946
D
67.0507
65.4731
71.3213
62.9082
69.9743
69.9740
77.0436
64.1671
59.3398
67.4882
67.4877
76.2186
76.2183
85.0806
61.1262
69.0340
n0 = 3
A
G
50.9072 77.4084
54.4865 70.6496
52.8052 73.9937
55.6316 62.8028
57.4793 66.6270
57.4795 66.7579
55.3859 65.7722
52.5697 66.0325
55.1222 56.1923
59.4202 59.8434
59.4197 59.8430
61.8463 58.2986
61.8466 58.4131
59.0995 67.3059
53.5154 58.5887
55.4759 58.2986
IV
9.2126
7.8650
8.7435
6.7570
7.3959
6.9740
8.2744
7.0831
5.4849
6.2879
5.9284
6.9268
6.5049
7.8053
5.9751
6.6140
IV
4.4075
4.2650
3.9509
5.4308
3.4942
5.0047
3.5988
3.1727
2.6128
1.2724
308
Table of Criteria Values for Hybrid 311B (K = 3)
Dsgn
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
p
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
3
3
3
3
3
2
3
3
3
3
2
3
3
2
3
3
2
3
2
3
2
2
2
1
1
l
3
3
3
3
3
2
2
2
2
2
3
3
2
2
2
2
2
1
3
2
2
2
1
1
2
1
1
1
c
2
1
1
1
0
3
2
2
1
1
1
0
2
1
1
1
0
2
0
1
1
0
2
1
0
1
0
0
q
0
1
1
1
2
0
1
1
2
2
0
1
0
1
1
1
2
1
0
0
0
1
0
1
0
0
1
0
D
73.3139
83.4924
83.4915
83.4915
92.7113
65.3158
74.3845
74.3837
48.1039
82.5978
80.6672
94.2866
70.2263
50.4595
82.0818
82.0829
57.2171
71.4588
93.1030
52.1946
78.2919
63.4332
65.8370
48.2000
71.6173
49.6583
32.2021
55.0489
n0 = 1
A
G
68.9705 65.5899
71.9311 60.6197
71.9301 60.6192
71.9301 62.4345
64.4227 68.7533
61.2293 62.7968
63.5522 60.6197
63.5514 60.6192
34.8912 58.7166
57.6189 66.1718
76.9273 56.9692
81.4122 57.4168
65.7947 56.9692
40.4813 50.5164
69.0469 57.4149
69.0480 55.6924
40.8265 57.2944
59.9442 55.2282
93.0250 93.0250
43.2465 45.5753
74.0786 52.3795
54.0395 45.9335
61.5439 51.7106
37.3782 44.5539
69.7688 69.7687
39.4771 39.2846
21.1145 34.4501
46.5126 46.5126
IV
4.5517
5.1081
5.1082
5.1082
6.5152
4.3738
4.9302
4.9303
5.8129
6.3374
4.0945
4.6509
3.9166
4.5145
4.4731
4.4730
5.1129
4.2952
3.6373
3.6698
3.4594
3.8145
3.2815
3.7367
2.9698
2.8920
1.4575
1.0500
D
63.7863
74.1173
74.1166
74.1166
85.4197
56.8276
66.0321
66.0314
44.3205
76.1015
70.5759
84.5059
61.4411
45.2252
73.5673
73.5682
53.6224
64.0461
82.1392
46.0482
69.0723
57.6768
58.0841
43.8260
64.0694
44.4248
29.9902
50.6376
n0 = 3
A
G
59.4103 56.4485
65.3536 51.2944
65.3529 51.2940
65.3529 55.4751
68.8176 58.5698
52.6358 54.0054
57.2490 51.2944
57.2484 51.2940
34.5993 49.9703
59.8898 56.2192
66.6704 49.0647
75.9745 49.2313
56.8228 49.0647
36.3617 42.7453
63.4441 49.2298
63.4448 47.4325
43.5081 48.8081
54.4629 46.8540
81.6343 81.6342
37.2122 39.2518
64.5202 45.2323
50.6235 39.3850
53.3382 44.6430
33.8961 37.9460
61.2257 61.2257
34.0939 33.9242
19.9586 29.5388
40.8172 40.8172
IV
5.0158
5.8188
5.4593
5.4593
6.4577
4.7031
5.5060
5.1465
6.2005
6.1449
4.5467
5.3497
4.2340
5.2593
4.6774
5.0369
5.4824
4.7242
4.0776
4.0475
3.7648
4.5411
3.4521
4.3017
3.3293
3.0899
1.9492
1.1771
rs = 2, n0 = 1
A
G
25.1869 45.4545
55.5566 81.0391
24.3511 42.4242
62.4504 75.3608
56.1870 78.0934
56.1870 78.0505
23.4531 39.3939
23.4531 39.3939
53.7895 77.6874
63.8042 69.5677
63.9863 74.1817
63.9863 72.4391
63.9863 73.0491
56.9408 72.0862
56.9408 72.0467
56.9408 79.7427
22.4857 36.3636
22.4857 36.3636
22.4857 36.3636
60.6512 72.5989
54.2845 72.0862
62.8571 64.1278
65.6911 68.0832
65.6911 67.0016
65.9017 67.9999
65.9017 66.9617
65.9017 75.1846
65.9017 73.1121
57.8581 73.0974
57.8581 73.3697
57.8581 66.0790
57.8581 66.0428
57.8581 73.0974
57.8581 73.0974
21.4405 33.3333
21.4405 33.3333
61.8801 66.5511
62.0671 67.9999
62.0671 66.8206
54.8813 66.0790
54.8813 73.0974
64.7773 63.3559
68.1081 61.8938
68.1081 60.9105
68.1081 69.3570
68.3573 66.4656
68.3573 68.5957
68.3573 61.8181
68.3573 68.3496
68.3573 68.3496
IV
42.4875
14.0038
41.9375
10.5050
13.4538
13.4538
41.3875
41.3875
13.3163
8.9851
9.9550
9.9550
9.9550
12.9038
12.9038
12.9038
40.8375
40.8375
40.8375
9.8175
12.7663
8.0500
8.4351
8.4351
9.4050
9.4050
9.4050
9.4050
12.3538
12.3538
12.3538
12.3538
12.3538
12.3538
40.2875
40.2875
8.2976
9.2675
9.2675
12.2163
12.2163
7.5000
7.8851
7.8851
7.8851
8.8550
8.8550
8.8550
8.8550
8.8550
Table of Criteria Values for CCDs (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
D
76.7266
79.6850
77.7270
79.8387
81.0400
81.0400
78.8976
78.8976
78.5514
78.9083
81.3235
81.3235
81.3235
82.6500
82.6500
82.6500
80.2855
80.2855
80.2855
78.6216
79.9040
77.2384
80.4248
80.4248
83.1139
83.1139
83.1139
83.1139
84.5940
84.5940
84.5940
84.5940
84.5940
84.5940
81.9573
81.9573
77.5143
80.1061
80.1061
81.5326
81.5326
78.7043
82.2832
82.2832
82.2832
85.3146
85.3146
85.3146
85.3146
85.3146
rs = 1, n0 = 1
A
G
31.6484 60.0000
56.5545 96.1473
30.5455 56.0000
67.3944 89.3343
56.0528 89.2797
56.0528 89.2032
29.3647 52.0000
29.3647 52.0000
54.8217 89.8983
72.8759 82.5053
67.6936 83.0416
67.6936 82.4624
67.6936 82.6908
55.4788 82.4120
55.4788 82.3415
55.4788 82.4120
28.0976 48.0000
28.0976 48.0000
28.0976 48.0000
65.7615 83.0416
54.1743 82.9831
75.6447 77.0363
73.8065 76.1297
73.8065 76.1035
68.0506 76.1215
68.0506 75.7999
68.0506 76.1215
68.0506 76.1215
54.8153 75.5443
54.8153 75.5443
54.8153 76.0678
54.8153 75.4797
54.8153 75.5443
54.8153 75.5443
26.7342 44.0000
26.7342 44.0000
71.3143 76.1297
65.9264 76.1215
65.9264 76.0683
53.4286 76.0678
53.4286 76.0678
77.0465 75.5198
74.9550 69.2088
74.9550 69.3333
74.9550 69.3333
68.4841 69.2014
68.4841 69.2014
68.4841 69.2014
68.4841 68.9090
68.4841 69.2014
IV
33.6111
15.0572
33.1944
10.7540
14.6405
14.6405
32.7778
32.7778
14.3627
8.6432
10.3373
10.3373
10.3373
14.2239
14.2239
14.2239
32.3611
32.3611
32.3611
10.0595
13.9461
7.2778
8.2265
8.2265
9.9206
9.9206
9.9206
9.9206
13.8072
13.8072
13.8072
13.8072
13.8072
13.8072
31.9444
31.9444
7.9487
9.6429
9.6429
13.5294
13.5294
6.8611
7.8098
7.8098
7.8098
9.5040
9.5040
9.5040
9.5040
9.5040
D
73.4893
76.7825
75.7051
74.7909
79.5463
79.5463
78.3450
78.3450
75.4161
71.2251
77.5418
77.5418
77.5418
82.8969
82.8969
82.8969
81.5415
81.5415
81.5415
73.1897
78.2442
66.6274
73.7593
73.7593
80.9236
80.9236
80.9236
80.9236
87.0390
87.0390
87.0390
87.0390
87.0390
87.0390
85.4876
85.4876
69.2549
75.9816
75.9816
81.7236
81.7236
68.7793
76.9198
76.9198
76.9198
85.1769
85.1769
85.1769
85.1769
85.1769
309
Table of Criteria Values for CCDs (K = 4)
Dsgn
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
p
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
l
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
c
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
q
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
D
85.3146
85.3146
86.9872
86.9872
86.9872
86.9872
84.0095
75.5769
79.0136
79.0136
81.9246
81.9246
81.9246
83.5307
54.2548
83.5307
78.6693
80.5338
80.5338
84.6129
84.6129
84.6129
84.6129
84.6129
88.0835
88.0835
88.0835
88.0835
88.0835
88.0835
90.0044
90.0044
86.5877
76.9862
80.8855
80.8855
80.8855
84.2033
84.2033
55.5728
84.2033
84.2033
56.7847
86.0395
86.0395
86.0395
77.3224
80.4940
80.4940
82.8806
82.8806
82.8806
87.6181
87.6181
87.6181
87.6181
91.6715
91.6715
91.6715
93.9235
78.7847
78.7847
83.2880
83.2880
56.0779
83.2880
83.2880
58.6721
58.6721
87.1410
87.1410
87.1410
87.1410
87.1410
87.1410
87.1410
60.1135
89.2818
74.8911
79.1719
79.1719
79.1719
79.1719
53.8026
82.8345
rs = 1, n0 = 1
A
G
68.4841 69.2014
68.4841 69.2014
54.0397 69.1525
54.0397 68.6767
54.0397 68.6767
54.0397 68.6767
25.2632 40.0000
74.0741 73.8854
72.1387 69.2088
72.1387 69.1850
66.1255 69.2014
66.1255 69.1530
66.1255 69.2014
52.5604 69.1525
37.4312 69.1525
52.5604 69.1525
63.9239 69.2784
78.8321 67.9678
78.8321 79.7276
76.4082 62.4000
76.4082 62.4000
76.4082 62.2879
76.4082 62.4000
76.4082 62.4000
69.0214 62.2812
69.0214 62.2812
69.0214 62.2812
69.0214 62.2812
69.0214 62.2812
69.0214 62.3505
53.1212 61.8090
53.1212 62.2373
23.6712 36.0000
75.3927 67.9678
73.1726 62.2879
73.1726 62.2879
73.1726 62.4000
66.3704 62.2812
66.3704 62.2812
45.1343 62.2812
66.3704 62.2812
66.3704 62.2812
37.5973 62.2373
51.5368 62.2373
51.5368 62.2373
51.5368 62.2373
70.2000 62.4000
63.9155 62.3505
63.9155 62.3505
81.1839 70.2667
81.1839 60.4158
81.1839 70.8690
78.3059 55.4667
78.3059 55.4667
78.3059 55.4667
78.3059 55.4667
69.7050 55.4227
69.7050 55.3611
69.7050 55.4227
52.0159 55.3220
77.1084 60.4158
77.1084 73.9382
74.5075 55.4667
74.5075 55.4667
49.3714 55.3670
74.5075 55.4667
74.5075 55.4667
46.6127 55.3611
46.6127 55.3611
66.6791 55.4227
66.6791 55.3611
66.6791 55.3611
66.6791 55.3611
66.6791 55.3611
66.6791 55.4227
66.6791 55.4227
37.8069 55.3220
50.3121 55.3220
73.4226 67.1559
71.0605 55.4667
71.0605 55.4667
71.0605 55.4667
71.0605 55.4667
43.1230 55.4227
63.9049 55.4227
IV
9.5040
9.5040
13.3905
13.3905
13.3905
13.3905
31.5278
6.5833
7.5321
7.5321
9.2262
9.2262
9.2262
13.1127
12.0847
13.1127
8.9484
6.4444
6.4444
7.3932
7.3932
7.3932
7.3932
7.3932
9.0873
9.0873
9.0873
9.0873
9.0873
9.0873
12.9739
12.9739
31.1111
6.1667
7.1154
7.1154
7.1154
8.8095
8.8095
8.8390
8.8095
8.8095
11.5074
12.6961
12.6961
12.6961
6.8376
8.5317
8.5317
6.0278
6.0278
6.0278
6.9765
6.9765
6.9765
6.9765
8.6706
8.6706
8.6706
12.5572
5.7500
5.7500
6.6987
6.6987
7.0984
6.6987
6.6987
8.2616
8.2616
8.3929
8.3929
8.3929
8.3929
8.3929
8.3929
8.3929
10.9300
12.2794
5.4722
6.4209
6.4209
6.4209
6.4209
8.0371
8.1151
D
85.1769
85.1769
92.2836
92.2836
92.2836
92.2836
90.4759
64.1733
71.7687
71.7687
79.4729
79.4729
79.4729
86.1036
55.9260
86.1036
74.1508
71.5039
71.5039
80.9671
80.9671
80.9671
80.9671
80.9671
90.6802
90.6802
90.6802
90.6802
90.6802
90.6802
99.1248
99.1248
96.9697
66.2037
74.9654
74.9654
74.9654
83.9585
83.9585
55.4112
83.9585
83.9585
60.5714
91.7771
91.7771
91.7771
69.4085
77.7351
77.7351
75.0620
75.0620
75.0620
86.3269
86.3269
86.3269
86.3269
98.0620
98.0620
98.0620
108.3937
68.8322
68.8322
79.1621
79.1621
53.2999
79.1621
79.1621
60.5454
60.5454
89.9233
89.9233
89.9233
89.9233
89.9233
89.9233
89.9233
66.9244
99.3975
63.1194
72.5920
72.5920
72.5920
72.5920
53.5593
82.4600
rs = 2, n0 = 1
A
G
68.3573 66.4656
68.3573 66.4656
58.9987 66.4522
58.9987 66.4522
58.9987 66.6997
58.9987 66.4522
20.3078 30.3030
60.7211 61.3429
63.6383 61.8938
63.6383 62.8994
63.8559 61.8181
63.8559 60.7460
63.8559 68.5957
55.6150 66.4522
38.4853 60.0718
55.6150 66.4522
59.9107 64.9040
67.2897 57.0203
67.2897 68.1522
71.3151 62.4213
71.3151 62.4213
71.3151 55.7044
71.3151 54.8195
71.3151 62.4213
71.6189 59.8190
71.6189 61.7361
71.6189 61.5146
71.6189 61.7361
71.6189 61.7361
71.6189 59.8190
60.4554 60.0297
60.4554 59.8070
19.0760 27.2727
62.4729 57.0203
65.9278 55.7044
65.9278 55.7044
65.9278 62.4213
66.1873 59.8190
66.1873 61.7361
43.6918 55.6363
66.1873 59.8190
66.1873 59.8190
40.3274 59.8070
56.5388 59.8070
56.5388 59.8070
56.5388 66.3354
61.2973 59.1886
61.5216 58.5226
61.5216 68.4254
70.7182 59.9994
70.7182 50.6847
70.7182 60.5797
75.7752 55.4856
75.7752 55.4856
75.7752 57.8060
75.7752 55.4856
76.1613 55.4948
76.1613 54.8766
76.1613 53.1725
62.3805 70.6882
64.8101 50.6847
64.8101 62.9835
69.0322 55.4856
69.0322 55.4856
44.5517 49.5150
69.0322 55.4856
69.0322 55.4856
47.2474 53.1725
47.2474 54.8766
69.3525 53.1725
69.3525 54.8766
69.3525 54.8766
69.3525 54.8766
69.3525 54.8766
69.3525 60.8225
69.3525 60.8225
42.8940 53.1618
57.7377 58.9648
59.8131 55.7153
63.3912 53.3185
63.3912 53.3185
63.3912 61.5716
63.3912 61.5716
41.6151 52.0201
63.6612 53.1725
IV
8.8550
8.8550
11.8038
11.8038
11.8038
11.8038
39.7375
7.3625
7.7476
7.7476
8.7175
8.7175
8.7175
11.6663
11.0969
11.6663
8.5800
6.9500
6.9500
7.3351
7.3351
7.3351
7.3351
7.3351
8.3050
8.3050
8.3050
8.3050
8.3050
8.3050
11.2538
11.2538
39.1875
6.8125
7.1976
7.1976
7.1976
8.1675
8.1675
8.4820
8.1675
8.1675
10.3348
11.1163
11.1163
11.1163
7.0601
8.0300
8.0300
6.4000
6.4000
6.4000
6.7851
6.7851
6.7851
6.7851
7.7550
7.7550
7.7550
10.7038
6.2625
6.2625
6.6476
6.6476
7.2384
6.6476
6.6476
7.7199
7.7199
7.6175
7.6175
7.6175
7.6175
7.6175
7.6175
7.6175
9.5727
10.5663
6.1250
6.5101
6.5101
6.5101
6.5101
7.6882
7.4800
310
Table of Criteria Values for CCDs (K = 4)
Dsgn
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
p
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
dv
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
l
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
c
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
q
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
D
82.8345
82.8345
53.8026
85.9987
91.6392
91.6392
96.5001
81.1589
56.0661
81.1589
59.7434
59.7434
86.4819
86.4819
86.4819
86.4819
86.4819
86.4819
86.4819
62.9124
62.9124
91.0692
91.0692
64.6818
76.5915
76.5915
54.1110
81.6149
81.6149
81.6149
81.6149
81.6149
81.6149
54.1110
56.9812
56.9812
85.9440
56.9812
90.3395
97.2893
60.3625
84.4363
84.4363
84.4363
65.0062
65.0062
90.9319
90.9319
69.0465
53.7769
78.9188
78.9188
78.9188
53.7769
57.9139
57.9139
57.9139
57.9139
84.9900
84.9900
57.9139
61.5135
35.8246
79.4364
96.7870
66.9360
89.2480
73.1619
58.2712
58.2712
82.2963
58.2712
63.6911
63.6911
39.1534
39.1534
42.0915
75.8861
55.4463
78.1627
65.7267
43.8178
48.9702
55.2694
37.2093
rs = 1, n0 = 1
A
G
63.9049 55.4227
63.9049 55.4227
43.1230 55.4227
84.4221 64.6959
80.8889 48.5333
80.8889 48.5333
70.6041 48.4948
79.4326 62.0104
51.8519 52.8639
79.4326 62.0104
52.1379 48.5333
52.1379 48.5333
76.2969 48.5333
76.2969 48.5333
76.2969 48.5333
76.2969 48.5333
76.2969 48.5333
76.2969 48.5333
76.2969 48.5333
48.6621 48.4948
48.6621 48.4410
67.0802 48.4948
67.0802 48.4948
38.0800 48.4068
75.0000 59.3119
75.0000 73.6450
47.2500 48.5333
72.1983 48.5333
72.1983 48.5333
72.1983 48.5333
72.1983 48.5333
72.1983 48.5333
72.1983 48.5333
47.2500 48.5333
44.3774 48.4948
44.3774 48.4948
63.8913 48.4948
44.3774 48.4948
89.1641 55.4537
84.6102 41.6000
55.9585 53.1517
82.7586 55.4537
82.7586 55.4537
82.7586 63.9639
56.3478 41.6000
56.3478 41.6000
78.8211 41.6000
78.8211 41.6000
51.6923 41.5670
49.5413 50.8388
77.2118 55.4537
77.2118 63.1243
77.2118 55.4537
49.5413 50.8388
49.8462 41.6000
49.8462 41.6000
49.8462 41.6000
49.8462 41.6000
73.7734 41.6000
73.7734 41.6000
49.8462 41.6000
46.1679 41.5670
26.4393 41.5670
69.3333 41.6000
96.7742 96.7742
62.9371 46.2114
87.9121 53.3032
63.5294 34.6667
53.5714 46.2114
53.5714 46.2114
80.5369 53.3032
53.5714 53.3032
54.0000 34.6667
54.0000 34.6667
30.5882 34.6667
30.5882 34.6667
30.4072 34.6392
74.3034 52.6035
46.9565 34.6667
77.4194 77.4194
61.0169 42.6426
35.0365 36.9691
39.6190 27.7333
50.3497 42.0828
28.0449 27.7333
IV
8.1151
8.1151
8.0371
5.6111
6.5598
6.5598
8.2540
5.3333
5.8697
5.3333
6.5211
6.5211
6.2821
6.2821
6.2821
6.2821
6.2821
6.2821
6.2821
7.6843
7.6843
7.9762
7.9762
10.3527
5.0556
5.0556
6.2966
6.0043
6.0043
6.0043
6.0043
6.0043
6.0043
6.2966
7.4597
7.4597
7.6984
7.4597
5.1944
6.1432
5.2924
4.9167
4.9167
4.9167
5.9437
5.9437
5.8654
5.8654
7.1069
5.0679
4.6389
4.6389
4.6389
5.0679
5.7192
5.7192
5.7192
5.7192
5.5876
5.5876
5.7192
6.8824
6.8887
5.3098
4.7778
4.7150
4.5000
5.3664
4.4905
4.4905
4.2222
4.4905
5.1419
5.1419
5.4544
5.4544
6.0048
3.9444
4.9173
4.1377
3.9132
4.2623
4.5705
3.6886
4.4723
D
82.4600
82.4600
53.5593
79.8981
93.7427
93.7427
108.4426
72.3656
49.9916
72.3656
58.6540
58.6540
84.9050
84.9050
84.9050
84.9050
84.9050
84.9050
84.9050
67.8516
67.8516
98.2191
98.2191
76.0814
65.5433
65.5433
50.9853
76.9005
76.9005
76.9005
76.9005
76.9005
76.9005
50.9853
58.9804
58.9804
88.9593
58.9804
86.8344
104.6308
55.3042
77.3606
77.3606
77.3606
66.6386
66.6386
93.2155
93.2155
78.9827
46.9638
68.9205
68.9205
68.9205
46.9638
56.5889
56.5889
56.5889
56.5889
83.0455
83.0455
56.5889
67.0714
39.0615
73.9852
97.5683
63.7036
84.9382
79.6756
52.3565
52.3565
73.9430
52.3565
65.4835
65.4835
40.2552
40.2552
49.3616
64.3711
53.8193
78.7541
61.6284
41.0856
54.3427
48.2268
38.4261
rs = 2, n0 = 1
A
G
63.6612 60.8225
63.6612 53.1725
41.6151 52.0201
75.6757 55.6121
82.4009 48.5499
82.4009 52.9466
82.9235 61.8687
68.0851 53.0073
43.4109 44.3491
68.0851 53.0073
48.9852 48.5499
48.9852 48.5499
73.4808 48.5499
73.4808 48.5499
73.4808 52.4919
73.4808 48.5499
73.4808 52.4919
73.4808 53.8752
73.4808 59.3436
52.7686 46.5259
52.7686 48.0170
73.8960 53.2197
73.8960 53.8851
46.7166 61.8522
61.8785 48.7509
61.8785 61.3353
42.2153 46.6537
66.3033 48.5499
66.3033 48.5499
66.3033 53.8752
66.3033 53.8752
66.3033 48.5499
66.3033 48.5499
42.2153 46.6537
44.9954 46.5259
44.9954 46.5259
66.6412 53.2197
44.9954 53.2197
83.4783 47.6675
93.2756 53.0358
48.1605 45.4348
73.0038 47.2376
73.0038 47.2376
73.0038 53.7536
56.4794 41.6142
56.4794 45.3828
80.3880 50.8660
80.3880 46.1787
62.5078 53.0303
40.6780 41.7865
64.8649 47.2376
64.8649 52.5731
64.8649 47.2376
40.6780 41.7865
46.4575 41.6142
46.4575 41.6142
46.4575 41.6142
46.4575 41.6142
70.6294 46.1787
70.6294 46.1787
46.4575 46.1787
50.4606 45.6169
28.3616 39.8793
62.9835 45.9026
97.5610 97.5610
56.8720 39.7229
81.2183 44.7947
71.8733 44.1965
45.1128 39.3647
45.1128 39.3647
69.5652 44.7947
45.1128 44.7947
54.0637 42.3883
54.0637 38.4823
29.9326 34.6785
29.9326 34.6785
38.7436 44.1919
60.8365 43.7932
43.3275 38.4823
78.0488 78.0488
53.9326 35.8357
29.9065 31.7783
47.3137 35.3572
41.2017 35.0488
27.3192 30.7858
IV
7.4800
7.4800
7.6882
5.8500
6.2351
6.2351
7.2050
5.7125
6.3999
5.7125
6.4763
6.4763
6.0976
6.0976
6.0976
6.0976
6.0976
6.0976
6.0976
6.9578
6.9578
7.0675
7.0675
8.8106
5.5750
5.5750
6.4445
5.9601
5.9601
5.9601
5.9601
5.9601
5.9601
6.4445
6.9261
6.9261
6.9300
6.9261
5.3000
5.6851
5.6378
5.1625
5.1625
5.1625
5.7142
5.7142
5.5476
5.5476
6.1957
5.6061
5.0250
5.0250
5.0250
5.6061
5.6824
5.6824
5.6824
5.6824
5.4101
5.4101
5.6824
6.1640
6.3392
5.2726
4.7500
4.8757
4.6125
4.9521
4.8440
4.8440
4.4750
4.8440
4.9203
4.9203
5.3331
5.3331
5.1725
4.3375
4.8886
4.1136
4.0819
4.5255
4.1663
4.0501
4.3608
311
Table of Criteria Values for CCDs (K = 4)
Dsgn
221
222
223
224
p
3
3
3
2
dv
2
2
1
1
l
2
1
1
1
c
0
1
0
0
q
0
0
1
0
D
61.3048
42.5063
24.6490
48.9898
rs = 1, n0 = 1
A
G
58.0645 58.0645
32.1429 31.9819
13.5652 20.8000
38.7097 38.7097
IV
3.3784
3.2802
1.9765
1.1944
D
61.7169
38.8792
28.2243
49.2366
rs = 2, n0 = 1
A
G
58.5366 58.5366
26.5193 26.8768
18.1369 26.5179
39.0244 39.0244
IV
3.3588
3.5532
1.7113
1.1875
rs = 2, n0 = 3
A
G
44.1210 81.4780
56.7699 76.8503
44.0115 79.1691
60.3485 71.3694
57.8460 73.7724
57.8460 73.6830
43.8857 73.5141
43.8857 81.3086
55.1614 73.2559
60.7093 65.9200
62.0026 69.9717
62.0026 68.6150
62.0026 69.3835
59.1542 68.0976
59.1542 68.0150
59.1542 75.3458
43.7399 75.0541
43.7399 67.8592
43.7399 75.0541
58.6860 68.5266
56.1280 68.0976
59.4595 60.6654
62.5753 64.2506
62.5753 63.7444
64.0781 64.1408
64.0781 63.6015
64.0781 70.9774
64.0781 69.1457
60.7786 69.0670
60.7786 69.6386
60.7786 62.4228
60.7786 62.3471
60.7786 69.0670
60.7786 69.0670
43.5688 68.7996
43.5688 68.7996
58.9099 62.8290
60.2400 64.1408
60.2400 63.0039
57.3149 62.4228
57.3149 69.0670
61.3027 59.9526
64.9716 58.4097
64.9716 57.9494
64.9716 65.3952
66.7600 62.8597
66.7600 64.7057
66.7600 58.3098
66.7600 57.8196
66.7600 64.5249
66.7600 62.8597
66.7600 62.8597
62.8497 62.7882
62.8497 62.7882
62.8497 63.3078
62.8497 62.7882
43.3653 62.5451
57.4506 58.0411
60.6609 58.4097
60.6609 59.4152
62.2170 58.3098
62.2170 57.2763
62.2170 64.7057
58.8072 62.7882
39.7190 56.7480
58.8072 62.7882
58.2529 61.2088
63.7168 53.9573
63.7168 64.5371
68.1620 58.8557
68.1620 58.8557
68.1620 52.5687
68.1620 52.1545
68.1620 58.8557
IV
20.1736
12.8884
19.5903
10.5498
12.3050
12.3050
19.0069
19.0069
12.1592
9.2808
9.9665
9.9665
9.9665
11.7217
11.7217
11.7217
18.4236
18.4236
18.4236
9.8206
11.5759
8.4167
8.6974
8.6974
9.3831
9.3831
9.3831
9.3831
11.1384
11.1384
11.1384
11.1384
11.1384
11.1384
17.8403
17.8403
8.5516
9.2373
9.2373
10.9925
10.9925
7.8333
8.1141
8.1141
8.1141
8.7998
8.7998
8.7998
8.7998
8.7998
8.7998
8.7998
10.5550
10.5550
10.5550
10.5550
17.2569
7.6875
7.9683
7.9683
8.6540
8.6540
8.6540
10.4092
10.3343
10.4092
8.5081
7.2500
7.2500
7.5308
7.5308
7.5308
7.5308
7.5308
Table of Criteria Values for CCDs (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
D
76.4417
76.2613
77.8445
75.3677
77.7554
77.7554
79.4953
79.4953
75.3677
73.9397
76.8932
76.8932
76.8932
79.5357
79.5357
79.5357
81.4655
81.4655
81.4655
74.3384
76.8932
72.0191
75.4425
75.4425
78.7358
78.7358
78.7358
78.7358
81.6922
81.6922
81.6922
81.6922
81.6922
81.6922
83.8569
83.8569
72.7123
75.8864
75.8864
78.7358
78.7358
73.4374
77.2862
77.2862
77.2862
81.0054
81.0054
81.0054
81.0054
81.0054
81.0054
81.0054
84.3574
84.3574
84.3574
84.3574
86.8194
70.5193
74.2152
74.2152
77.7866
77.7866
77.7866
81.0054
52.6146
81.0054
74.6957
75.2087
75.2087
79.6009
79.6009
79.6009
79.6009
79.6009
rs = 1, n0 = 3
A
G
52.2876 95.2381
60.8696 89.1470
51.8519 88.8889
65.8749 82.9504
60.9971 82.7793
60.9971 82.6454
51.3580 82.5397
51.3580 82.5397
59.4286 83.8710
68.8172 76.7054
66.4935 77.4317
66.4935 76.5696
66.4935 76.5669
61.1465 76.4117
61.1465 76.2880
61.1465 76.4117
50.7937 76.1905
50.7937 76.1905
50.7937 76.1905
64.4836 77.4317
59.4427 77.4194
70.4000 71.7019
69.8413 71.0005
69.8413 71.2940
67.2397 70.9791
67.2397 70.1864
67.2397 70.9791
67.2397 70.9791
61.3240 70.0440
61.3240 70.0440
61.3240 70.9677
61.3240 69.9307
61.3240 70.0440
61.3240 70.0440
50.1425 69.8413
50.1425 69.8413
67.4330 71.0005
65.0046 70.9791
65.0046 70.9709
59.4595 70.9677
59.4595 70.9677
71.7489 70.3191
71.1111 64.5459
71.1111 65.0474
71.1111 65.0474
68.1576 64.5264
68.1576 64.5264
68.1576 64.5264
68.1576 63.8058
68.1576 64.5264
68.1576 64.5264
68.1576 64.5264
61.5385 64.5161
61.5385 63.6764
61.5385 63.6764
61.5385 63.6764
49.3827 63.4921
68.9655 68.7889
68.3761 64.5459
68.3761 64.8127
65.6410 64.5264
65.6410 64.5190
65.6410 64.5264
59.4796 64.5161
40.8163 64.5161
59.4796 64.5161
63.3037 65.0407
73.4694 63.2872
73.4694 74.3095
72.7273 58.5427
72.7273 58.5427
72.7273 58.0913
72.7273 58.8235
72.7273 58.5427
IV
17.1000
12.8500
16.6500
10.5000
12.4000
12.4000
16.2000
16.2000
12.1000
8.9100
10.0500
10.0500
10.0500
11.9500
11.9500
11.9500
15.7500
15.7500
15.7500
9.7500
11.6500
7.7000
8.4600
8.4600
9.6000
9.6000
9.6000
9.6000
11.5000
11.5000
11.5000
11.5000
11.5000
11.5000
15.3000
15.3000
8.1600
9.3000
9.3000
11.2000
11.2000
7.2500
8.0100
8.0100
8.0100
9.1500
9.1500
9.1500
9.1500
9.1500
9.1500
9.1500
11.0500
11.0500
11.0500
11.0500
14.8500
6.9500
7.7100
7.7100
8.8500
8.8500
8.8500
10.7500
10.5655
10.7500
8.5500
6.8000
6.8000
7.5600
7.5600
7.5600
7.5600
7.5600
D
74.5552
73.9985
77.2060
71.3807
76.7915
76.7915
80.3820
80.3820
72.8043
67.6855
74.0812
74.0812
74.0812
80.1835
80.1835
80.1835
84.2528
84.2528
84.2528
69.9234
75.6831
63.1570
70.1440
70.1440
77.4048
77.4048
77.4048
77.4048
84.3859
84.3859
84.3859
84.3859
84.3859
84.3859
89.0684
89.0684
65.8603
72.6777
72.6777
79.2325
79.2325
65.2317
73.2123
73.2123
73.2123
81.5904
81.5904
81.5904
81.5904
81.5904
81.5904
81.5904
89.7205
89.7205
89.7205
89.7205
95.2119
60.8633
68.3095
68.3095
76.1266
76.1266
76.1266
83.7122
54.3726
83.7122
71.0286
67.8602
67.8602
77.1454
77.1454
77.1454
77.1454
77.1454
312
Table of Criteria Values for CCDs (K = 4)
Dsgn
75
76
77
78
79
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
p
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
l
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
c
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
q
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
D
83.8684
83.8684
83.8684
83.8684
83.8684
87.7332
87.7332
90.5828
71.8956
76.0944
76.0944
76.0944
80.1738
80.1738
52.9134
80.1738
80.1738
55.3517
83.8684
83.8684
83.8684
72.7423
76.6420
76.6420
77.4832
77.4832
77.4832
82.5921
82.5921
82.5921
82.5921
87.5898
87.5898
87.5898
92.1435
73.6539
73.6539
78.5104
78.5104
52.8611
78.5104
78.5104
56.0598
56.0598
83.2611
83.2611
83.2611
83.2611
83.2611
83.2611
83.2611
58.9743
87.5898
70.0139
74.6304
74.6304
74.6304
74.6304
51.4070
79.1463
79.1463
79.1463
51.4070
80.5088
86.6036
86.6036
92.6181
75.9779
52.4870
75.9779
56.4605
56.4605
81.7297
81.7297
81.7297
81.7297
81.7297
81.7297
81.7297
60.3816
60.3816
87.4057
87.4057
63.9824
71.7020
rs = 1, n0 = 3
A
G
69.3141 58.0738
69.3141 58.0738
69.3141 58.0738
69.3141 58.0738
69.3141 58.0738
61.8026 57.3088
61.8026 58.0645
48.4848 57.1429
70.2439 63.2872
69.5652 58.0913
69.5652 58.0913
69.5652 58.5427
66.4360 58.0738
66.4360 58.0738
44.4444 58.0738
66.4360 58.0738
66.4360 58.0738
41.8605 58.0645
59.5041 58.0645
59.5041 58.0645
59.5041 58.0645
66.6667 58.5427
63.7874 58.5366
63.7874 58.5366
75.7396 65.4878
75.7396 56.2553
75.7396 66.0529
74.8538 52.0379
74.8538 52.0379
74.8538 52.2876
74.8538 52.0379
70.8160 52.0325
70.8160 51.6211
70.8160 52.0325
62.1359 51.6129
71.9101 56.2553
71.9101 68.9332
71.1111 52.0379
71.1111 52.0379
46.7836 51.6367
71.1111 52.0379
71.1111 52.0379
46.3768 51.6211
46.3768 51.6211
67.4572 52.0325
67.4572 51.6211
67.4572 51.6211
67.4572 51.6211
67.4572 51.6211
67.4572 52.0325
67.4572 52.0325
43.2432 51.6129
59.5349 51.6129
68.4492 62.5705
67.7249 52.0379
67.7249 52.0379
67.7249 52.2876
67.7249 52.2876
42.6667 52.0325
64.4025 52.0325
64.4025 52.0325
64.4025 52.0325
42.6667 52.0325
78.8732 60.3166
77.7778 45.5332
77.7778 45.7516
72.8455 45.5285
74.1722 57.7963
48.2759 49.2234
74.1722 57.7963
49.6454 45.5332
49.6454 45.5332
73.2026 45.5332
73.2026 45.5332
73.2026 45.7516
73.2026 45.5332
73.2026 45.7516
73.2026 45.7516
73.2026 45.7516
49.1228 45.5285
49.1228 45.1685
68.8172 45.5285
68.8172 45.5285
45.1613 45.1613
70.0000 55.2653
IV
8.7000
8.7000
8.7000
8.7000
8.7000
10.6000
10.6000
14.4000
6.5000
7.2600
7.2600
7.2600
8.4000
8.4000
8.6949
8.4000
8.4000
9.9420
10.3000
10.3000
10.3000
6.9600
8.1000
8.1000
6.3500
6.3500
6.3500
7.1100
7.1100
7.1100
7.1100
8.2500
8.2500
8.2500
10.1500
6.0500
6.0500
6.8100
6.8100
7.3231
6.8100
6.8100
8.0714
8.0714
7.9500
7.9500
7.9500
7.9500
7.9500
7.9500
7.9500
9.3184
9.8500
5.7500
6.5100
6.5100
6.5100
6.5100
7.8289
7.6500
7.6500
7.6500
7.8289
5.9000
6.6600
6.6600
7.8000
5.6000
6.2007
5.6000
6.6996
6.6996
6.3600
6.3600
6.3600
6.3600
6.3600
6.3600
6.3600
7.4478
7.4478
7.5000
7.5000
8.6949
5.3000
D
87.0149
87.0149
87.0149
87.0149
87.0149
96.7006
96.7006
103.2988
62.8300
71.4270
71.4270
71.4270
80.5649
80.5649
53.1715
80.5649
80.5649
59.0901
89.5327
89.5327
89.5327
66.1324
74.5930
74.5930
71.2952
71.2952
71.2952
82.3602
82.3602
82.3602
82.3602
94.3053
94.3053
94.3053
106.1944
65.3780
65.3780
75.5246
75.5246
50.8508
75.5246
75.5246
58.2260
58.2260
86.4784
86.4784
86.4784
86.4784
86.4784
86.4784
86.4784
65.5665
97.3807
59.9519
69.2564
69.2564
69.2564
69.2564
51.5075
79.3010
79.3010
79.3010
51.5075
75.9684
89.5862
89.5862
104.5833
68.8064
47.5328
68.8064
56.0533
56.0533
81.1403
81.1403
81.1403
81.1403
81.1403
81.1403
81.1403
65.4369
65.4369
94.7236
94.7236
74.9470
62.3196
rs = 2, n0 = 3
A
G
70.3590 56.5737
70.3590 58.2352
70.3590 58.0724
70.3590 58.2352
70.3590 56.5737
65.5811 56.9770
65.5811 56.5094
43.1192 74.8052
59.1376 53.9573
62.9477 52.5687
62.9477 52.5687
62.9477 58.8557
64.8168 56.5737
64.8168 58.2352
42.3833 52.4788
64.8168 56.5737
64.8168 56.5737
42.2773 56.5094
60.7402 56.5094
60.7402 56.5094
60.7402 63.0062
58.4745 55.8175
60.0840 55.2416
60.0840 64.5483
67.0157 56.8144
67.0157 47.9621
67.0157 57.3663
72.6195 52.3162
72.6195 52.3162
72.6195 55.0886
72.6195 52.3162
75.4429 53.8705
75.4429 51.7646
75.4429 50.2878
69.3483 66.8374
61.3909 47.9621
61.3909 59.6528
66.0607 52.3162
66.0607 52.3162
42.4528 46.7277
66.0607 52.3162
66.0607 52.3162
46.1205 50.2878
46.1205 51.7646
68.3890 50.2878
68.3890 51.7646
68.3890 51.7646
68.3890 51.7646
68.3890 51.7646
68.3890 57.3762
68.3890 57.3762
45.9794 50.2305
63.3427 56.0055
56.6372 52.7415
60.5885 50.2731
60.5885 50.2731
60.5885 58.0547
60.5885 58.0547
40.4522 49.1036
62.5413 50.2878
62.5413 57.3762
62.5413 50.2878
40.4522 49.1036
71.7949 52.6734
79.2857 45.7767
79.2857 50.4831
83.1696 58.5714
64.5533 50.1955
41.0758 41.9668
64.5533 50.1955
46.8053 45.7767
46.8053 45.7767
70.5462 45.7767
70.5462 45.7767
70.5462 50.0447
70.5462 45.7767
70.5462 50.0447
70.5462 50.7979
70.5462 56.6595
52.0176 44.0018
52.0176 45.2940
73.6045 50.2042
73.6045 52.4781
51.8126 58.4828
58.6387 46.1488
IV
8.2165
8.2165
8.2165
8.2165
8.2165
9.9717
9.9717
16.6736
7.1042
7.3849
7.3849
7.3849
8.0706
8.0706
8.5421
8.0706
8.0706
9.5260
9.8259
9.8259
9.8259
7.2391
7.9248
7.9248
6.6667
6.6667
6.6667
6.9474
6.9474
6.9474
6.9474
7.6331
7.6331
7.6331
9.3884
6.5208
6.5208
6.8016
6.8016
7.4752
6.8016
6.8016
7.7338
7.7338
7.4873
7.4873
7.4873
7.4873
7.4873
7.4873
7.4873
8.7177
9.2425
6.3750
6.6558
6.6558
6.6558
6.6558
7.7001
7.3415
7.3415
7.3415
7.7001
6.0833
6.3641
6.3641
7.0498
5.9375
6.6828
5.9375
6.6669
6.6669
6.2183
6.2183
6.2183
6.2183
6.2183
6.2183
6.2183
6.9255
6.9255
6.9040
6.9040
7.9094
5.7917
313
Table of Criteria Values for CCDs (K = 4)
Dsgn
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
71.7020
51.1376
77.1301
77.1301
77.1301
77.1301
77.1301
77.1301
51.1376
54.6890
54.6890
82.4867
54.6890
84.7275
92.2570
56.6127
79.1910
79.1910
79.1910
61.6437
61.6437
86.2285
86.2285
66.6667
50.4362
74.0163
74.0163
74.0163
50.4362
54.9183
54.9183
54.9183
54.9183
80.5939
80.5939
54.9183
59.3932
34.5899
75.3275
91.0077
62.9392
83.9189
69.7093
54.7917
54.7917
77.3823
54.7917
60.6855
60.6855
37.3057
37.3057
40.9826
71.3548
52.8298
73.7788
62.0403
41.3602
46.9943
52.1695
35.7079
58.2387
40.3805
23.9382
47.1405
rs = 1, n0 = 3
A
G
70.0000 68.7254
44.8718 45.5332
69.1358 45.5332
69.1358 45.5332
69.1358 45.7516
69.1358 45.7516
69.1358 45.5332
69.1358 45.5332
44.8718 45.5332
44.4444 45.5285
44.4444 45.5285
65.2111 45.5285
44.4444 45.5285
83.4783 51.6999
82.0513 39.2157
52.1739 49.5396
77.4194 51.6999
77.4194 51.6999
77.4194 59.6972
54.0541 39.0284
54.0541 39.2157
76.1905 39.2157
76.1905 39.2157
53.3333 39.0244
46.1538 47.3703
72.1805 51.6999
72.1805 58.9075
72.1805 51.6999
46.1538 47.3703
47.6190 39.0284
47.6190 39.0284
47.6190 39.0284
47.6190 39.0284
71.1111 39.2157
71.1111 39.2157
47.6190 39.2157
47.0588 39.0244
26.4463 39.0244
66.6667 39.2157
90.9091 90.9091
58.8235 43.0833
82.4742 49.7477
61.7284 32.6797
50.0000 43.0833
50.0000 43.0833
75.4717 49.7477
50.0000 49.7477
52.0833 32.6797
52.0833 32.6797
29.2398 32.5237
29.2398 32.5237
31.3725 32.5203
69.5652 49.0895
45.0450 32.6797
72.7273 72.7273
57.1429 39.7981
32.6531 34.4666
38.6473 26.1438
47.0588 39.2716
26.9360 26.1438
54.5455 54.5455
30.0000 29.8486
13.3333 19.6078
36.3636 36.3636
IV
5.3000
6.4571
6.0600
6.0600
6.0600
6.0600
6.0600
6.0600
6.4571
7.2053
7.2053
7.2000
7.2053
5.4500
6.2100
5.5772
5.1500
5.1500
5.1500
6.0760
6.0760
5.9100
5.9100
6.8243
5.3347
4.8500
4.8500
4.8500
5.3347
5.8335
5.8335
5.8335
5.8335
5.6100
5.6100
5.8335
6.5818
6.8943
5.3100
5.0000
4.9537
4.7000
5.4525
4.7112
4.7112
4.4000
4.7112
5.2100
5.2100
5.6427
5.6427
5.9397
4.1000
4.9675
4.3301
4.0876
4.4901
4.6881
3.8452
4.5821
3.5355
3.4295
2.0700
1.2500
D
62.3196
48.7246
73.4907
73.4907
73.4907
73.4907
73.4907
73.4907
48.7246
56.8814
56.8814
85.7934
56.8814
82.6793
100.2165
52.6578
73.6589
73.6589
73.6589
63.8272
63.8272
89.2827
89.2827
76.4594
44.7166
65.6226
65.6226
65.6226
44.7166
54.2014
54.2014
54.2014
54.2014
79.5419
79.5419
54.2014
64.9286
37.8136
70.8637
93.0820
60.7744
81.0326
76.5547
49.9490
49.9490
70.5429
49.9490
62.9185
62.9185
38.6784
38.6784
48.0373
61.4112
51.7112
75.3542
58.9678
39.3119
52.4611
46.1448
37.0956
59.3428
37.3836
27.4623
47.8091
rs = 2, n0 = 3
A
G
58.6387 58.1215
40.2620 43.9890
63.5420 45.7767
63.5420 45.7767
63.5420 50.7979
63.5420 50.7979
63.5420 45.7767
63.5420 45.7767
40.2620 43.9890
44.0597 44.0018
44.0597 44.0018
66.0126 50.2042
44.0597 50.2042
79.3388 45.1486
90.3434 50.2405
45.6177 43.0247
69.3141 44.7396
69.3141 44.7396
69.3141 50.9428
54.2167 39.2371
54.2167 43.2712
77.5687 48.5653
77.5687 43.5410
62.7084 50.2041
38.5027 39.5561
61.5385 44.7396
61.5385 49.8184
61.5385 44.7396
38.5027 39.5561
44.4535 39.2371
44.4535 39.2371
44.4535 39.2371
44.4535 39.2371
67.9592 43.5410
67.9592 43.5410
44.4535 43.5410
50.0056 43.0322
27.7672 37.7158
60.4682 43.2961
93.0233 93.0233
53.9730 37.6238
77.2947 42.4523
69.6592 41.8670
42.7553 37.2830
42.7553 37.2830
66.1157 42.4523
42.7553 42.4523
52.0380 40.4711
52.0380 36.2842
28.6359 32.6976
28.6359 32.6976
38.8856 41.8367
57.7617 41.4984
41.5320 36.2842
74.4186 74.4186
51.2456 33.9618
28.3186 30.0991
45.9211 33.4936
39.0773 33.2123
26.1899 29.0273
55.8140 55.8140
25.1309 25.4714
17.6599 25.1202
37.2093 37.2093
IV
5.7917
6.6332
6.0724
6.0724
6.0724
6.0724
6.0724
6.0724
6.6332
6.8918
6.8918
6.7581
6.8918
5.5000
5.7808
5.8745
5.3542
5.3542
5.3542
5.8586
5.8586
5.6349
5.6349
6.1172
5.8409
5.2083
5.2083
5.2083
5.8409
5.8249
5.8249
5.8249
5.8249
5.4891
5.4891
5.8249
6.0835
6.4337
5.3433
4.9167
5.0662
4.7708
5.0503
5.0326
5.0326
4.6250
5.0326
5.0166
5.0166
5.5109
5.5109
5.1962
4.4792
4.9829
4.2580
4.2243
4.7140
4.2734
4.1906
4.4797
3.4766
3.6828
1.7802
1.2292
Table of Criteria Values for BBDs (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
D
76.7262
79.6846
77.7266
79.8382
81.0395
81.0395
78.8971
78.8971
78.5511
78.9078
81.3230
81.3230
81.3230
82.6495
n0 = 1
A
G
31.6483 60.0000
56.5543 96.1468
30.5454 56.0000
67.3941 89.3338
56.0526 89.2792
56.0526 89.2027
29.3647 52.0000
29.3647 52.0000
54.8216 89.8978
72.8756 82.5048
67.6933 83.0412
67.6933 82.4619
67.6933 82.6903
55.4785 82.4116
IV
33.6112
15.0572
33.1945
10.7540
14.6406
14.6406
32.7778
32.7778
14.3628
8.6432
10.3374
10.3374
10.3374
14.2239
D
76.4413
76.2608
77.8441
75.3673
77.7550
77.7550
79.4948
79.4948
75.3674
73.9392
76.8927
76.8927
76.8927
79.5352
n0 = 3
A
G
52.2874 95.2376
60.8693 89.1465
51.8517 88.8884
65.8746 82.9500
60.9968 82.7789
60.9968 82.6449
51.3578 82.5392
51.3578 82.5392
59.4284 83.8705
68.8169 76.7050
66.4932 77.4313
66.4932 76.5692
66.4932 76.5665
61.1462 76.4113
IV
17.1001
12.8501
16.6501
10.5001
12.4001
12.4001
16.2001
16.2001
12.1000
8.9101
10.0501
10.0501
10.0501
11.9501
314
Table of Criteria Values for BBDs (K = 4)
Dsgn
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
p
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
l
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
c
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
q
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
D
82.6495
82.6495
80.2850
80.2850
80.2850
78.6212
79.9036
77.2378
80.4242
80.4242
83.1134
83.1134
83.1134
83.1134
84.5934
84.5934
84.5934
84.5934
84.5934
84.5934
81.9568
81.9568
77.5139
80.1057
80.1057
81.5321
81.5321
78.7037
82.2825
82.2825
82.2825
85.3139
85.3139
85.3139
85.3139
85.3139
85.3139
85.3139
86.9865
86.9865
86.9865
86.9865
84.0089
75.5765
79.0131
79.0131
81.9241
81.9241
81.9241
83.5302
54.2545
83.5302
78.6690
80.5331
80.5331
84.6122
84.6122
84.6122
84.6122
84.6122
88.0828
88.0828
88.0828
88.0828
88.0828
88.0828
90.0036
90.0036
86.5870
76.9857
80.8850
80.8850
80.8850
84.2028
84.2028
55.5724
84.2028
84.2028
56.7843
86.0390
86.0390
86.0390
77.3221
80.4936
80.4936
n0 = 1
A
G
55.4785 82.3410
55.4785 82.4116
28.0975 48.0000
28.0975 48.0000
28.0975 48.0000
65.7613 83.0412
54.1741 82.9826
75.6443 77.0359
73.8061 76.1293
73.8061 76.1031
68.0503 76.1211
68.0503 75.7995
68.0503 76.1211
68.0503 76.1211
54.8150 75.5439
54.8150 75.5439
54.8150 76.0674
54.8150 75.4792
54.8150 75.5439
54.8150 75.5439
26.7341 44.0000
26.7341 44.0000
71.3140 76.1293
65.9262 76.1211
65.9262 76.0679
53.4284 76.0674
53.4284 76.0674
77.0461 75.5193
74.9545 69.2084
74.9545 69.3329
74.9545 69.3329
68.4837 69.2010
68.4837 69.2010
68.4837 69.2010
68.4837 68.9086
68.4837 69.2010
68.4837 69.2010
68.4837 69.2010
54.0395 69.1522
54.0395 68.6763
54.0395 68.6763
54.0395 68.6763
25.2631 40.0000
74.0737 73.8850
72.1384 69.2084
72.1384 69.1846
66.1252 69.2010
66.1252 69.1527
66.1252 69.2010
52.5602 69.1522
37.4311 69.1522
52.5602 69.1522
63.9237 69.2780
78.8316 67.9674
78.8316 79.7270
76.4076 62.3997
76.4076 62.3997
76.4076 62.2876
76.4076 62.3997
76.4076 62.3997
69.0210 62.2809
69.0210 62.2809
69.0210 62.2809
69.0210 62.2809
69.0210 62.2809
69.0210 62.3502
53.1209 61.8087
53.1209 62.2369
23.6712 36.0000
75.3923 67.9674
73.1723 62.2876
73.1723 62.2876
73.1723 62.3997
66.3701 62.2809
66.3701 62.2809
45.1341 62.2809
66.3701 62.2809
66.3701 62.2809
37.5971 62.2369
51.5367 62.2369
51.5367 62.2369
51.5367 62.2369
70.1998 62.3997
63.9153 62.3502
63.9153 62.3502
IV
14.2239
14.2239
32.3612
32.3612
32.3612
10.0596
13.9461
7.2778
8.2265
8.2265
9.9207
9.9207
9.9207
9.9207
13.8072
13.8072
13.8072
13.8072
13.8072
13.8072
31.9445
31.9445
7.9488
9.6429
9.6429
13.5295
13.5295
6.8612
7.8099
7.8099
7.8099
9.5040
9.5040
9.5040
9.5040
9.5040
9.5040
9.5040
13.3906
13.3906
13.3906
13.3906
31.5278
6.5834
7.5321
7.5321
9.2262
9.2262
9.2262
13.1128
12.0848
13.1128
8.9484
6.4445
6.4445
7.3932
7.3932
7.3932
7.3932
7.3932
9.0874
9.0874
9.0874
9.0874
9.0874
9.0874
12.9739
12.9739
31.1112
6.1667
7.1154
7.1154
7.1154
8.8096
8.8096
8.8390
8.8096
8.8096
11.5074
12.6961
12.6961
12.6961
6.8376
8.5318
8.5318
D
79.5352
79.5352
81.4649
81.4649
81.4649
74.3381
76.8928
72.0186
75.4420
75.4420
78.7353
78.7353
78.7353
78.7353
81.6916
81.6916
81.6916
81.6916
81.6916
81.6916
83.8563
83.8563
72.7119
75.8860
75.8860
78.7354
78.7354
73.4368
77.2856
77.2856
77.2856
81.0048
81.0048
81.0048
81.0048
81.0048
81.0048
81.0048
84.3567
84.3567
84.3567
84.3567
86.8188
70.5189
74.2148
77.7862
77.7862
77.7862
77.7862
81.0049
52.6142
81.0049
74.6954
75.2081
75.2081
79.6003
79.6003
79.6003
79.6003
79.6003
83.8676
83.8676
83.8676
83.8676
83.8676
83.8676
87.7324
87.7324
90.5821
71.8952
76.0939
76.0939
76.0939
80.1733
80.1733
52.9130
80.1733
80.1733
55.3514
83.8678
83.8678
83.8678
72.7419
76.6417
76.6417
n0 = 3
A
G
61.1462 76.2876
61.1462 76.4113
50.7935 76.1901
50.7935 76.1901
50.7935 76.1901
64.4834 77.4313
59.4425 77.4189
70.3996 71.7015
69.8409 71.0001
69.8409 71.2936
67.2394 70.9787
67.2394 70.1860
67.2394 70.9787
67.2394 70.9787
61.3237 70.0436
61.3237 70.0436
61.3237 70.9673
61.3237 69.9303
61.3237 70.0436
61.3237 70.0436
50.1423 69.8409
50.1423 69.8409
67.4327 71.0001
65.0044 70.9787
65.0044 70.9705
59.4593 70.9673
59.4593 70.9673
71.7484 70.3187
71.1107 64.5456
71.1107 65.0470
71.1107 65.0470
68.1572 64.5261
68.1572 64.5261
68.1572 64.5261
68.1572 63.8054
68.1572 64.5261
68.1572 64.5261
68.1572 64.5261
61.5381 64.5158
61.5381 63.6760
61.5381 63.6760
61.5381 63.6760
49.3825 63.4917
68.9652 68.7886
68.3758 64.5456
65.6407 64.5186
65.6407 64.5261
65.6407 64.5186
65.6407 64.5261
59.4793 64.5158
40.8162 64.5158
59.4793 64.5158
63.3035 65.0403
73.4689 63.2868
73.4689 74.3089
72.7268 58.5423
72.7268 58.5423
72.7268 58.0910
72.7268 58.8232
72.7268 58.5423
69.3136 58.0734
69.3136 58.0734
69.3136 58.0734
69.3136 58.0734
69.3136 58.0734
69.3136 58.5363
61.8022 57.3084
61.8022 58.0642
48.4846 57.1425
70.2435 63.2868
69.5649 58.0910
69.5649 58.0910
69.5649 58.5423
66.4357 58.0734
66.4357 58.0734
44.4443 58.0734
66.4357 58.0734
66.4357 58.0734
41.8603 58.0642
59.5039 58.0642
59.5039 58.0642
59.5039 58.0642
66.6665 58.5423
63.7872 58.5363
63.7872 58.5363
IV
11.9501
11.9501
15.7501
15.7501
15.7501
9.7500
11.6500
7.7001
8.4601
8.4601
9.6001
9.6001
9.6001
9.6001
11.5001
11.5001
11.5001
11.5001
11.5001
11.5001
15.3001
15.3001
8.1600
9.3000
9.3000
11.2000
11.2000
7.2501
8.0101
8.0101
8.0101
9.1501
9.1501
9.1501
9.1501
9.1501
9.1501
9.1501
11.0501
11.0501
11.0501
11.0501
14.8501
6.9500
7.7100
8.8500
8.8500
8.8500
8.8500
10.7500
10.5656
10.7500
8.5500
6.8001
6.8001
7.5601
7.5601
7.5601
7.5601
7.5601
8.7001
8.7001
8.7001
8.7001
8.7001
8.7001
10.6001
10.6001
14.4001
6.5000
7.2600
7.2600
7.2600
8.4000
8.4000
8.6949
8.4000
8.4000
9.9420
10.3000
10.3000
10.3000
6.9600
8.1000
8.1000
315
Table of Criteria Values for BBDs (K = 4)
Dsgn
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
p
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
l
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
c
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
q
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
D
82.8798
82.8798
82.8798
87.6172
87.6172
87.6172
87.6172
91.6706
91.6706
91.6706
93.9226
78.7841
78.7841
83.2874
83.2874
56.0775
83.2874
83.2874
58.6717
58.6717
87.1404
87.1404
87.1404
87.1404
87.1404
87.1404
87.1404
60.1131
89.2811
74.8908
79.1715
79.1715
79.1715
79.1715
53.8023
82.8341
82.8341
82.8341
53.8023
85.9978
91.6382
91.6382
96.4990
81.1582
56.0657
81.1582
59.7429
59.7429
86.4812
86.4812
86.4812
86.4812
86.4812
86.4812
86.4812
62.9119
62.9119
91.0685
91.0685
64.6813
76.5910
76.5910
54.1107
81.6145
81.6145
81.6145
81.6145
81.6145
81.6145
54.1107
56.9809
56.9809
85.9436
56.9809
90.3383
97.2880
60.3619
84.4354
84.4354
84.4354
65.0055
65.0055
90.9311
90.9311
69.0459
n0 = 1
A
G
81.1833 70.2662
81.1833 60.4155
81.1833 70.8685
78.3053 55.4664
78.3053 55.4664
78.3053 55.4664
78.3053 55.4664
69.7045 55.4224
69.7045 55.3608
69.7045 55.4224
52.0157 55.3217
77.1080 60.4155
77.1080 73.9377
74.5070 55.4664
74.5070 55.4664
49.3712 55.3667
74.5070 55.4664
74.5070 55.4664
46.6125 55.3608
46.6125 55.3608
66.6787 55.4224
66.6787 55.3608
66.6787 55.3608
66.6787 55.3608
66.6787 55.3608
66.6787 55.4224
66.6787 55.4224
37.8068 55.3217
50.3119 55.3217
73.4223 67.1556
71.0602 55.4664
71.0602 55.4664
71.0602 55.4664
71.0602 55.4664
43.1229 55.4224
63.9047 55.4224
63.9047 55.4224
63.9047 55.4224
43.1229 55.4224
84.4213 64.6955
80.8881 48.5331
80.8881 48.5331
70.6036 48.4946
79.4321 62.0099
51.8515 52.8635
79.4321 62.0099
52.1376 48.5331
52.1376 48.5331
76.2964 48.5331
76.2964 48.5331
76.2964 48.5331
76.2964 48.5331
76.2964 48.5331
76.2964 48.5331
76.2964 48.5331
48.6618 48.4946
48.6618 48.4407
67.0798 48.4946
67.0798 48.4946
38.0798 48.4065
74.9997 59.3117
74.9997 73.6446
47.2498 48.5331
72.1980 48.5331
72.1980 48.5331
72.1980 48.5331
72.1980 48.5331
72.1980 48.5331
72.1980 48.5331
47.2498 48.5331
44.3772 48.4946
44.3772 48.4946
63.8911 48.4946
44.3772 48.4946
89.1630 55.4533
84.6092 41.5998
55.9581 53.1514
82.7579 55.4533
82.7579 55.4533
82.7579 63.9635
56.3474 41.5998
56.3474 41.5998
78.8204 41.5998
78.8204 41.5998
51.6920 41.5668
IV
6.0278
6.0278
6.0278
6.9765
6.9765
6.9765
6.9765
8.6707
8.6707
8.6707
12.5572
5.7500
5.7500
6.6988
6.6988
7.0985
6.6988
6.6988
8.2617
8.2617
8.3929
8.3929
8.3929
8.3929
8.3929
8.3929
8.3929
10.9301
12.2795
5.4722
6.4210
6.4210
6.4210
6.4210
8.0371
8.1151
8.1151
8.1151
8.0371
5.6112
6.5599
6.5599
8.2540
5.3334
5.8698
5.3334
6.5211
6.5211
6.2821
6.2821
6.2821
6.2821
6.2821
6.2821
6.2821
7.6843
7.6843
7.9762
7.9762
10.3527
5.0556
5.0556
6.2966
6.0043
6.0043
6.0043
6.0043
6.0043
6.0043
6.2966
7.4598
7.4598
7.6984
7.4598
5.1945
6.1432
5.2924
4.9167
4.9167
4.9167
5.9438
5.9438
5.8654
5.8654
7.1070
D
77.4824
77.4824
77.4824
82.5913
82.5913
82.5913
82.5913
87.5889
87.5889
87.5889
92.1426
73.6534
73.6534
78.5098
78.5098
52.8607
78.5098
78.5098
56.0594
56.0594
83.2605
83.2605
83.2605
83.2605
83.2605
83.2605
83.2605
58.9739
87.5891
70.0136
74.6300
74.6300
74.6300
74.6300
51.4068
79.1459
79.1459
79.1459
51.4068
80.5079
86.6026
86.6026
92.6171
75.9773
52.4866
75.9773
56.4600
56.4600
81.7290
81.7290
81.7290
81.7290
81.7290
81.7290
81.7290
60.3811
60.3811
87.4050
87.4050
63.9818
71.7016
71.7016
51.1373
77.1297
77.1297
77.1297
77.1297
77.1297
77.1297
51.1373
54.6887
54.6887
82.4863
54.6887
84.7264
92.2558
56.6122
79.1902
79.1902
79.1902
61.6431
61.6431
86.2276
86.2276
66.6660
n0 = 3
A
G
75.7390 65.4873
75.7390 56.2549
75.7390 66.0524
74.8532 52.0376
74.8532 52.0376
74.8532 52.2873
74.8532 52.0376
70.8155 52.0322
70.8155 51.6208
70.8155 52.0322
62.1355 51.6126
71.9097 56.2549
71.9097 68.9327
71.1107 52.0376
71.1107 52.0376
46.7834 51.6365
71.1107 52.0376
71.1107 52.0376
46.3766 51.6208
46.3766 51.6208
67.4568 52.0322
67.4568 51.6208
67.4568 51.6208
67.4568 51.6208
67.4568 51.6208
67.4568 52.0322
67.4568 52.0322
43.2430 51.6126
59.5346 51.6126
68.4489 62.5701
67.7246 52.0376
67.7246 52.0376
67.7246 52.2873
67.7246 52.2873
42.6665 52.0322
64.4023 52.0322
64.4023 52.0322
64.4023 52.0322
42.6665 52.0322
78.8725 60.3161
77.7770 45.5329
77.7770 45.7514
72.8449 45.5282
74.1717 57.7958
48.2756 49.2231
74.1717 57.7958
49.6451 45.5329
49.6451 45.5329
73.2021 45.5329
73.2021 45.5329
73.2021 45.7514
73.2021 45.5329
73.2021 45.7514
73.2021 45.7514
73.2021 45.7514
49.1225 45.5282
49.1225 45.1682
68.8168 45.5282
68.8168 45.5282
45.1610 45.1610
69.9997 55.2651
69.9997 68.7250
44.8716 45.5329
69.1355 45.5329
69.1355 45.5329
69.1355 45.7514
69.1355 45.7514
69.1355 45.5329
69.1355 45.5329
44.8716 45.5329
44.4443 45.5282
44.4443 45.5282
65.2108 45.5282
44.4443 45.5282
83.4773 51.6995
82.0503 39.2155
52.1735 49.5393
77.4187 51.6995
77.4187 51.6995
77.4187 59.6969
54.0536 39.0282
54.0536 39.2155
76.1898 39.2155
76.1898 39.2155
53.3329 39.0242
IV
6.3501
6.3501
6.3501
7.1101
7.1101
7.1101
7.1101
8.2501
8.2501
8.2501
10.1501
6.0500
6.0500
6.8100
6.8100
7.3232
6.8100
6.8100
8.0714
8.0714
7.9500
7.9500
7.9500
7.9500
7.9500
7.9500
7.9500
9.3185
9.8500
5.7500
6.5100
6.5100
6.5100
6.5100
7.8289
7.6500
7.6500
7.6500
7.8289
5.9001
6.6601
6.6601
7.8001
5.6000
6.2008
5.6000
6.6996
6.6996
6.3600
6.3600
6.3600
6.3600
6.3600
6.3600
6.3600
7.4479
7.4479
7.5000
7.5000
8.6949
5.3000
5.3000
6.4571
6.0600
6.0600
6.0600
6.0600
6.0600
6.0600
6.4571
7.2054
7.2054
7.2000
7.2054
5.4501
6.2101
5.5773
5.1500
5.1500
5.1500
6.0761
6.0761
5.9100
5.9100
6.8243
316
Table of Criteria Values for BBDs (K = 4)
Dsgn
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
53.7765
78.9183
78.9183
78.9183
53.7765
57.9135
57.9135
57.9135
57.9135
84.9895
84.9895
57.9135
61.5131
35.8244
79.4361
96.7855
66.9353
89.2470
73.1610
58.2708
58.2708
82.2957
58.2708
63.6906
63.6906
39.1531
39.1531
42.0911
75.8858
55.4461
78.1615
65.7261
43.8174
48.9697
55.2691
37.2091
61.3040
42.5061
24.6488
48.9893
n0 = 1
A
G
49.5411 50.8386
77.2114 55.4533
77.2114 63.1239
77.2114 55.4533
49.5411 50.8386
49.8459 41.5998
49.8459 41.5998
49.8459 41.5998
49.8459 41.5998
73.7730 41.5998
73.7730 41.5998
49.8459 41.5998
46.1677 41.5668
26.4393 41.5668
69.3332 41.5998
96.7727 96.7727
62.9364 46.2110
87.9112 53.3029
63.5288 34.6665
53.5711 46.2110
53.5711 46.2110
80.5364 53.3029
53.5711 53.3029
53.9997 34.6665
53.9997 34.6665
30.5881 34.6665
30.5881 34.6665
30.4071 34.6390
74.3032 52.6033
46.9564 34.6665
77.4182 77.4182
61.0165 42.6423
35.0363 36.9688
39.6187 27.7332
50.3495 42.0826
28.0449 27.7332
58.0636 58.0636
32.1427 31.9818
13.5652 20.7999
38.7091 38.7091
IV
5.0679
4.6389
4.6389
4.6389
5.0679
5.7193
5.7193
5.7193
5.7193
5.5876
5.5876
5.7193
6.8824
6.8887
5.3098
4.7778
4.7151
4.5000
5.3664
4.4905
4.4905
4.2222
4.4905
5.1419
5.1419
5.4544
5.4544
6.0048
3.9445
4.9174
4.1377
3.9132
4.2623
4.5705
3.6886
4.4723
3.3784
3.2802
1.9765
1.1945
D
50.4359
74.0158
74.0158
74.0158
50.4359
54.9179
54.9179
54.9179
54.9179
80.5934
80.5934
54.9179
59.3929
34.5896
75.3273
91.0063
62.9384
83.9179
69.7085
54.7913
54.7913
77.3817
54.7913
60.6850
60.6850
37.3054
37.3054
40.9823
71.3545
52.8296
73.7777
62.0397
41.3598
46.9938
52.1692
35.7078
58.2380
40.3802
23.9380
47.1400
n0 = 3
A
G
46.1536 47.3701
72.1801 51.6995
72.1801 58.9072
72.1801 51.6995
46.1536 47.3701
47.6188 39.0282
47.6188 39.0282
47.6188 39.0282
47.6188 39.0282
71.1107 39.2155
71.1107 39.2155
47.6188 39.2155
47.0586 39.0242
26.4462 39.0242
66.6665 39.2155
90.9077 90.9077
58.8229 43.0830
82.4733 49.7474
61.7277 32.6796
49.9997 43.0830
49.9997 43.0830
75.4712 49.7474
49.9997 49.7474
52.0830 32.6796
52.0830 32.6796
29.2396 32.5235
29.2396 32.5235
31.3724 32.5201
69.5650 49.0893
45.0449 32.6796
72.7261 72.7261
57.1424 39.7979
32.6528 34.4664
38.6470 26.1436
47.0587 39.2714
26.9359 26.1436
54.5446 54.5446
29.9999 29.8484
13.3333 19.6077
36.3631 36.3631
IV
5.3347
4.8500
4.8500
4.8500
5.3347
5.8336
5.8336
5.8336
5.8336
5.6100
5.6100
5.8336
6.5818
6.8943
5.3100
5.0001
4.9537
4.7000
5.4525
4.7112
4.7112
4.4000
4.7112
5.2100
5.2100
5.6428
5.6428
5.9397
4.1000
4.9675
4.3302
4.0877
4.4902
4.6882
3.8452
4.5821
3.5356
3.4295
2.0700
1.2500
Table of Criteria Values for SCDs (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
D
65.0312
64.9997
69.9291
62.2534
70.2842
70.2842
72.0925
72.0925
66.6349
58.3659
67.5112
67.5112
67.5112
72.6735
72.6735
72.6735
74.7012
79.1432
74.7012
63.7220
72.6735
53.6744
63.3909
63.3909
69.7628
69.7628
69.7628
69.7628
75.6021
75.6021
80.5193
80.5193
75.6021
75.6021
82.9733
82.9733
59.5197
69.7628
rs = 1, n0 = 1
A
G
30.1982 29.3713
40.7337 27.4906
31.5604 37.1184
42.3058 25.5317
44.7395 34.5990
44.7395 34.7176
30.7806 34.4671
30.7806 34.4671
43.1613 33.7870
41.5975 23.7481
47.2141 31.9455
47.2141 32.0006
47.2141 32.5126
44.5566 31.9375
44.5566 32.0470
44.5566 31.9375
29.9182 31.8158
32.5011 48.8217
29.9182 31.8158
45.3196 31.1956
48.3733 47.7646
39.9093 21.9457
46.7382 29.5881
46.7382 29.8079
47.2283 29.2833
47.2283 29.8032
47.2283 29.2833
47.2283 29.3339
44.3422 29.2760
44.3422 29.3764
50.8798 44.7579
50.8798 44.9433
44.3422 29.2760
44.3422 29.2002
31.6120 44.7533
31.6120 43.5947
44.7193 28.8865
51.9699 43.8006
IV
29.4667
17.1540
27.6250
14.3382
15.3123
15.3123
27.0583
27.0583
15.1707
12.9558
12.4965
12.4965
12.4965
14.7457
14.7457
14.7457
26.4917
25.2167
26.4917
12.3548
13.3290
12.0583
11.1142
11.1142
11.9298
11.9298
11.9298
11.9298
14.1790
14.1790
12.9040
12.9040
14.1790
14.1790
24.6500
24.6500
10.9725
10.5132
D
61.5916
62.3351
66.4373
58.4942
67.6945
67.6945
70.2775
70.2775
62.2088
53.3295
63.6235
63.6235
63.6235
72.0560
72.0560
72.0560
75.0393
77.6181
75.0393
58.0574
68.0118
47.3816
57.9617
57.9617
67.7253
67.7253
67.7253
67.7253
77.5743
77.5743
80.4870
80.4870
77.5743
77.5743
84.1290
84.1290
52.4525
63.5893
rs = 2, n0 = 1
A
G
24.2526 32.6890
38.9613 30.5680
24.7172 40.5074
39.8128 28.3939
42.3112 37.7093
42.3112 37.7794
24.2920 37.6140
24.2920 37.6140
39.6244 35.4755
38.5259 26.3822
43.7246 34.8239
43.7246 34.8832
43.7246 35.3316
43.4787 34.8086
43.4787 34.8733
43.4787 34.8086
23.8140 34.7206
24.8410 46.6777
23.8140 34.7206
40.6397 32.7602
43.4787 46.3460
36.4641 24.3388
42.4063 32.2016
42.4063 32.3903
45.2312 31.9219
45.2312 32.3873
45.2312 31.9219
45.2312 31.9762
44.9443 31.9079
44.9443 31.9672
49.1260 42.8026
49.1260 43.0412
44.9443 31.9079
44.9443 31.8354
24.3458 44.0000
24.3458 44.0000
39.2536 30.2775
45.2312 42.7871
IV
37.7778
16.4889
36.1806
13.8941
14.8917
14.8917
35.3472
35.3472
15.0306
12.7655
12.2969
12.2969
12.2969
14.0584
14.0584
14.0584
34.5139
33.7500
34.5139
12.4358
13.4334
12.0694
11.1683
11.1683
11.4636
11.4636
11.4636
11.4636
13.2250
13.2250
12.4612
12.4612
13.2250
13.2250
32.9167
32.9167
11.3072
10.8386
317
Table of Criteria Values for SCDs (K = 4)
Dsgn
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
p
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
l
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
c
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
q
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
D
69.7628
75.6021
75.6021
58.2885
65.3079
65.3079
65.3079
72.5642
72.5642
77.7724
77.7724
72.5642
72.5642
72.5642
84.9622
84.9622
84.9622
84.9622
94.1176
54.3851
65.3079
65.3079
72.5642
72.5642
72.5642
79.2726
55.1846
79.2726
72.5642
59.6911
59.6911
67.7298
67.7298
73.1522
73.1522
67.7298
82.2372
82.2372
82.2372
82.2372
82.2372
82.2372
97.9903
97.9903
101.6527
59.6911
67.7298
67.7298
67.7298
76.1413
76.1413
54.2752
76.1413
76.1413
59.8781
84.0016
84.0016
84.0016
67.7298
76.1413
76.1413
61.4920
67.0575
61.4920
77.2993
77.2993
77.2993
77.2993
96.1613
96.1613
96.1613
107.3991
61.4920
61.4920
70.8838
70.8838
52.0457
70.8838
70.8838
59.3719
59.3719
80.8617
80.8617
80.8617
80.8617
rs = 1, n0 = 1
A
G
51.9699 44.0138
48.4965 43.7842
48.4965 43.7842
44.9438 27.1957
46.7063 26.8982
46.7063 27.0981
46.7063 26.8982
47.2453 26.6212
47.2453 27.0939
55.6216 41.0564
55.6216 41.3896
47.2453 26.6212
47.2453 26.7077
47.2453 26.6672
51.2963 40.6890
51.2963 40.8576
51.2963 40.8576
51.2963 39.8586
33.9163 58.8235
42.8954 26.5439
51.8528 40.4417
51.8528 41.1163
52.5179 39.8187
52.5179 40.0125
52.5179 39.8187
48.6451 39.8038
39.3106 58.3695
48.6451 39.8038
59.1152 63.0026
44.7205 24.4761
44.7205 24.4761
46.6675 24.2084
46.6675 24.3883
55.9080 37.2782
55.9080 38.8339
46.6675 24.2084
56.7694 36.9508
56.7694 36.9508
56.7694 37.2506
56.7694 37.2506
56.7694 36.1886
56.7694 36.6580
63.4604 58.4874
63.4604 58.1190
32.8955 52.9412
50.0000 37.0060
52.4464 36.3975
52.4464 36.1096
52.4464 36.3975
53.2037 35.8368
53.2037 36.0113
43.0580 54.0120
53.2037 35.8368
53.2037 36.3615
41.5500 58.1190
48.8280 38.1027
48.8280 35.8235
48.8280 38.1027
59.8588 57.4537
60.8473 56.8364
60.8473 66.4311
44.4444 21.7565
54.0084 34.8394
44.4444 21.7565
57.2538 33.1362
57.2538 34.2482
57.2538 34.8307
57.2538 32.4467
75.8933 54.7101
75.8933 53.2758
75.8933 51.6938
66.3511 68.7208
50.3937 32.8942
50.3937 32.8942
53.2078 32.3534
53.2078 32.8930
43.4810 48.0814
53.2078 32.3534
53.2078 32.8930
46.7173 51.6938
46.7173 53.2758
54.0865 34.1860
54.0865 34.4106
54.0865 34.6577
54.0865 31.8550
IV
10.5132
12.7623
12.7623
10.2167
10.5475
10.5475
10.5475
11.3632
11.3632
10.0882
10.0882
11.3632
11.3632
11.3632
12.3373
12.3373
12.3373
12.3373
22.8083
10.0750
9.1308
9.1308
9.9465
9.9465
9.9465
12.1957
10.6300
12.1957
8.5298
9.6500
9.6500
9.9808
9.9808
8.7058
8.7058
9.9808
9.5215
9.5215
9.5215
9.5215
9.5215
9.5215
10.4957
10.4957
22.2417
8.2333
8.5642
8.5642
8.5642
9.3798
9.3798
8.5011
9.3798
9.3798
9.8448
11.6290
11.6290
11.6290
7.1475
7.9632
7.9632
9.0833
7.8083
9.0833
8.1392
8.1392
8.1392
8.1392
7.6798
7.6798
7.6798
9.9290
7.6667
7.6667
7.9975
7.9975
7.3550
7.9975
7.9975
7.7159
7.7159
8.8132
8.8132
8.8132
8.8132
D
63.5893
72.8369
72.8369
51.3175
61.5092
61.5092
61.5092
72.9980
72.9980
76.0186
76.0186
72.9980
72.9980
72.9980
88.2640
88.2640
88.2640
88.2640
96.5016
45.9783
57.3902
57.3902
68.1095
68.1095
68.1095
79.0809
53.4900
79.0809
63.5484
54.0824
54.0824
66.1413
66.1413
69.1892
69.1892
66.1413
83.6897
83.6897
83.6897
83.6897
83.6897
83.6897
103.3495
103.3495
109.0937
50.0735
61.2385
61.2385
61.2385
74.0728
74.0728
51.1396
74.0728
74.0728
60.3709
87.4439
87.4439
87.4439
56.6992
68.5821
68.5821
57.7489
60.7513
57.7489
76.1902
76.1902
76.1902
76.1902
99.2828
99.2828
99.2828
119.6612
52.9560
52.9560
66.4139
66.4139
47.0414
66.4139
66.4139
58.2697
58.2697
82.2664
82.2664
82.2664
82.2664
rs = 2, n0 = 1
A
G
45.2312 42.5358
44.9443 43.4445
44.9443 43.4445
40.0668 29.5248
43.8317 29.2742
43.8317 29.4457
43.8317 29.2742
47.1821 29.0199
47.1821 29.4430
52.3255 39.4884
52.3255 39.2215
47.1821 29.0199
47.1821 29.0616
47.1821 29.0693
51.9038 41.6321
51.9038 41.7686
51.9038 41.7686
51.9038 40.2962
25.0163 40.0000
36.9800 27.7464
43.8317 38.8987
43.8317 38.7784
47.1821 39.5187
47.1821 39.5959
47.1821 39.5187
46.8389 39.4950
34.2029 45.4559
46.8389 39.4950
47.1821 49.2850
41.2214 26.5723
41.2214 26.5723
45.7095 26.3467
45.7095 26.5011
51.1183 35.5641
51.1183 35.3120
45.7095 26.3467
56.2988 37.9221
56.2988 37.9221
56.2988 38.0639
56.2988 38.0639
56.2988 36.5368
56.2988 36.7295
64.0201 49.2120
64.0201 49.0015
24.4240 36.0000
41.2214 35.2525
45.7095 35.9924
45.7095 35.7343
45.7095 35.9924
49.8077 35.5668
49.8077 35.6363
35.7192 41.6207
49.8077 35.5668
49.8077 35.9673
38.0040 48.1937
49.3834 38.4999
49.3834 35.5455
49.3834 38.4999
45.7095 44.6277
49.8077 44.5891
49.8077 53.8515
42.7617 23.6198
48.1203 31.9789
42.7617 23.6198
55.2440 33.9328
55.2440 34.7309
55.2440 35.0457
55.2440 32.6853
74.2266 44.8644
74.2266 44.4660
74.2266 43.5593
73.1724 80.9143
42.7617 32.3686
42.7617 32.3686
48.2959 31.9933
48.2959 32.3665
34.4654 37.0261
48.2959 31.9933
48.2959 32.3665
40.7312 43.1265
40.7312 44.4660
53.5315 34.4929
53.5315 34.6625
53.5315 34.6862
53.5315 31.6149
IV
10.8386
12.6000
12.6000
10.4722
10.3350
10.3350
10.3350
10.6302
10.6302
9.8664
9.8664
10.6302
10.6302
10.6302
11.6278
11.6278
11.6278
11.6278
31.3194
10.6111
9.7100
9.7100
10.0052
10.0052
10.0052
11.7667
11.0952
11.7667
9.3802
9.6389
9.6389
9.5016
9.5016
8.7377
8.7377
9.5016
9.0330
9.0330
9.0330
9.0330
9.0330
9.0330
10.0306
10.0306
30.4861
9.0139
8.8766
8.8766
8.8766
9.1719
9.1719
9.1541
9.1719
9.1719
9.9405
10.9334
10.9334
10.9334
8.2516
8.5469
8.5469
8.8056
8.0417
8.8056
7.9044
7.9044
7.9044
7.9044
7.4358
7.4358
7.4358
9.1973
8.1806
8.1806
8.0433
8.0433
8.2282
8.0433
8.0433
7.9994
7.9994
8.3386
8.3386
8.3386
8.3386
318
Table of Criteria Values for SCDs (K = 4)
Dsgn
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
p
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
dv
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
l
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
c
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
q
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
D
80.8617
80.8617
80.8617
66.3103
98.4854
61.4920
70.8838
70.8838
70.8838
70.8838
52.5212
80.8617
80.8617
80.8617
52.5212
70.5374
91.6158
91.6158
106.4970
63.8874
48.7286
63.8874
57.3232
57.3232
75.1557
75.1557
75.1557
75.1557
75.1557
75.1557
75.1557
66.6342
66.6342
96.4568
96.4568
75.6058
63.8874
63.8874
49.8285
75.1557
75.1557
75.1557
75.1557
75.1557
75.1557
49.8285
57.9222
57.9222
87.3632
57.9222
84.7008
102.3746
53.9454
67.2271
67.2271
67.2271
65.2016
65.2016
91.2054
91.2054
77.7177
45.8099
67.2271
67.2271
67.2271
45.8099
55.3686
55.3686
55.3686
55.3686
81.2547
81.2547
55.3686
65.9971
38.4359
72.3897
95.2658
62.2002
82.9337
78.0832
51.1209
51.1209
72.1979
51.1209
64.1747
rs = 1, n0 = 1
A
G
54.0865 32.0100
54.0865 34.1860
54.0865 34.6577
44.7356 51.6613
60.9776 57.4730
58.1818 54.1876
61.9657 51.7511
61.9657 51.7511
61.9657 59.7615
61.9657 59.7615
41.0564 50.5212
63.1606 51.6938
63.1606 59.0499
63.1606 51.6938
41.0564 50.5212
55.1724 30.0841
80.8279 47.1225
80.8279 51.6907
83.1737 60.1810
50.9091 28.7825
42.2111 43.1251
50.9091 28.7825
47.8761 47.1225
47.8761 47.1225
54.2199 30.1888
54.2199 31.2451
54.2199 31.6379
54.2199 28.3092
54.2199 28.0852
54.2199 30.1888
54.2199 31.6379
52.4482 45.2321
52.4482 46.6163
73.8504 51.6687
73.8504 53.2150
49.6275 60.1307
60.2151 47.4142
60.2151 59.6852
41.2197 45.2822
64.9027 47.1225
64.9027 47.1225
64.9027 52.2913
64.9027 52.2913
64.9027 47.1225
64.9027 47.1225
41.2197 45.2822
44.5644 45.2321
44.5644 45.2321
66.4066 51.6687
44.5644 51.6687
81.3559 46.3739
91.8080 51.6024
46.8547 44.1969
51.6129 27.5232
51.6129 24.6707
51.6129 27.5232
55.3338 40.3907
55.3338 44.3063
78.9694 49.6946
78.9694 44.8211
62.7044 51.5837
39.5604 40.6407
63.1579 45.9547
63.1579 51.1587
63.1579 45.9547
39.5604 40.6407
45.4394 40.3907
45.4394 40.3907
45.4394 40.3907
45.4394 40.3907
69.2810 44.8211
69.2810 44.8211
45.4394 44.8211
50.2941 44.2874
28.0875 38.7704
61.7101 44.5614
95.2381 95.2381
55.3846 38.6449
79.2079 43.5920
70.7663 43.0020
43.9024 38.2956
43.9024 38.2956
67.7966 43.5920
43.9024 43.5920
53.0413 41.4121
IV
8.8132
8.8132
8.8132
9.0596
9.7873
6.2500
6.5808
6.5808
6.5808
6.5808
7.6832
7.3965
7.3965
7.3965
7.6832
7.2417
6.2975
6.2975
7.1132
7.1000
6.5414
7.1000
6.5698
6.5698
7.4308
7.4308
7.4308
7.4308
7.4308
7.4308
7.4308
6.9307
6.9307
6.9715
6.9715
8.2744
5.6833
5.6833
6.5371
6.0142
6.0142
6.0142
6.0142
6.0142
6.0142
6.5371
6.8980
6.8980
6.8298
6.8980
5.4000
5.7308
5.7562
6.5333
6.5333
6.5333
5.7847
5.7847
5.5892
5.5892
6.1455
5.7235
5.1167
5.1167
5.1167
5.7235
5.7519
5.7519
5.7519
5.7519
5.4475
5.4475
5.7519
6.1128
6.3795
5.3058
4.8333
4.9710
4.6917
4.9995
4.9383
4.9383
4.5500
4.9383
4.9667
D
82.2664
82.2664
82.2664
70.2300
104.3069
48.5609
60.9018
60.9018
60.9018
60.9018
48.9988
75.4386
75.4386
75.4386
48.9988
66.5775
91.3854
91.3854
116.7132
56.9072
41.6570
56.9072
53.9611
53.9611
73.7158
73.7158
73.7158
73.7158
73.7158
73.7158
73.7158
68.9167
68.9167
99.7608
99.7608
85.3074
51.5422
51.5422
44.2662
66.7661
66.7661
66.7661
66.7661
66.7661
66.7661
44.2662
56.5348
56.5348
85.2707
56.5348
80.4833
108.8507
47.9097
62.6379
62.6379
62.6379
64.7961
64.7961
90.6381
90.6381
86.1985
38.0260
55.8040
55.8040
55.8040
38.0260
51.4287
51.4287
51.4287
51.4287
75.4728
75.4728
51.4287
68.4158
39.8445
62.8450
96.7870
58.2712
77.6949
83.7152
44.1613
44.1613
62.3690
44.1613
63.4443
rs = 2, n0 = 1
A
G
53.5315 31.6767
53.5315 34.4929
53.5315 34.6862
44.1351 43.5569
61.4608 47.4814
42.7617 40.2477
48.2959 39.6691
48.2959 39.6691
48.2959 48.3215
48.2959 48.3215
33.6026 39.6348
53.5315 43.5593
53.5315 47.8680
53.5315 43.5593
33.6026 39.6348
51.8519 29.8336
75.4890 39.1570
75.4890 41.3135
91.4697 70.8072
44.9198 28.3225
32.0611 32.8385
44.9198 28.3225
39.6463 38.3441
39.6463 39.1570
52.0851 30.3868
52.0851 31.1187
52.0851 31.4164
52.0851 27.9941
52.0851 27.7933
52.0851 30.3868
52.0851 31.4164
49.6967 38.1144
49.6967 38.9078
71.8970 42.2917
71.8970 42.7038
55.6854 70.8000
44.9198 35.2167
44.9198 45.0351
32.0765 34.7104
52.0851 39.1570
52.0851 39.1570
52.0851 42.5863
52.0851 42.2814
52.0851 39.1570
52.0851 39.1570
32.0765 34.7104
38.3518 38.1144
38.3518 38.1144
59.2243 42.2917
38.3518 42.2917
72.3618 36.4232
97.5947 60.6942
36.8601 34.4625
48.1605 27.2431
48.1605 24.2764
48.1605 27.2431
49.5844 33.5631
49.5844 35.4116
72.8933 38.6995
72.8933 36.5025
70.3410 60.6918
29.3478 30.1858
48.1605 35.6168
48.1605 38.6015
48.1605 35.6178
29.3478 30.1858
36.8839 33.5631
36.8839 33.5631
36.8839 33.5631
36.8839 33.5631
58.1703 36.5025
58.1703 36.5025
36.8839 36.5025
47.2570 36.2500
25.4931 32.6695
48.3954 36.2412
96.7742 96.7742
46.6321 30.3527
68.9655 33.2591
76.3939 50.5785
33.5821 29.6815
33.5821 29.6815
53.5714 33.2591
33.5821 33.2591
46.6780 32.2496
IV
8.3386
8.3386
8.3386
8.7858
9.3362
7.5556
7.4183
7.4183
7.4183
7.4183
8.3522
7.7136
7.7136
7.7136
8.3522
7.2083
6.3072
6.3072
6.6025
7.3472
7.6018
7.3472
7.0735
7.0735
7.2100
7.2100
7.2100
7.2100
7.2100
7.2100
7.2100
6.8447
6.8447
6.7414
6.7414
7.6311
6.7222
6.7222
7.4263
6.5850
6.5850
6.5850
6.5850
6.5850
6.5850
7.4263
7.1975
7.1975
6.8802
7.1975
5.6111
5.4739
6.4471
6.5139
6.5139
6.5139
5.9188
5.9188
5.6127
5.6127
5.6900
6.7999
5.8889
5.8889
5.8889
6.7999
6.2716
6.2716
6.2716
6.2716
5.7516
5.7516
6.2716
6.0428
6.5087
5.8905
4.7778
5.2924
4.9167
4.7641
5.6452
5.6452
5.0556
5.6452
5.1169
319
Table of Criteria Values for SCDs (K = 4)
Dsgn
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
64.1747
39.4507
39.4507
48.7044
62.8520
52.7436
77.0104
60.2639
40.1759
53.3854
47.1590
37.7492
60.5007
38.1131
27.8390
48.5071
rs = 1, n0 = 1
A
G
53.0413 37.3509
29.2737 33.6589
29.2737 33.6589
38.8747 42.9864
59.2593 42.6149
42.4170 37.3509
76.1905 76.1905
52.5547 34.8736
29.0909 30.9159
46.6190 34.4016
40.1114 34.1058
26.7466 29.8807
57.1429 57.1429
25.8065 26.1552
17.9004 25.8012
38.0952 38.0952
IV
4.9667
5.4207
5.4207
5.1774
4.4083
4.9340
4.1858
4.1531
4.6198
4.2186
4.1204
4.4190
3.4177
3.6180
1.7454
1.2083
D
63.4443
39.0016
39.0016
54.9318
50.0662
48.0818
78.1627
55.2694
36.8462
57.9537
39.0813
37.0291
61.3048
33.7373
30.8554
48.9898
rs = 2, n0 = 1
A
G
46.6780 30.4188
24.7773 27.9693
24.7773 27.9693
45.3072 50.5765
43.7956 31.9117
33.6059 30.2010
77.4194 77.4194
42.8571 26.6073
22.6415 24.2821
52.0851 40.4628
29.6296 25.7344
22.1038 24.3350
58.0645 58.0645
19.2513 19.9555
21.7278 30.3471
38.7097 38.7097
IV
5.1169
5.7539
5.7539
4.7409
5.1944
5.4697
4.1377
4.4905
5.1462
3.9861
4.8433
4.7718
3.3784
4.1641
1.5931
1.1944
rs = 2, n0 = 3
A
G
33.8624 30.2676
37.6999 28.3678
36.2020 37.5068
37.3693 26.3415
41.2562 35.0212
41.2562 35.1643
36.8305 34.8278
36.8305 34.8278
38.5066 32.9407
35.8702 24.4433
41.1481 32.3273
41.1481 32.3010
41.1481 32.7501
42.6509 32.3273
42.6509 32.4593
42.6509 32.3273
37.5918 32.1487
40.4423 43.2201
37.5918 32.1487
38.2008 30.4069
42.6509 43.0853
33.8462 22.5730
39.5272 29.8411
39.5272 30.0308
42.6556 29.6334
42.6556 30.0209
42.6556 29.6334
42.6556 29.6093
44.4257 29.6333
44.4257 29.7544
48.8663 39.6481
48.8663 40.1347
44.4257 29.6333
44.4257 29.4861
41.8301 42.3874
41.8301 40.8985
36.5704 28.0567
42.6556 39.6661
42.6556 39.6500
44.4257 40.3915
44.4257 40.3915
37.2093 27.3977
40.8932 27.1283
40.8932 27.3007
40.8932 27.1283
44.6171 26.9394
44.6171 27.2917
49.5959 36.5882
49.5959 36.4999
44.6171 26.9394
44.6171 27.0652
44.6171 26.9175
52.2591 38.5650
52.2591 38.8426
52.2591 38.8426
52.2591 37.4677
48.3749 50.3889
34.3348 25.7440
40.8932 36.0680
40.8932 36.0880
44.6171 36.7196
44.6171 36.9537
IV
21.6000
16.1583
19.8750
14.4429
14.4333
14.4333
18.9750
18.9750
14.5833
13.5118
12.7179
12.7179
12.7179
13.5333
13.5333
13.5333
18.0750
17.2500
18.0750
12.8679
12.8583
12.8750
11.7868
11.7868
11.8179
11.8179
11.8179
11.8179
12.6333
12.6333
11.8083
11.8083
12.6333
12.6333
16.3500
16.3500
11.9368
11.1429
11.1429
11.9583
11.9583
11.1500
10.8868
10.8868
10.8868
10.9179
10.9179
10.0929
10.0929
10.9179
10.9179
10.9179
10.9083
10.9083
10.9083
10.9083
14.6250
11.3000
10.2118
10.2118
10.2429
10.2429
Table of Criteria Values for SCDs (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
D
62.6073
60.1665
67.6758
56.8887
65.2283
65.2283
70.1919
70.1919
61.8414
52.9797
61.8021
61.8021
61.8021
67.6515
67.6515
67.6515
73.2459
77.6013
73.2459
58.3334
67.6515
48.5125
57.6164
57.6164
63.9963
63.9963
63.9963
63.9963
70.6316
70.6316
75.2256
75.2256
70.6316
70.6316
82.0367
82.0367
54.0979
63.9963
63.9963
70.6316
70.6316
52.7362
59.4521
59.4521
59.4521
66.7323
66.7323
71.5220
71.5220
66.7323
66.7323
66.7323
79.7202
79.7202
79.7202
79.7202
93.9893
49.2046
59.4521
59.4521
66.7323
66.7323
rs = 1, n0 = 3
A
G
37.8002 26.2796
39.2276 24.6570
41.4879 33.2112
38.8480 22.8974
43.6950 31.0597
43.6950 31.2596
41.4411 30.8390
41.4411 30.8390
42.0179 30.3285
37.6038 21.2589
43.5948 28.6732
43.5948 28.6326
43.5948 29.1190
43.8329 28.6705
43.8329 28.8550
43.8329 28.6705
41.3867 28.4668
47.1841 43.6826
41.3867 28.4668
41.7918 27.9981
47.9968 42.9493
35.8452 19.6770
42.3497 26.4915
42.3497 26.7071
43.7354 26.2838
43.7354 26.6924
43.7354 26.2838
43.7354 26.2466
43.9970 26.2813
43.9970 26.4504
51.3079 40.0502
51.3079 40.3607
43.9970 26.2813
43.9970 26.1543
47.7073 40.0424
47.7073 39.0058
40.4982 25.8629
48.2959 39.3757
48.2959 39.8823
48.6152 39.3702
48.6152 39.3702
40.4040 24.4028
42.3743 24.0832
42.3743 24.2791
42.3743 24.0832
43.9054 23.8943
43.9054 24.2658
52.0461 36.7778
52.0461 37.0396
43.9054 23.8943
43.9054 24.0987
43.9054 23.8605
52.4544 36.4093
52.4544 36.6915
52.4544 36.6915
52.4544 35.8405
58.4119 57.6073
38.5542 23.8163
47.1160 36.2216
47.1160 36.9457
49.0166 35.7961
49.0166 36.2566
IV
19.4222
16.5822
17.3639
15.0543
14.5239
14.5239
16.7306
16.7306
14.3655
14.0263
12.9960
12.9960
12.9960
13.8905
13.8905
13.8905
16.0972
14.6722
16.0972
12.8377
12.3072
13.2417
11.9680
11.9680
12.3627
12.3627
12.3627
12.3627
13.2572
13.2572
11.8322
11.8322
13.2572
13.2572
14.0389
14.0389
11.8097
10.7793
10.7793
11.6739
11.6739
11.1833
11.3347
11.3347
11.3347
11.7293
11.7293
10.3043
10.3043
11.7293
11.7293
11.7293
11.1989
11.1989
11.1989
11.1989
11.9806
11.0250
9.7513
9.7513
10.1460
10.1460
D
61.3629
59.0391
66.5377
54.8862
64.2269
64.2269
70.8100
70.8100
59.0222
49.8390
59.7653
59.7653
59.7653
68.5040
68.5040
68.5040
76.1421
78.7588
76.1421
54.5367
64.6592
44.1799
54.2137
54.2137
63.7017
63.7017
63.7017
63.7017
73.9275
73.9275
76.7033
76.7033
73.9275
73.9275
86.0789
86.0789
49.0607
59.8115
59.8115
69.4128
69.4128
47.8833
57.5900
57.5900
57.5900
68.7691
68.7691
71.6148
71.6148
68.7691
68.7691
68.7691
84.3574
84.3574
84.3574
84.3574
99.7293
42.9014
53.7334
53.7334
64.1639
64.1639
320
Table of Criteria Values for SCDs (K = 4)
Dsgn
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
p
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
dv
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
l
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
c
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
q
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
D
66.7323
74.3816
51.7798
74.3816
66.7323
54.0720
54.0720
61.7754
61.7754
66.7211
66.7211
61.7754
75.8589
75.8589
75.8589
75.8589
75.8589
75.8589
92.4315
92.4315
102.7608
54.0720
61.7754
61.7754
61.7754
70.2358
70.2358
50.0656
70.2358
70.2358
56.4813
79.2363
79.2363
79.2363
61.7754
70.2358
70.2358
55.7894
60.8388
55.7894
70.6731
70.6731
70.6731
70.6731
89.0418
89.0418
89.0418
101.9777
55.7894
55.7894
64.8075
64.8075
47.5842
64.8075
64.8075
54.9762
54.9762
74.8749
74.8749
74.8749
74.8749
74.8749
74.8749
74.8749
62.9631
93.5140
55.7894
64.8075
64.8075
64.8075
64.8075
48.6327
74.8749
74.8749
74.8749
48.6327
64.1232
84.0213
84.0213
99.0967
58.0779
44.2976
58.0779
52.5714
52.5714
rs = 1, n0 = 3
A
G
49.0166 35.7961
49.3786 35.7911
39.5405 52.3474
49.3786 35.7911
55.4747 56.3897
40.2235 21.9625
40.2235 21.9625
42.4044 21.6749
42.4044 21.8512
50.9577 33.3552
50.9577 35.0222
42.4044 21.6749
53.4481 33.1001
53.4481 33.1001
53.4481 33.3357
53.4481 33.3357
53.4481 32.4165
53.4481 33.5388
68.5626 52.7570
68.5626 52.1357
61.0385 68.8995
45.0000 33.2546
47.7474 32.5994
47.7474 32.6420
47.7474 32.5994
49.9272 32.2165
49.9272 32.6310
40.2564 48.3568
49.9272 32.2165
49.9272 32.5477
42.7182 52.1357
50.3449 34.2723
50.3449 32.2120
50.3449 34.2723
54.6309 51.4231
57.5034 50.9416
57.5034 59.4842
40.0000 19.5223
48.6692 31.3157
40.0000 19.5223
52.3331 29.6490
52.3331 30.9808
52.3331 31.4142
52.3331 29.0321
73.3861 51.2071
73.3861 47.7010
73.3861 46.4673
74.4069 61.6988
45.3901 29.5596
45.3901 29.5596
48.5608 28.9773
48.5608 29.5566
39.6079 43.0854
48.5608 28.9773
48.5608 29.5566
44.1149 46.4673
44.1149 47.7010
51.1141 30.6249
51.1141 30.8023
51.1141 31.7542
51.1141 28.6369
51.1141 29.0053
51.1141 30.6249
51.1141 31.7542
47.4887 46.3429
67.0062 51.8866
52.4590 48.8318
56.7408 46.3057
56.7408 46.3057
56.7408 53.4731
56.7408 53.4731
38.5109 45.2814
60.2580 46.4673
60.2580 52.8749
60.2580 46.4673
38.5109 45.2814
49.7778 27.0397
74.8727 42.1639
74.8727 47.1356
82.0994 54.1796
45.9016 25.8647
38.0090 38.8375
45.9016 25.8647
43.8164 42.1639
43.8164 42.1639
IV
10.1460
11.0405
9.9883
11.0405
8.5627
10.5500
10.5500
10.7013
10.7013
9.2763
9.2763
10.7013
9.6710
9.6710
9.6710
9.6710
9.6710
9.6710
9.1405
9.1405
11.3472
8.9667
9.1180
9.1180
9.1180
9.5127
9.5127
8.7567
9.5127
9.5127
9.1107
10.4072
10.4072
10.4072
7.5347
7.9293
7.9293
9.9167
8.4917
9.9167
8.6430
8.6430
8.6430
8.6430
7.6127
7.6127
7.6127
8.5072
8.3333
8.3333
8.4847
8.4847
7.8523
8.4847
8.4847
7.8791
7.8791
8.8793
8.8793
8.8793
8.8793
8.8793
8.8793
8.8793
8.2332
8.3489
6.7500
6.9013
6.9013
6.9013
6.9013
7.8426
7.2960
7.2960
7.2960
7.8426
7.8583
6.5847
6.5847
6.9793
7.7000
7.1072
7.7000
6.9747
6.9747
D
64.1639
75.5807
51.1225
75.5807
59.8670
50.5063
50.5063
62.0035
62.0035
64.8607
64.8607
62.0035
78.9931
78.9931
78.9931
78.9931
78.9931
78.9931
99.1237
99.1237
114.1272
46.7625
57.4075
57.4075
57.4075
69.9158
69.9158
48.2696
69.9158
69.9158
57.9024
83.8684
83.8684
83.8684
53.1521
64.7333
64.7333
53.9881
56.7949
53.9881
71.5342
71.5342
71.5342
71.5342
93.9362
93.9362
93.9362
115.2746
49.5073
49.5073
62.3553
62.3553
44.1666
62.3553
62.3553
55.1318
55.1318
77.8362
77.8362
77.8362
77.8362
77.8362
77.8362
77.8362
67.6555
100.4832
45.3984
57.1801
57.1801
57.1801
57.1801
46.3601
71.3761
71.3761
71.3761
46.3601
62.3274
85.9713
85.9713
110.7694
53.2744
38.9977
53.2744
50.7642
50.7642
rs = 2, n0 = 3
A
G
44.6171 36.7196
46.7608 36.7196
33.7079 42.1088
46.7608 36.7196
44.6171 45.6346
38.2979 24.6579
38.2979 24.6579
42.6966 24.4155
42.6966 24.5707
47.7987 32.9668
47.7987 32.8961
42.6966 24.4155
53.6099 35.1596
53.6099 35.1596
53.6099 35.2701
53.6099 35.2701
53.6099 33.8735
53.6099 34.4593
66.6153 45.8261
66.6153 45.3976
52.0325 84.2105
38.2979 32.7362
42.6966 33.3643
42.6966 33.2919
42.6966 33.3643
47.2740 33.0477
47.2740 33.2583
33.7349 38.5684
47.2740 33.0477
47.2740 33.3260
38.0282 44.6488
49.9711 35.8067
49.9711 33.0476
49.9711 35.8067
42.6966 41.3804
47.2740 41.3215
47.2740 49.9563
39.7516 21.9181
44.7552 29.6980
39.7516 21.9181
51.7667 31.4284
51.7667 32.3758
51.7667 32.6713
51.7667 30.2727
71.6657 42.4319
71.6657 41.2117
71.6657 40.3656
78.9311 75.0000
39.7516 30.0610
39.7516 30.0610
45.1877 29.6572
45.1877 30.0609
32.1693 34.3286
45.1877 29.6572
45.1877 30.0609
38.6874 39.9378
38.6874 41.2117
51.0760 31.9812
51.0760 32.1189
51.0760 32.5692
51.0760 29.3757
51.0760 29.5630
51.0760 31.9812
51.0760 32.5692
45.2830 40.3535
64.5921 44.2362
39.7516 37.4058
45.1877 36.7826
45.1877 36.7826
45.1877 44.7607
45.1877 44.7607
31.7730 36.7302
51.0760 40.3656
51.0760 44.4056
51.0760 40.3656
31.7730 36.7302
48.2759 27.7112
71.2182 36.2616
71.2182 38.6060
89.8825 65.6616
41.7910 26.3034
29.7872 30.5120
41.7910 26.3034
37.0990 35.5172
37.0990 36.2616
IV
10.2429
11.0583
10.7734
11.0583
9.5679
10.2500
10.2500
9.9868
9.9868
9.1618
9.1618
9.9868
9.1929
9.1929
9.1929
9.1929
9.1929
9.1929
9.1833
9.1833
13.7250
9.5750
9.3118
9.3118
9.3118
9.3429
9.3429
9.4520
9.3429
9.3429
9.5263
10.1583
10.1583
10.1583
8.6368
8.6679
8.6679
9.3500
8.5250
9.3500
8.2618
8.2618
8.2618
8.2618
7.4679
7.4679
7.4679
8.2833
8.6750
8.6750
8.4118
8.4118
8.6621
8.4118
8.4118
8.2050
8.2050
8.4429
8.4429
8.4429
8.4429
8.4429
8.4429
8.4429
8.2792
8.4333
8.0000
7.7368
7.7368
7.7368
7.7368
8.5860
7.7679
7.7679
7.7679
8.5860
7.6250
6.5368
6.5368
6.5679
7.7750
8.0714
7.7750
7.4150
7.4150
321
Table of Criteria Values for SCDs (K = 4)
Dsgn
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
68.9256
68.9256
68.9256
68.9256
68.9256
68.9256
68.9256
62.0040
62.0040
89.7542
89.7542
72.4014
58.0779
58.0779
45.6980
68.9256
68.9256
68.9256
68.9256
68.9256
68.9256
45.6980
53.8973
53.8973
81.2925
53.8973
77.2029
94.2753
49.1700
61.2760
61.2760
61.2760
60.0433
60.0433
83.9898
83.9898
72.7913
41.7547
61.2760
61.2760
61.2760
41.7547
50.9882
50.9882
50.9882
50.9882
74.8264
74.8264
50.9882
61.8137
35.9995
66.6627
87.1552
56.9047
75.8730
72.3212
46.7686
46.7686
66.0513
46.7686
59.4391
59.4391
36.5395
36.5395
46.0363
57.5010
48.8515
70.8469
55.4407
36.9605
49.8751
43.3847
35.2670
56.1769
35.3892
26.3858
45.8831
rs = 1, n0 = 3
A
G
49.6482 27.0118
49.6482 28.2774
49.6482 28.5739
49.6482 25.3551
49.6482 25.3882
49.6482 27.0118
49.6482 28.5739
50.3155 40.6589
50.3155 41.7384
72.0628 46.2655
72.0628 50.0653
55.4505 53.9865
54.3689 42.7279
54.3689 53.8862
37.6048 40.5175
59.7055 42.1639
59.7055 42.1639
59.7055 46.7889
59.7055 46.7889
59.7055 42.1639
59.7055 42.1639
37.6048 40.5175
42.2932 40.6589
42.2932 40.6589
64.2128 46.2655
42.2932 46.2655
73.8462 41.8327
86.0313 46.5374
42.2701 39.8536
46.6019 24.7456
46.6019 22.1697
46.6019 24.7456
51.0485 36.1405
51.0485 40.4020
73.5140 45.4212
73.5140 40.1048
61.9195 46.4396
35.6436 36.6239
57.1429 41.4516
57.1429 46.1881
57.1429 41.4516
35.6436 36.6239
41.6886 36.1405
41.6886 36.1405
41.6886 36.1405
41.6886 36.1405
64.1766 40.1048
64.1766 40.1048
41.6886 40.1048
48.6662 39.6562
26.6489 34.8505
56.9438 39.8963
86.9565 86.9565
50.1393 34.8606
72.0721 39.3647
66.3896 38.7812
39.6476 34.5430
39.6476 34.5430
61.5385 39.3647
39.6476 39.3647
49.1633 37.8510
49.1633 33.4207
26.8485 30.1171
26.8485 30.1171
38.4172 38.6997
53.6913 38.4743
39.0348 33.4207
69.5652 69.5652
47.6821 31.4917
26.2295 27.8885
43.8452 31.0249
36.2720 30.7921
24.6184 26.7365
52.1739 52.1739
23.3010 23.6188
16.9317 23.2687
34.7826 34.7826
IV
7.8513
7.8513
7.8513
7.8513
7.8513
7.8513
7.8513
7.0016
7.0016
6.8210
6.8210
7.3556
6.1167
6.1167
6.9382
6.2680
6.2680
6.2680
6.2680
6.2680
6.2680
6.9382
6.9650
6.9650
6.6627
6.9650
5.8000
5.9513
6.2296
7.0667
7.0667
7.0667
6.0972
6.0972
5.7930
5.7930
6.1240
6.1930
5.4833
5.4833
5.4833
6.1930
6.0606
6.0606
6.0606
6.0606
5.6347
5.6347
6.0606
6.0874
6.6548
5.4763
5.1667
5.3520
5.0083
5.2196
5.3155
5.3155
4.8500
5.3155
5.1830
5.1830
5.7934
5.7934
5.3113
4.6917
5.1465
4.4745
4.4379
4.9969
4.4499
4.4013
4.6738
3.6534
3.8773
1.8874
1.2917
D
69.3486
69.3486
69.3486
69.3486
69.3486
69.3486
69.3486
65.4070
65.4070
94.6803
94.6803
82.6466
48.2519
48.2519
41.6437
62.8106
62.8106
62.8106
62.8106
62.8106
62.8106
41.6437
53.6557
53.6557
80.9281
53.6557
75.4836
102.6734
44.9335
58.7468
58.7468
58.7468
61.1190
61.1190
85.4944
85.4944
82.1461
35.6638
52.3374
52.3374
52.3374
35.6638
48.5102
48.5102
48.5102
48.5102
71.1898
71.1898
48.5102
65.1994
37.9713
59.2785
91.0077
54.7917
73.0556
79.2577
41.5244
41.5244
58.6448
41.5244
60.0661
60.0661
36.9250
36.9250
52.6518
47.0766
45.5216
73.7788
52.1695
34.7797
55.1739
36.8894
35.2530
58.2387
32.0500
29.6489
47.1405
rs = 2, n0 = 3
A
G
48.8522 28.1443
48.8522 29.0132
48.8522 29.2601
48.8522 25.9500
48.8522 25.8937
48.8522 28.1443
48.8522 29.2601
47.6886 35.3199
47.6886 36.0602
69.7354 39.1998
69.7354 40.4647
60.0000 65.6250
41.7910 32.7301
41.7910 41.8989
29.9550 32.1847
48.8522 36.2616
48.8522 36.2616
48.8522 39.4378
48.8522 39.1657
48.8522 36.2616
48.8522 36.2616
29.9550 32.1847
36.4991 35.3199
36.4991 35.3199
56.9664 39.1998
36.4991 39.1998
67.6056 33.8775
92.9664 56.2963
34.2857 32.0461
44.8598 25.3102
44.8598 22.5458
44.8598 25.3102
46.6258 31.0813
46.6258 33.0908
68.9342 36.1943
68.9342 33.8038
69.1358 56.2814
27.2727 28.0543
44.8598 33.1242
44.8598 35.9133
44.8598 33.1251
27.2727 28.0543
34.5455 31.0813
34.5455 31.0813
34.5455 31.0813
34.5455 31.0813
54.7748 33.8038
54.7748 33.8038
34.5455 33.8038
45.5285 33.5998
24.2687 30.2742
45.4410 33.5706
90.9091 90.9091
43.4783 28.2313
64.5161 30.9480
72.7969 46.9136
31.2500 27.6043
31.2500 27.6043
50.0000 30.9480
31.2500 30.9480
43.9815 30.1619
43.9815 28.1698
23.1990 25.9011
23.1990 25.9011
44.5505 46.9012
40.8163 29.6882
31.5091 27.9755
72.7273 72.7273
40.0000 24.7584
21.0526 22.5850
49.6732 37.5309
27.5862 23.9422
20.7226 22.5359
54.5455 54.5455
17.9104 18.5688
20.7650 28.1481
36.3636 36.3636
IV
7.5118
7.5118
7.5118
7.5118
7.5118
7.5118
7.5118
6.9579
6.9579
6.7179
6.7179
7.0321
7.1000
7.1000
7.7961
6.8368
6.8368
6.8368
6.8368
6.8368
6.8368
7.7961
7.3389
7.3389
6.8679
7.3389
5.9000
5.6368
6.8243
6.8750
6.8750
6.8750
6.1679
6.1679
5.7868
5.7868
5.7108
7.2053
6.2000
6.2000
6.2000
7.2053
6.5490
6.5490
6.5490
6.5490
5.9368
5.9368
6.5490
6.0919
6.7503
6.0868
5.0000
5.5772
5.1500
4.9208
5.9583
5.9583
5.3000
5.9583
5.3019
5.3019
6.0513
6.0513
4.8412
5.4500
5.6829
4.3301
4.7112
5.4447
4.1422
5.0922
4.9907
3.5355
4.3841
1.6816
1.2500
rs = 2, n0 = 1
A
G
24.4177 39.3544
IV
40.3332
Table of Criteria Values for PBCDs (K = 4)
Dsgn
1
p
15
dv
4
l
4
c
6
q
4
D
69.8808
rs = 1, n0 = 1
A
G
31.0800 44.2317
IV
31.3200
D
66.4403
322
Table of Criteria Values for PBCDs (K = 4)
Dsgn
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
p
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
c
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
q
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
D
71.2300
72.4898
69.8184
74.2071
74.2071
75.6126
74.7573
71.0536
67.2566
72.8634
72.8634
72.8634
77.8281
77.8281
76.8749
79.4168
79.4168
78.4314
69.5150
74.2233
63.8654
70.2238
70.2238
76.6231
76.6231
75.5999
75.5999
82.3252
82.3252
82.3252
82.3252
81.2115
81.2115
82.9979
81.8429
66.7108
72.7598
72.7598
78.1441
76.5795
66.6260
73.9464
73.9464
72.8609
81.3801
81.3801
81.3801
81.3801
80.1699
80.1699
80.1699
86.7292
86.7292
86.7292
85.4025
87.4480
62.9689
69.8557
69.8557
76.8455
76.8455
75.1546
82.4101
53.9847
82.4101
72.6296
70.1505
69.0073
78.7528
78.7528
78.7528
78.7528
77.4526
86.1208
86.1208
86.1208
86.1208
84.6583
84.6583
92.3614
92.3614
93.1510
65.8521
73.8924
73.8924
rs = 1, n0 = 1
A
G
47.7961 41.4550
30.9144 48.4574
52.5507 38.5126
49.4015 45.0132
49.4015 45.0132
30.7216 50.3995
30.2435 53.1169
47.6251 42.6033
53.6044 35.5545
55.1422 41.9125
55.1422 41.6992
55.1422 41.9281
51.4047 46.5424
51.4047 46.9004
49.9729 49.3147
30.4960 56.3471
30.4960 51.6920
29.9766 57.1429
52.7625 39.6500
49.2487 44.2524
52.9323 32.6975
56.6715 38.4212
56.6715 38.4570
58.5384 43.0804
58.5384 43.0985
56.5265 45.2356
56.5265 45.0153
53.9796 51.6781
53.9796 52.8348
53.9796 47.5542
53.9796 47.4280
52.2320 53.8191
52.2320 53.6671
29.6646 52.3810
29.1005 52.3810
53.9438 36.3472
55.5190 40.9411
55.5190 40.6418
51.3125 45.1825
49.2743 46.9769
56.1635 35.2989
60.8299 39.1657
60.8299 39.2066
58.4519 41.1303
63.1913 46.9913
63.1913 48.1778
63.1913 44.2763
63.1913 43.7459
60.5812 49.1617
60.5812 49.3106
60.5812 48.7897
55.2185 56.4000
55.2185 56.1720
55.2185 57.5420
53.1106 51.1653
28.6845 47.6190
53.2293 33.3743
57.2698 37.2207
57.2698 37.5363
59.2270 41.5000
59.2270 41.8907
56.2717 43.1659
53.1636 56.0913
37.6860 51.1718
53.1636 56.0913
55.9815 42.0639
60.6701 35.6689
58.0528 37.1844
66.7996 42.7783
66.7996 43.3957
66.7996 39.8938
66.7996 39.4120
63.5819 44.6966
66.3874 50.7935
66.3874 53.9495
66.3874 51.2729
66.3874 51.9962
63.0449 47.3724
63.0449 46.0903
56.6371 57.9107
56.6371 57.6957
27.5387 42.8571
56.7600 33.8775
61.9307 37.3517
61.9307 37.7993
IV
15.8868
30.4308
12.3277
14.9976
14.9976
29.5445
29.7325
14.8093
10.5813
11.4384
11.4384
11.4384
14.1114
14.1114
14.2993
28.6609
28.6609
28.8545
11.2501
13.9335
9.4500
9.6921
9.6921
10.5522
10.5522
10.7402
10.7402
13.2278
13.2278
13.2278
13.2278
13.4213
13.4213
27.9783
28.1867
9.5037
10.3743
10.3743
13.0594
13.3461
8.5608
8.8058
8.8058
8.9938
9.6686
9.6686
9.6686
9.6686
9.8621
9.8621
9.8621
12.5452
12.5452
12.5452
12.7535
27.3162
8.3724
8.6280
8.6280
9.5002
9.5002
9.7869
12.3118
11.5598
12.3118
9.3110
7.6745
7.8625
7.9222
7.9222
7.9222
7.9222
8.1158
8.9860
8.9860
8.9860
8.9860
9.1943
9.1943
11.8830
11.8830
26.6700
7.4967
7.7539
7.7539
D
68.3041
69.9645
65.2863
72.3675
72.3675
74.2597
73.4617
67.4194
60.8158
69.2429
69.2429
69.2429
77.4101
77.4101
76.5094
79.6008
79.6008
78.6690
64.1288
71.6676
55.4243
64.4306
64.4306
74.2243
74.2243
73.2826
73.2826
83.8194
83.8194
83.8194
83.8194
82.7496
82.7496
85.2964
84.1940
59.2571
68.2382
68.2382
77.0313
75.7145
58.5129
69.0481
69.0481
68.0851
80.6719
80.6719
80.6719
80.6719
79.5401
79.5401
79.5401
90.9080
90.9080
90.9080
89.6163
92.6430
53.3661
62.9480
62.9480
73.5152
73.5152
72.1340
83.5226
54.5573
83.5226
67.0499
62.5177
61.5496
75.1379
75.1379
75.1379
75.1379
73.9675
87.9124
87.9124
87.9124
87.9124
86.5256
86.5256
100.3564
100.3564
102.4553
56.4117
67.7691
67.7691
rs = 2, n0 = 1
A
G
44.7350 36.8121
24.2392 43.8332
47.3350 34.2008
47.0635 40.7026
47.0635 40.7202
24.0353 44.8276
23.6732 44.8276
44.0934 37.6361
46.6801 31.5709
50.4399 37.7917
50.4399 37.6658
50.4399 37.7845
50.1009 42.7187
50.1009 42.8962
48.4281 44.1336
23.8007 41.3793
23.8007 41.3793
23.4134 41.3793
46.7813 34.9291
46.4075 39.5099
44.7837 29.0030
49.9234 34.6431
49.9234 34.6437
54.6718 39.4198
54.6718 39.4113
52.5124 40.4858
52.5124 40.3793
54.2318 48.9991
54.2318 49.5993
54.2318 43.4789
54.2318 43.4857
52.0903 49.2734
52.0903 49.2137
23.1129 37.9310
22.7064 37.9310
46.0360 32.0190
49.9403 36.4406
49.9403 36.2723
49.4762 40.9895
47.3781 42.3160
47.8712 31.6956
54.4570 35.8370
54.4570 35.8378
52.1091 36.8068
60.7839 44.5457
60.7839 45.3157
60.7839 40.1980
60.7839 39.8590
57.8517 44.9364
57.8517 44.9628
57.8517 44.9548
57.2833 51.4355
57.2833 51.6138
57.2833 51.7976
54.6176 48.2560
22.3258 34.4828
43.9560 29.2805
49.3355 33.1285
49.3355 33.2988
54.3440 37.5233
54.3440 37.7394
51.5840 38.7463
53.0840 51.2149
36.3442 44.6039
53.0840 51.2149
49.3577 36.5551
52.2686 32.4888
49.8722 33.2271
61.2462 40.4173
61.2462 40.9298
61.2462 36.2053
61.2462 35.8866
57.9574 40.6545
66.0526 46.6890
66.0526 47.9369
66.0526 46.5008
66.0526 46.8005
62.1653 44.3349
62.1653 43.4344
61.4090 53.2009
61.4090 52.9905
21.4229 31.0345
47.0588 30.0168
54.0724 33.7718
54.0724 34.0135
IV
15.4471
39.2927
12.4004
14.4066
14.4066
38.2535
38.4825
14.4687
11.0762
11.3598
11.3598
11.3598
13.3673
13.3673
13.5964
37.2154
37.2154
37.4471
11.4220
13.4425
10.2606
10.0356
10.0356
10.3206
10.3206
10.5497
10.5497
12.3293
12.3293
12.3293
12.3293
12.5610
12.5610
36.4126
36.6510
10.0978
10.3958
10.3958
12.4168
12.7120
9.2201
8.9964
8.9964
9.2255
9.2826
9.2826
9.2826
9.2826
9.5143
9.5143
9.5143
11.5265
11.5265
11.5265
11.7649
35.6195
9.2822
9.0716
9.0716
9.3701
9.3701
9.6653
11.4805
11.1514
11.4805
9.4316
8.1808
8.4099
7.9584
7.9584
7.9584
7.9584
8.1901
8.4798
8.4798
8.4798
8.4798
8.7182
8.7182
10.7334
10.7334
34.8345
8.2560
8.0459
8.0459
323
Table of Criteria Values for PBCDs (K = 4)
Dsgn
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
p
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
dv
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
l
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
c
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
q
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
D
72.0881
81.3689
81.3689
54.2126
81.3689
81.3689
57.6263
86.5457
86.5457
85.8319
69.4021
76.3924
74.5910
74.8063
74.8063
73.4183
83.5888
83.5888
83.5888
81.9936
92.3564
92.3564
92.3564
99.8447
69.6326
67.7228
78.4183
78.4183
53.3641
78.4183
78.4183
58.3745
58.3745
85.8404
85.8404
85.8404
85.8404
85.8404
85.0443
85.0443
62.5256
92.0080
64.8907
73.0438
73.0438
71.1089
71.1089
52.8293
81.2840
79.2157
79.9604
52.8293
77.7614
90.1556
90.1556
100.9559
73.8984
51.6753
73.8984
57.9405
57.9405
82.9235
82.9235
82.9235
82.9235
82.9235
82.0453
82.0453
64.1982
64.1982
91.9519
91.9519
69.4417
68.1390
66.0801
51.6941
77.9126
77.9126
75.6510
75.6510
76.4644
76.4644
51.6941
57.2772
57.2772
rs = 1, n0 = 1
A
G
58.3691 38.8511
63.1279 50.4966
63.1279 53.4672
43.1046 47.3786
63.1279 50.4966
63.1279 50.9965
38.9581 55.1923
53.4916 56.3111
53.4916 56.3111
52.7554 57.7909
58.0225 37.8592
58.9561 47.7477
55.8554 41.7550
67.4071 39.0901
67.4071 36.2445
63.7450 40.3321
71.3969 46.3126
71.3969 48.4146
71.3969 46.2819
67.0932 42.5849
70.5620 54.6698
70.5620 53.1455
70.5620 51.3431
58.3521 61.3398
61.8840 33.6208
57.9117 34.9881
67.1989 45.5033
67.1989 47.5329
45.0462 42.1710
67.1989 45.5033
67.1989 47.5329
45.8555 49.0950
45.8555 52.3414
65.1892 50.3176
65.1892 50.8475
65.1892 53.4766
65.1892 50.8475
65.1892 53.4766
63.9653 53.0321
63.9653 51.8825
40.6743 53.0601
53.9060 56.7774
57.5281 34.0835
61.9493 42.9940
61.9493 43.7544
58.1344 37.1174
58.1344 37.1174
41.7597 48.2857
62.8064 53.2259
59.1908 48.9260
60.2548 49.4655
41.7597 48.2857
67.8377 42.5504
77.9013 47.1679
77.9013 52.4743
76.4461 53.7450
67.9612 41.7763
45.1613 37.8730
67.9612 41.7763
48.8615 44.1639
48.8615 46.1369
70.5634 45.2954
70.5634 47.4660
70.5634 48.1734
70.5634 45.2954
70.5634 48.1205
68.9318 46.5055
68.9318 46.0589
49.9544 46.4819
49.9544 48.1723
68.0434 51.4478
68.0434 52.4441
43.1165 53.6723
61.8986 38.3169
57.5835 32.9369
43.6482 43.4128
67.4002 48.9873
67.4002 47.3548
62.7027 43.2850
62.7027 42.8131
64.0724 43.9393
64.0724 44.1853
43.6482 43.4128
44.5183 47.2854
44.5183 47.2854
IV
8.0405
8.7527
8.7527
8.8728
8.7527
8.7527
10.8092
11.7473
11.7473
11.8452
7.5646
8.5708
8.8443
6.7909
6.7909
6.9845
7.2396
7.2396
7.2396
7.4480
8.3238
8.3238
8.3238
11.2368
6.6226
6.9092
7.0063
7.0063
7.4297
7.0063
7.0063
8.1222
8.1222
8.1882
8.1882
8.1882
8.1882
8.1882
8.2860
8.2860
10.0587
11.1830
6.4333
6.8244
6.8244
7.0980
7.0980
7.9606
7.8571
8.1041
8.0051
7.9606
6.3167
6.5775
6.5775
7.6777
5.8750
6.4086
5.8750
6.6791
6.6791
6.4418
6.4418
6.4418
6.4418
6.4418
6.5396
6.5396
7.3717
7.3717
7.6238
7.6238
9.3081
5.6931
5.9667
6.5174
6.1108
6.1108
6.3578
6.3578
6.2588
6.2588
6.5174
7.2100
7.2100
D
66.3558
80.0136
80.0136
53.1406
80.0136
80.0136
60.2993
90.8191
90.8191
90.3016
61.1801
72.2044
70.7438
67.9055
67.9055
66.7167
82.0343
82.0343
82.0343
80.5799
97.8458
97.8458
97.8458
113.5203
60.4608
59.0442
73.7901
73.7901
50.0355
73.7901
73.7901
59.2776
59.2776
87.4487
87.4487
87.4487
87.4487
87.4487
86.8883
86.8883
68.3340
100.8424
53.8890
65.7390
65.7390
64.2449
64.2449
51.4370
79.1677
77.4081
77.9094
51.4370
72.4990
91.7985
91.7985
112.2395
65.5605
45.6579
65.5605
56.2066
56.2066
80.7378
80.7378
80.7378
80.7378
80.7378
80.1468
80.1468
68.2205
68.2205
98.0321
98.0321
80.2570
57.4512
55.9613
47.7937
72.0609
72.0609
70.2333
70.2333
70.7534
70.7534
47.7937
58.0094
58.0094
rs = 2, n0 = 1
A
G
51.0524 34.8726
59.9742 46.0947
59.9742 47.9939
39.4706 40.9151
59.9742 46.0947
59.9742 46.4212
39.2421 49.7295
55.4562 51.0426
55.4562 51.0426
54.8066 52.6093
48.6399 32.9004
53.2451 41.3045
50.4382 37.0499
59.0388 36.7188
59.0388 32.6173
55.6162 36.4706
67.4049 42.1043
67.4049 42.8466
67.4049 41.6208
62.8901 39.6770
73.9083 48.7796
73.9083 48.3051
73.9083 47.1080
67.3642 75.4784
51.6129 30.2489
48.5302 31.2427
60.3795 41.3567
60.3795 42.6622
39.0466 36.3998
60.3795 41.3567
60.3795 42.6622
43.9038 44.2604
43.9038 45.8959
64.5295 45.4709
64.5295 45.8549
64.5295 48.0582
64.5295 45.8549
64.5295 48.0582
63.5436 47.7565
63.5436 46.8189
43.5863 48.4820
58.7367 51.6606
46.0857 29.4627
52.8191 36.7162
52.8191 37.0381
49.7306 32.9342
49.7306 32.9342
37.8483 41.5681
58.8836 47.1709
55.2246 42.7236
55.9797 43.7992
37.8483 41.5681
60.4687 37.7451
77.2005 42.6202
77.2005 45.5761
87.0197 66.0556
57.8266 37.4895
37.0697 32.3373
57.8266 37.4895
44.0025 39.4578
44.0025 40.1656
65.7864 40.5579
65.7864 43.1406
65.7864 42.0512
65.7864 40.5579
65.7864 42.0512
64.6182 41.8334
64.6182 41.2887
51.3140 42.4267
51.3140 43.5372
71.5124 46.2655
71.5124 46.7633
50.8195 66.0436
49.9940 32.4277
46.8468 29.0594
37.1875 36.4072
59.1756 43.8185
59.1756 41.7201
54.9910 37.7665
54.9910 37.3845
55.8482 38.7082
55.8482 38.9570
37.1875 36.4072
42.2786 42.3449
42.2786 42.3449
IV
8.3411
8.4338
8.4338
8.8735
8.4338
8.4338
10.1740
10.7617
10.7617
10.8360
8.1075
8.5059
8.7872
7.1428
7.1428
7.3745
7.1556
7.1556
7.1556
7.3940
7.6866
7.6866
7.6866
9.9484
7.2303
7.5255
7.1096
7.1096
7.7888
7.1096
7.1096
7.8960
7.8960
7.7149
7.7149
7.7149
7.7149
7.7149
7.7893
7.7893
9.1965
10.0429
7.2919
7.1817
7.1817
7.4630
7.4630
7.9917
7.5946
7.8614
7.7861
7.9917
6.5784
6.3625
6.3625
6.9017
6.2940
7.0564
6.2940
6.8113
6.8113
6.3908
6.3908
6.3908
6.3908
6.3908
6.4651
6.4651
6.9185
6.9185
6.9962
6.9962
8.2190
6.3661
6.6474
6.9070
6.2704
6.2704
6.5373
6.5373
6.4619
6.4619
6.9070
7.0142
7.0142
324
Table of Criteria Values for PBCDs (K = 4)
Dsgn
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
83.8919
56.7155
86.1013
99.6222
56.5774
78.0984
78.0984
77.1342
64.6585
64.6585
89.3353
89.3353
72.8769
49.5274
72.6206
70.1673
71.0483
49.5274
56.6014
56.6014
56.6014
56.6014
80.2679
80.2679
55.9543
63.0664
37.1539
77.1670
96.1720
64.2314
84.3819
75.3922
54.7507
54.7507
74.2116
54.0004
63.3835
63.3835
39.5057
39.5057
44.9805
70.7847
54.7786
77.6970
62.5484
42.4244
50.9446
52.1207
37.7033
60.9800
40.8220
26.0520
48.7950
rs = 1, n0 = 1
A
G
60.7272 45.9262
43.6243 50.5189
81.2923 47.9783
88.1710 46.0943
49.7238 40.2336
72.1571 43.5556
72.1571 42.5444
70.1754 40.6738
55.0819 41.9031
55.0819 45.0559
75.6074 44.9779
75.6074 44.2164
56.7138 46.0671
43.5484 38.3712
68.3311 44.3839
62.7692 37.3840
64.3765 40.7050
43.5484 38.3712
47.6040 42.0969
47.6040 42.6656
47.6040 42.0969
47.6040 42.6656
65.3941 39.9216
65.3941 39.9216
46.4173 43.5651
47.5709 44.7537
27.7211 41.0752
67.3782 45.4629
96.1538 96.1538
57.9151 41.7229
78.9793 44.8092
67.0282 38.4119
48.3351 39.7893
48.3351 39.7893
66.0487 39.3871
46.8750 38.6865
52.6711 37.5466
52.6711 37.3963
30.6189 36.0581
30.6189 36.0581
34.1433 38.3893
68.4932 48.7833
45.9484 37.8858
76.9231 76.9231
55.2995 37.7997
32.7869 34.6874
42.8735 30.7295
46.3918 39.0267
28.1101 30.3086
57.6923 57.6923
29.7030 29.7604
15.4408 23.0471
38.4615 38.4615
IV
7.5385
7.3139
5.4462
5.9313
5.6580
5.3105
5.3105
5.4083
5.9285
5.9285
5.8775
5.8775
6.6211
5.4964
4.9795
5.2265
5.1275
5.4964
5.7669
5.7669
5.7669
5.7669
5.7921
5.7921
5.8708
6.5634
6.5768
5.2313
4.8000
4.9075
4.7462
5.1780
4.7458
4.7458
4.6608
4.8497
5.1203
5.1203
5.3805
5.3805
5.5869
4.1000
4.8547
4.1569
4.0992
4.3841
4.3905
3.8336
4.3905
3.3941
3.3941
1.8595
1.2000
D
85.2802
57.6065
82.9328
106.5996
51.5011
71.3961
71.3961
70.7867
65.6346
65.6346
91.0294
91.0294
82.2776
42.6249
62.5270
60.6809
61.2054
42.6249
54.3225
54.3225
54.3225
54.3225
77.3704
77.3704
53.8826
67.5457
39.6589
69.7858
97.2317
60.9588
80.4488
81.5485
48.5795
48.5795
66.1898
48.1078
64.3569
64.3569
39.9506
39.9506
51.8877
58.4820
51.7904
78.4993
58.3902
39.4045
56.0045
44.5049
38.1041
61.5394
36.8257
29.4233
49.1304
rs = 2, n0 = 1
A
G
58.9512 41.2420
41.5660 44.9944
75.7295 40.5219
95.4316 56.6231
41.9487 34.6318
63.1836 37.2803
63.1836 37.2803
61.9291 36.5817
52.9660 37.6389
52.9660 40.3342
74.7051 43.0799
74.7051 40.0191
66.2154 56.6191
34.8462 31.7535
56.1543 37.5967
51.7911 32.3938
52.6796 34.7536
34.8462 31.7535
42.1249 37.0450
42.1249 38.3648
42.1249 37.0450
42.1249 38.3648
59.3044 35.6384
59.3044 35.6384
41.3017 38.6987
48.9375 40.9644
27.7338 37.3556
57.5259 42.6365
97.2222 97.2222
51.4244 34.9074
72.5983 37.3818
74.0975 47.1859
39.5621 33.2635
39.5621 33.2635
55.7231 32.8549
38.6930 33.0033
50.2661 37.1902
50.2661 34.4599
28.1998 32.2502
28.1998 32.2502
41.7796 47.1826
53.8462 38.9517
39.7502 35.5304
77.7778 77.7778
47.9498 31.0263
27.0096 28.8051
49.6033 37.7487
36.4513 31.2697
25.5441 28.4243
58.3333 58.3333
23.5514 24.0865
19.7745 28.3116
38.8889 38.8889
IV
7.1417
7.0987
5.5469
5.5775
6.0789
5.5752
5.5752
5.6496
5.8338
5.8338
5.6720
5.6720
5.9411
6.1746
5.4548
5.7217
5.6463
6.1746
5.9295
5.9295
5.9295
5.9295
5.8175
5.8175
6.0140
6.1213
6.3223
5.4394
4.7619
5.1014
4.8564
4.8563
5.1971
5.1971
5.0019
5.2816
5.0365
5.0365
5.4419
5.4419
4.9552
4.6238
5.0477
4.1239
4.3041
4.7342
4.0748
4.3153
4.4654
3.3672
3.7578
1.6517
1.1905
Table of Criteria Values for PBCDs (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
D
68.6527
67.2460
71.5894
65.0386
70.2369
70.2369
75.1255
74.2757
67.2521
62.2078
67.9888
67.9888
67.9888
73.8854
73.8854
72.9804
79.4628
79.4628
78.4768
64.8644
70.4632
58.7962
65.0287
65.0287
71.6383
rs = 1, n0 = 3
A
G
44.5206 40.3854
48.5224 37.9894
46.0031 44.2437
49.7945 35.2767
50.7949 41.1127
50.7949 41.1128
47.8311 46.0170
46.5759 48.4980
48.7474 38.9109
49.6213 32.5936
52.5216 38.2908
52.5216 38.1647
52.5216 38.4500
53.7168 42.5111
53.7168 43.1300
52.0114 45.2566
50.1456 51.4473
50.1456 47.1971
48.6281 53.2396
50.1615 36.2227
51.1539 40.4187
48.5326 29.9316
52.5722 35.1074
52.5722 35.3268
56.1397 39.3607
IV
17.9473
14.4024
16.9734
12.4642
13.4285
13.4285
16.0027
16.2086
13.2222
11.1560
11.4903
11.4903
11.4903
12.4578
12.4578
12.6637
15.0350
15.0350
15.2470
11.2840
12.2630
10.1595
10.1821
10.1821
10.5196
D
66.8769
65.3333
70.7936
61.8523
69.3384
69.3384
75.5947
74.7824
64.5974
57.3764
65.6699
65.6699
65.6699
74.3180
74.3180
73.4533
81.6044
81.6044
80.6492
60.8197
68.8049
52.1639
60.8335
60.8335
70.4821
rs = 2, n0 = 3
A
G
37.6562 36.8154
44.3243 34.5286
38.9889 41.0053
45.0575 32.0629
47.0011 38.0769
47.0011 38.1132
40.6455 43.2925
39.5518 44.5887
43.8480 35.2082
43.9600 29.6055
48.1433 35.3801
48.1433 35.3005
48.1433 35.4483
50.5579 39.9630
50.5579 40.3298
48.7416 41.4399
42.7616 50.0045
42.7616 44.3416
41.4452 50.2094
44.5856 32.6984
46.5605 36.9612
42.0048 27.1781
47.0667 32.4363
47.0667 32.5221
52.3764 36.9082
IV
21.0704
14.7070
19.9581
12.6804
13.5947
13.5947
18.8472
19.0921
13.6612
11.5806
11.5681
11.5681
11.5681
12.4838
12.4838
12.7287
17.7376
17.7376
17.9852
11.6345
12.5642
10.8303
10.4683
10.4683
10.4572
325
Table of Criteria Values for PBCDs (K = 4)
Dsgn
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
p
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
c
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
q
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
D
71.6383
70.6817
70.6817
78.4321
78.4321
78.4321
78.4321
77.3710
77.3710
83.7400
82.5747
61.7756
68.0263
68.0263
74.4488
72.9581
61.3883
68.5726
68.5726
67.5660
76.2664
76.2664
76.2664
76.2664
75.1322
75.1322
75.1322
82.9800
82.9800
82.9800
81.7106
89.1156
58.0188
64.7791
64.7791
72.0166
72.0166
70.4321
78.8476
51.6510
78.8476
68.0657
64.7012
63.6468
73.1558
73.1558
73.1558
73.1558
71.9480
80.9433
80.9433
80.9433
80.9433
79.5687
79.5687
88.8293
88.8293
96.0931
60.7366
68.6408
68.6408
66.9647
76.4770
76.4770
50.9534
76.4770
76.4770
55.4225
83.2360
83.2360
82.5495
64.4696
71.7998
70.1066
69.0825
69.0825
67.8007
77.8157
77.8157
77.8157
76.3306
87.1189
87.1189
87.1189
96.6523
rs = 1, n0 = 3
A
G
56.1397 39.5443
54.1166 41.4865
54.1166 41.1144
57.6195 47.2057
57.6195 48.6674
57.6195 43.5562
57.6195 43.3389
55.4505 49.2894
55.4505 49.1752
51.2913 56.3816
49.4753 51.1707
50.0032 33.2109
53.1061 37.4048
53.1061 37.1078
54.3187 41.2699
51.8329 42.9096
51.5314 32.3286
56.5936 35.7910
56.5936 36.0420
54.3409 37.7642
61.1795 42.9463
61.1795 44.2548
61.1795 40.4316
61.1795 40.2662
58.5065 45.1265
58.5065 45.5284
58.5065 44.7274
60.2172 51.7082
60.2172 51.3153
60.2172 53.3083
57.4919 46.8910
52.5886 58.2225
48.8267 30.5608
53.2223 34.0121
53.2223 34.3332
57.1250 37.9182
57.1250 38.2986
54.1224 39.4414
57.5599 51.2415
39.4994 46.8970
57.5599 51.2415
53.8285 38.4338
55.7211 32.6799
53.3037 34.0734
62.4080 39.0588
62.4080 39.9557
62.4080 36.4448
62.4080 36.3889
59.3355 40.8548
64.9296 46.5376
64.9296 49.3296
64.9296 46.8600
64.9296 47.9852
61.4403 43.2599
61.4403 42.3686
63.4878 53.2995
63.4878 52.9259
54.1286 62.6087
52.1101 31.0332
57.7614 34.1351
57.7614 34.6580
54.3724 35.5067
61.5266 46.1585
61.5266 48.9605
41.4265 43.2656
61.5266 46.1585
61.5266 46.5856
41.7507 50.6192
59.2129 51.4455
59.2129 51.4455
58.2278 53.1883
54.0430 34.5993
57.2056 43.6430
54.0188 38.1543
61.9999 35.8432
61.9999 33.2237
58.6081 36.9871
67.0184 42.2890
67.0184 44.4411
67.0184 42.8865
62.8722 38.9008
69.9877 52.4908
69.9877 48.5340
69.9877 47.2786
67.8685 56.3206
IV
10.5196
10.7255
10.7255
11.4900
11.4900
11.4900
11.4900
11.7021
11.7021
14.2874
14.5156
9.9758
10.3248
10.3248
11.3057
11.6196
9.1856
9.2115
9.2115
9.4173
9.5518
9.5518
9.5518
9.5518
9.7638
9.7638
9.7638
10.7425
10.7425
10.7425
10.9706
13.5621
8.9793
9.0167
9.0167
9.3675
9.3675
9.6814
10.4869
10.4744
10.4869
9.1602
8.2150
8.4208
8.2437
8.2437
8.2437
8.2437
8.4557
8.8042
8.8042
8.8042
8.8042
9.0324
9.0324
10.0172
10.0172
12.8544
8.0202
8.0593
8.0593
8.3733
8.5487
8.5487
8.9242
8.5487
8.5487
9.6524
9.8686
9.8686
9.9758
7.8521
8.3495
8.6491
7.2472
7.2472
7.4592
7.4961
7.4961
7.4961
7.7243
8.0790
8.0790
8.0790
9.3095
D
70.4821
69.5879
69.5879
80.6612
80.6612
80.6612
80.6612
79.6317
79.6317
88.1742
87.0346
55.9489
64.7979
64.7979
74.1289
72.8617
55.1042
65.2536
65.2536
64.3435
76.7193
76.7193
76.7193
76.7193
75.6430
75.6430
75.6430
87.7304
87.7304
87.7304
86.4840
96.7300
50.2572
59.4887
59.4887
69.9133
69.9133
68.5998
80.6033
52.6504
80.6033
63.7648
58.9193
58.0069
71.0890
71.0890
71.0890
71.0890
69.9817
83.7581
83.7581
83.7581
83.7581
82.4369
82.4369
97.1840
97.1840
108.2889
53.1648
64.1173
64.1173
62.7802
76.2326
76.2326
50.6295
76.2326
76.2326
58.3932
87.9482
87.9482
87.4471
57.8834
68.7924
67.4008
64.0563
64.0563
62.9349
77.7235
77.7235
77.7235
76.3455
93.4354
93.4354
93.4354
110.4079
rs = 2, n0 = 3
A
G
52.3764 36.9891
50.2598 37.9876
50.2598 37.7968
55.5167 45.8384
55.5167 46.9441
55.5167 40.7065
55.5167 40.7206
53.1266 46.0950
53.1266 46.0547
43.9272 52.8825
42.3856 49.6162
43.3753 29.9773
47.7440 34.1166
47.7440 33.9374
50.2331 38.3454
47.9295 39.5863
44.9215 29.7115
51.4209 33.5582
51.4209 33.6593
49.1841 34.5488
58.5462 41.6731
58.5462 42.7061
58.5462 37.6212
58.5462 37.4720
55.6424 42.1916
55.6424 42.3888
55.6424 42.0807
59.5496 48.1676
59.5496 48.4333
59.5496 48.9199
56.4859 45.1871
45.3577 55.0246
41.2371 27.4433
46.5440 31.0196
46.5440 31.1958
52.1797 35.1339
52.1797 35.3660
49.4633 36.2802
54.7378 47.9113
36.6740 41.7642
54.7378 47.9113
47.2775 34.2267
49.0803 30.4637
46.8220 31.1577
57.9672 37.8211
57.9672 38.4831
57.9672 33.8973
57.9672 33.7734
54.8199 38.0792
64.0142 43.6972
64.0142 44.9141
64.0142 43.6845
64.0142 44.0641
60.1195 41.4970
60.1195 40.6897
65.2256 50.0670
65.2256 49.6313
47.1818 79.3076
44.1718 28.1408
51.1072 31.6257
51.1072 31.9074
48.2250 32.6577
57.9315 43.1223
57.9315 45.0808
37.8115 38.2945
57.9315 43.1223
57.9315 43.4646
40.2302 46.5736
58.1396 47.7501
58.1396 47.7501
57.3775 49.5069
45.9249 30.8089
51.2444 38.6821
48.4695 34.6938
55.4940 34.4519
55.4940 30.5934
52.2625 34.2183
63.9750 39.4302
63.9750 40.2261
63.9750 39.2420
59.6317 37.1176
72.3005 46.7543
72.3005 45.2186
72.3005 44.1449
73.8930 70.7448
IV
10.4572
10.7020
10.7020
11.3742
11.3742
11.3742
11.3742
11.6219
11.6219
16.8794
17.1342
10.5347
10.5375
10.5375
11.4677
11.7833
9.7180
9.3574
9.3574
9.6023
9.3476
9.3476
9.3476
9.3476
9.5952
9.5952
9.5952
10.5161
10.5161
10.5161
10.7709
16.0316
9.7844
9.4378
9.4378
9.4411
9.4411
9.7566
10.4669
10.5996
10.4669
9.5069
8.6071
8.8520
8.2478
8.2478
8.2478
8.2478
8.4954
8.4894
8.4894
8.4894
8.4894
8.7442
8.7442
9.6682
9.6682
15.1925
8.6875
8.3413
8.3413
8.6569
8.4402
8.4402
9.0432
8.4402
8.4402
9.5547
9.6985
9.6985
9.7780
8.4071
8.5173
8.8180
7.4975
7.4975
7.7451
7.3896
7.3896
7.3896
7.6444
7.6415
7.6415
7.6415
8.8291
326
Table of Criteria Values for PBCDs (K = 4)
Dsgn
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
p
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
dv
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
l
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
c
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
q
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
D
64.3047
62.5410
73.0022
73.0022
49.6785
73.0022
73.0022
55.0641
55.0641
80.9724
80.9724
80.9724
80.9724
80.9724
80.2215
80.2215
60.5264
89.0662
59.9256
67.9989
67.9989
66.1977
66.1977
49.8333
76.6744
74.7234
75.4259
49.8333
71.9282
84.1619
84.1619
95.6752
68.3550
47.7989
68.3550
54.0885
54.0885
77.4107
77.4107
77.4107
77.4107
77.4107
76.5908
76.5908
60.8402
60.8402
87.1422
87.1422
67.7853
63.0277
61.1233
48.2574
72.7328
72.7328
70.6216
70.6216
71.3809
71.3809
48.2574
54.2812
54.2812
79.5037
53.7488
79.8152
93.3435
52.4468
72.3966
72.3966
71.5028
60.5834
60.5834
83.7050
83.7050
69.4950
45.9115
67.3188
65.0446
65.8612
45.9115
53.0342
53.0342
53.0342
53.0342
75.2090
75.2090
rs = 1, n0 = 3
A
G
56.8854 30.8098
53.2108 32.0677
62.9738 41.5469
62.9738 43.5219
41.9480 38.5291
62.9738 41.5469
62.9738 43.5219
44.5313 44.9951
44.5313 47.7907
64.2364 46.1199
64.2364 46.4764
64.2364 48.8364
64.2364 46.4764
64.2364 48.8364
62.9368 48.4303
62.9368 47.7054
44.9534 48.6815
61.4158 52.2997
52.8562 31.2354
57.9350 39.2558
57.9350 40.1198
54.2861 33.9247
54.2861 33.9247
40.3247 44.2506
61.7098 48.9218
57.9036 44.7183
59.0202 45.2213
40.3247 44.2506
62.4651 39.0568
73.6343 43.0930
73.6343 48.5288
77.4156 49.5733
62.5798 38.3426
41.4669 34.7432
62.5798 38.3426
45.7191 40.3274
45.7191 42.1504
66.4777 41.3606
66.4777 43.3654
66.4777 44.7656
66.4777 41.3606
66.4777 44.5614
64.8928 42.5241
64.8928 42.7012
49.2799 42.8149
49.2799 43.9926
68.0896 46.9987
68.0896 48.3615
49.8721 49.2805
56.9540 35.1523
52.9551 30.1964
40.7333 39.6442
63.4071 44.7562
63.4071 43.3257
58.8635 39.5247
58.8635 39.1432
60.1863 40.1190
60.1863 40.3435
40.7333 39.6442
43.5360 43.5616
43.5360 43.5616
60.1477 41.9409
42.6009 46.1360
75.1083 44.1130
84.2937 42.6421
45.7295 36.9505
66.5788 40.0208
66.5788 39.0859
64.7316 37.3572
51.9454 38.2838
51.9454 41.6691
71.8010 41.5961
71.8010 40.6873
57.4478 42.4914
40.0141 35.2305
63.0133 40.7868
57.8372 34.3191
59.3321 37.3860
40.0141 35.2305
44.6939 38.4403
44.6939 38.9809
44.6939 38.4403
44.6939 38.9809
61.7673 36.4724
61.7673 36.4724
IV
7.0628
7.3768
7.2406
7.2406
7.7870
7.2406
7.2406
8.1022
8.1022
7.9304
7.9304
7.9304
7.9304
7.9304
8.0376
8.0376
8.8303
9.2505
6.8556
7.0414
7.0414
7.3410
7.3410
7.9251
7.5679
7.8384
7.7300
7.9251
6.7278
6.7709
6.7709
7.3713
6.2440
6.8540
6.2440
6.9650
6.9650
6.6223
6.6223
6.6223
6.6223
6.6223
6.7295
6.7295
7.2802
7.2802
7.3123
7.3123
8.0083
6.0449
6.3444
6.7879
6.2598
6.2598
6.5303
6.5303
6.4219
6.4219
6.7879
7.1031
7.1031
7.2189
7.2169
5.7744
6.0632
6.0319
5.6258
5.6258
5.7329
6.1429
6.1429
6.0042
6.0042
6.4581
5.8549
5.2632
5.5338
5.4253
5.8549
5.9659
5.9659
5.9659
5.9659
5.9107
5.9107
D
57.0336
55.6973
69.9125
69.9125
47.4062
69.9125
69.9125
56.6056
56.6056
83.5070
83.5070
83.5070
83.5070
83.5070
82.9718
82.9718
66.4605
98.0776
50.8343
62.2845
62.2845
60.8689
60.8689
49.1185
75.5992
73.9189
74.3976
49.1185
68.4709
87.1327
87.1327
107.4957
61.9179
43.1211
61.9179
53.3498
53.3498
76.6342
76.6342
76.6342
76.6342
76.6342
76.0732
76.0732
65.3372
65.3372
93.8888
93.8888
78.4914
54.2592
52.8521
45.3645
68.3982
68.3982
66.6636
66.6636
67.1572
67.1572
45.3645
55.5577
55.5577
81.6758
55.1718
78.4494
101.4267
48.7169
67.5364
67.5364
66.9600
62.4496
62.4496
86.6121
86.6121
79.1095
40.3206
59.1468
57.4004
57.8966
40.3206
51.6865
51.6865
51.6865
51.6865
73.6159
73.6159
rs = 2, n0 = 3
A
G
48.4848 28.3666
45.5776 29.3009
57.2202 38.7019
57.2202 40.0719
36.8626 34.0830
57.2202 38.7019
57.2202 40.0719
42.3140 41.4255
42.3140 43.0069
62.7617 42.5591
62.7617 42.9554
62.7617 45.3465
62.7617 42.9554
62.7617 45.3465
61.7654 44.7047
61.7654 44.0080
45.7791 45.4102
63.0362 48.6444
43.2732 27.6275
49.9738 34.3910
49.9738 34.7665
47.0204 30.8445
47.0204 30.8445
36.3262 38.9084
57.0723 44.1294
53.4062 40.0183
54.1614 40.9750
36.3262 38.9084
56.8845 35.4332
73.6034 39.8709
73.6034 43.0564
86.4932 61.9651
54.3857 35.1925
34.7970 30.3415
54.3857 35.1925
41.6514 36.9172
41.6514 37.6524
62.5440 37.9466
62.5440 40.3578
62.5440 39.6961
62.5440 37.9466
62.5440 39.6961
61.4156 39.1823
61.4156 38.9699
49.9635 39.7600
49.9635 40.7562
70.2968 43.3120
70.2968 44.9274
55.6474 61.9017
46.9851 30.4265
44.0145 27.2576
35.1360 34.0946
56.1671 41.1230
56.1671 39.0440
52.1413 35.4701
52.1413 35.0242
52.9654 36.2243
52.9654 36.4572
35.1360 34.0946
40.8720 39.6346
40.8720 39.6346
57.4367 38.6062
40.1605 42.1213
71.4254 38.0735
91.7624 53.1389
39.4202 32.5186
59.5115 35.0155
59.5115 35.0155
58.3221 34.3567
50.3773 35.2110
50.3773 38.1165
71.4007 40.7395
71.4007 37.4377
65.8313 53.1130
32.7207 29.8066
52.8506 35.3138
48.7211 30.4098
49.5617 32.6334
32.7207 29.8066
39.9294 34.6603
39.9294 35.8901
39.9294 34.6603
39.9294 35.8901
56.4289 33.3395
56.4289 33.3395
IV
7.5910
7.9066
7.3404
7.3404
8.1149
7.3404
7.3404
7.9983
7.9983
7.6718
7.6718
7.6718
7.6718
7.6718
7.7513
7.7513
8.5098
8.9301
7.6568
7.4175
7.4175
7.7182
7.7182
8.1005
7.5431
7.8284
7.7479
8.1005
6.8941
6.5418
6.5418
6.8024
6.5901
7.4236
6.5901
7.0700
7.0700
6.5720
6.5720
6.5720
6.5720
6.5720
6.6515
6.6515
6.9534
6.9534
6.9034
6.9034
7.4649
6.6672
6.9679
7.1723
6.4434
6.4434
6.7286
6.7286
6.6481
6.6481
7.1723
7.0556
7.0556
7.0590
7.1460
5.7915
5.7027
6.3787
5.8217
5.8217
5.9012
6.0252
6.0252
5.8037
5.8037
5.9085
6.4809
5.6931
5.9783
5.8978
6.4809
6.1274
6.1274
6.1274
6.1274
5.9592
5.9592
327
Table of Criteria Values for PBCDs (K = 4)
Dsgn
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
52.4278
60.1398
35.4297
72.3036
89.4215
59.7229
78.4589
71.0071
50.9077
50.9077
69.0025
50.2100
59.6969
59.6969
37.2079
37.2079
43.2675
65.8162
51.5925
72.5726
58.4231
39.6264
48.3554
48.6832
35.7870
57.3916
38.4198
25.0499
46.6252
rs = 1, n0 = 3
A
G
43.5490 39.7811
47.3513 41.0497
27.1284 37.8450
63.7080 41.8436
89.2857 89.2857
53.4170 38.3733
73.1159 41.2341
64.1826 35.5351
44.5062 36.5825
44.5062 36.5825
61.0061 36.2102
43.1507 35.5618
49.9139 34.7242
49.9139 34.4156
28.7724 32.9444
28.7724 32.9444
34.6011 35.4095
63.2911 44.9224
43.3335 34.8697
71.4286 71.4286
51.1053 34.7988
30.1508 31.9118
41.1899 28.4281
42.7892 35.9380
26.5291 27.8958
53.5714 53.5714
27.3556 27.4090
14.9375 21.3211
35.7143 35.7143
IV
6.0797
6.3949
6.6965
5.2965
5.0667
5.2099
5.0077
5.3209
5.0328
5.0328
4.9142
5.1467
5.2576
5.2576
5.6408
5.6408
5.6122
4.3000
4.9668
4.3879
4.3246
4.6669
4.5566
4.0338
4.5566
3.5827
3.5827
1.9743
1.2667
D
51.2679
64.9448
38.1318
66.3994
92.1801
57.7917
76.2691
77.8547
46.0556
46.0556
62.7509
45.6083
61.4418
61.4418
38.1410
38.1410
50.1641
55.4435
49.4446
74.6695
55.5414
37.4820
53.7403
42.3336
36.5636
58.8632
35.2243
28.4740
47.5191
rs = 2, n0 = 3
A
G
39.1390 36.2077
47.8696 38.3503
26.8277 35.0093
54.7086 39.8863
92.1053 92.1053
48.4280 32.8030
68.5567 35.1396
71.2850 44.2824
37.1996 31.2516
37.1996 31.2516
52.5056 30.8661
36.3783 31.0061
47.9205 35.1836
47.9205 32.2371
26.7295 30.1699
26.7295 30.1699
41.5573 44.2608
50.7246 36.6228
37.7447 33.2386
73.6842 73.6842
45.2058 29.1706
25.3776 27.0725
47.7749 35.4259
34.3013 29.4006
24.2547 26.5909
55.2632 55.2632
22.1441 22.6498
19.0980 26.5694
36.8421 36.8421
IV
6.2178
6.1011
6.4755
5.5551
4.9524
5.3338
5.0534
4.9803
5.4361
5.4361
5.2089
5.5264
5.1729
5.1729
5.6648
5.6648
5.0141
4.8048
5.1848
4.2889
4.4815
4.9632
4.2035
4.4934
4.6210
3.5019
3.9194
1.7294
1.2381
n0 = 3
A
G
48.0865 67.5553
52.9242 63.1855
48.4532 63.0517
55.4119 58.8072
53.8236 58.6723
54.4228 58.7199
48.8834 58.5480
49.5681 58.5480
51.3774 59.6757
56.0775 54.4793
57.5236 55.4474
56.7089 54.2836
56.7643 54.5133
54.9124 54.6891
56.1663 54.2030
55.8745 54.6891
49.9071 54.0443
49.3954 54.0443
50.1534 54.0443
53.7858 55.3842
52.1671 55.0852
56.7008 51.4913
57.9580 51.0590
57.5947 51.7327
58.9693 50.8267
58.3222 49.9706
59.2638 51.6457
59.4200 51.4311
57.1263 50.1317
58.4612 50.1317
56.2577 50.5476
56.2577 49.6860
57.3611 50.1317
57.3611 50.1317
50.5874 49.5406
51.8072 49.5406
54.3236 50.9555
55.8064 50.8267
54.9704 50.7688
53.1324 50.5265
54.1158 50.5265
58.4791 50.1802
60.5348 47.0258
59.5272 49.2269
59.9545 49.4824
61.9775 46.7555
61.1087 46.9025
60.3842 46.2061
60.3841 45.4278
IV
16.7645
13.4075
16.1511
11.2772
12.7942
12.7022
15.5378
15.4186
12.6409
9.9562
10.5446
10.6639
10.5906
12.1808
11.9856
12.0398
14.8325
14.9244
14.8052
10.5106
12.0275
8.7467
9.2921
9.3429
9.9986
10.0506
9.8401
9.9304
11.3957
11.2312
11.5675
11.5675
11.4264
11.4264
14.2191
14.0726
9.1896
9.7779
9.8973
11.4142
11.2731
8.1334
8.6014
8.7295
8.6788
9.2706
9.3361
9.4372
9.4372
Table of Criteria Values for UNFSDs (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
D
72.4060
73.7060
73.6400
71.8950
75.7740
75.1630
75.7740
76.8450
72.2670
69.8470
74.8230
73.9320
73.2840
78.3430
77.6580
78.8470
79.0340
78.7650
79.5420
70.2300
74.3440
67.4270
72.2400
71.1330
77.3740
75.7670
76.6790
77.6980
81.8480
81.4910
81.9700
81.1890
82.2240
82.2240
82.7590
84.2370
67.9050
73.2000
72.2490
76.9640
77.5050
68.5520
75.5900
73.5700
74.2710
81.3290
79.6940
80.2120
79.3700
n0 = 1
A
G
32.6220 71.4286
50.3822 69.1959
31.9042 66.6667
58.0194 64.4028
50.6080 64.2534
51.2364 64.3011
31.1143 61.9048
31.3661 61.9048
48.6206 64.9193
60.2667 59.6658
60.1934 60.1159
59.0949 59.4487
59.5760 59.7031
50.8740 59.5467
52.3577 59.3548
51.8098 59.5467
30.4153 57.1429
30.2410 57.1429
30.4987 57.1429
56.1894 60.1159
48.7059 59.9255
61.7975 56.1449
62.1977 55.8751
61.7577 56.4617
60.8849 55.1062
60.4184 54.7279
62.1548 56.5439
61.4938 55.1272
52.5787 54.5845
53.9393 54.5845
51.1924 54.9318
51.1924 54.4086
52.2275 54.5845
52.2275 54.5845
29.4482 52.3810
29.8214 52.3810
58.3196 55.8064
58.2448 55.1062
57.1239 55.0063
48.8071 54.9318
49.7474 54.9402
63.6938 54.6979
64.8755 51.4990
63.6473 52.9008
64.1618 53.9232
63.7362 50.1156
63.0337 51.3564
62.0876 50.0966
62.0876 49.7526
IV
30.2400
15.5120
29.6800
11.3011
14.9520
14.8680
29.1200
29.0111
14.8121
9.5348
10.6461
10.7411
10.6571
14.3920
14.2138
14.1997
28.4760
28.5600
28.4512
10.6011
14.2521
8.1600
8.9169
8.9748
10.1702
10.1811
9.9565
10.0759
13.4893
13.4615
13.8320
13.8320
13.6397
13.6397
27.9160
27.7823
8.8349
9.9461
10.0411
13.6920
13.4998
7.6000
8.2729
8.4148
8.3570
9.4462
9.4800
9.6210
9.6211
D
71.1331
69.3348
72.7253
66.9173
71.9580
70.8679
75.2862
76.3494
68.1377
64.6552
69.6983
69.0276
68.3193
74.6409
73.4167
74.5406
79.0794
78.8111
79.5880
65.4724
70.8313
62.0749
66.8855
65.9280
72.3882
70.7681
71.5572
72.6789
77.9410
77.2843
78.3999
76.9982
78.1274
78.1274
83.4993
84.9901
62.9364
68.3106
67.5934
73.6118
73.5042
63.1626
70.1209
68.2883
68.9248
76.2491
74.6050
75.2267
74.3013
328
Table of Criteria Values for UNFSDs (K = 4)
Dsgn
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
p
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
l
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
c
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
q
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
D
79.9030
80.9800
80.9800
86.7050
86.4040
85.9890
87.6790
88.3000
65.1380
70.2700
69.0870
75.7830
74.8400
75.0350
80.6180
52.4480
80.6180
71.2990
70.8740
71.3840
79.7800
77.1730
77.2130
77.2130
77.4740
85.9020
84.9960
84.6330
84.3870
85.7480
86.9220
92.8020
93.2590
95.8170
66.0920
73.6740
71.4890
72.2470
79.9150
78.1320
51.9380
79.9150
79.9150
56.6300
85.4750
85.4750
85.0190
67.9360
73.8840
73.0740
73.8870
74.4860
74.4860
82.3340
81.9740
81.9740
83.2840
91.7310
91.4070
93.3510
102.3700
68.3030
68.8560
78.0320
75.1700
50.6420
78.0320
75.1700
57.0960
56.4200
84.8010
83.3930
83.1210
83.3930
83.1210
83.7970
83.1220
62.6270
92.5030
63.1420
71.3470
68.9720
69.7950
69.7950
49.9110
n0 = 1
A
G
63.5492 51.4035
63.3392 50.1156
63.3392 50.1156
52.7382 49.9458
53.0795 49.6223
52.7382 49.6223
54.3140 49.6223
28.5942 47.6190
59.7010 53.1825
60.1123 51.4990
59.6601 51.3288
58.7652 50.0966
58.2874 50.0058
60.0682 51.4035
50.3244 49.9457
33.9109 49.9380
50.3244 49.9457
56.0672 50.1156
66.1757 49.7009
66.1758 58.1287
68.6761 48.5308
67.0247 48.5308
66.1208 46.3491
66.1206 47.6107
66.7379 48.5308
65.7518 45.1041
65.3860 46.2635
66.0031 46.2632
65.3861 46.2207
67.6227 46.2635
67.6224 45.1041
55.1763 44.6600
55.1763 44.9676
27.6610 42.8571
61.4327 49.6344
62.6561 46.3491
61.3850 46.3491
61.9171 48.5308
61.4766 45.1041
60.7510 46.2207
39.3758 45.0869
61.4766 45.1041
61.4766 45.1041
35.9052 44.9512
50.5910 44.9511
50.5910 44.9511
50.2466 48.1257
57.7461 47.4339
56.3667 45.1041
57.7009 48.3672
69.5645 51.1315
69.5649 44.1798
69.5642 51.6699
70.2648 43.1385
70.6229 43.1385
70.2646 48.4078
72.3788 43.1385
70.6171 46.1178
70.2412 41.1231
70.6045 40.0925
58.7088 60.6339
63.7442 44.1786
63.7441 53.5341
66.3622 43.1385
64.6310 43.1385
41.0921 41.1992
66.3622 43.1385
64.6310 43.1385
43.1239 40.0925
42.6317 41.1231
63.3021 40.0925
63.5638 41.1228
62.9205 41.0851
63.5638 41.1228
62.9205 41.0851
62.9209 44.4376
62.9210 44.3932
38.8947 39.9712
52.4198 42.7784
58.8228 47.9576
60.0863 42.1635
58.7737 42.1635
59.3229 46.8008
59.3229 46.8008
37.2846 40.0925
IV
9.4805
9.5160
9.5160
13.0797
12.9952
13.0797
12.8008
27.2618
7.4601
8.2169
8.2749
9.4702
9.4811
9.2565
12.7893
12.3765
12.7893
9.2461
7.0400
7.0400
7.6187
7.7587
7.8548
7.8548
7.7969
8.9559
8.9199
8.9205
8.9559
8.8089
8.8090
12.2407
12.2407
26.5929
6.9001
7.5729
7.7148
7.6569
8.7463
8.7800
9.2427
8.7463
8.7463
11.3963
12.2952
12.2952
12.3797
7.5169
8.7702
8.5565
6.4800
6.4800
6.4800
7.2369
7.1986
7.2369
7.1241
8.2489
8.2489
8.2430
11.2115
6.3401
6.3401
6.9187
7.0587
7.7590
6.9187
7.0587
8.3154
8.3375
8.2558
8.2205
8.2559
8.2205
8.2559
8.2199
8.2560
10.3607
11.5408
6.2001
6.8729
7.0148
6.9569
6.9569
8.4343
D
74.7289
75.9184
75.9184
83.1612
82.4928
81.8606
83.4698
89.9836
60.0171
65.1527
64.1274
71.0729
70.1886
70.1761
77.0926
50.3975
77.0926
66.6821
65.3681
65.8385
74.0098
71.7310
71.8017
71.8017
72.0269
80.7694
79.7249
79.3644
79.2182
80.4095
81.7113
88.7582
89.4004
98.8429
60.9579
68.4644
66.4790
67.1680
75.1434
73.3459
48.8554
75.1434
75.1434
54.6135
82.0116
82.0116
81.3136
63.0961
69.4982
68.5245
68.2339
68.7867
68.7863
76.7185
76.3641
76.3640
77.5829
86.4110
86.0027
88.0609
98.4790
63.0772
63.5879
72.5329
70.0252
47.2007
72.5329
70.0252
53.8827
53.0996
80.0281
78.4634
78.3008
78.4634
78.3008
78.8648
78.3012
60.4024
88.9865
58.3104
66.4480
64.2848
65.0350
65.0350
47.1231
n0 = 3
A
G
61.4964 46.9506
61.6815 46.7555
61.6815 46.7555
58.9780 45.9523
58.9780 45.5742
59.2535 45.5742
60.7806 45.5742
52.2572 45.0369
54.7941 48.7836
56.0876 47.0258
55.7132 47.0297
57.1305 46.2061
56.4629 46.2257
57.4348 46.9506
55.2314 45.9332
36.2251 45.9523
55.2314 45.9332
53.8766 46.7555
60.8101 45.5980
60.8102 53.3738
64.2032 44.5341
62.8124 44.5341
62.0737 42.3232
62.0736 44.6996
62.5898 44.5341
64.6914 42.0800
63.9924 42.2559
64.4653 42.2556
63.9925 42.2123
66.3369 42.2559
66.4502 42.0800
63.0328 41.0168
62.8253 41.3571
54.6487 62.0985
56.4257 45.5368
58.5579 42.3232
57.5116 42.3232
57.9550 44.5341
60.0602 42.0800
59.1547 42.2123
37.8779 41.5855
60.0602 42.0800
60.0602 42.0800
38.9831 41.3571
56.9420 41.3399
56.9420 41.3399
57.2275 44.2199
53.9595 43.4265
55.0332 42.0800
55.3472 44.1807
63.9994 46.9462
63.9997 40.5327
63.9990 47.4434
66.2294 39.5859
66.5101 39.5859
66.2293 45.1153
68.2026 39.5859
70.0686 45.2023
69.3990 37.5608
70.1625 37.4044
68.6835 55.5211
58.6073 40.5316
58.6072 49.1650
62.1694 39.5859
60.7052 39.5859
38.3893 37.6206
62.1694 39.5859
60.7052 39.5859
41.9624 37.4044
41.3034 37.5608
62.6854 37.4044
62.4464 37.5605
61.9472 37.5220
62.4464 37.5605
61.9472 37.5220
61.9477 40.5893
61.9478 40.5441
43.2455 36.7619
60.9368 39.3066
54.0534 44.0168
56.2609 38.6013
55.1763 38.6013
55.6358 42.9650
55.6358 42.9650
35.9920 37.4044
IV
9.3123
9.3171
9.3171
10.8131
10.7907
10.8131
10.6487
13.5026
7.9801
8.5254
8.5762
9.2319
9.2839
9.0734
10.6291
10.9292
10.6291
9.0112
7.5200
7.5200
7.9032
8.0340
8.1161
8.1162
8.0654
8.7037
8.7227
8.6989
8.7227
8.5767
8.5767
10.0353
10.0353
12.7700
7.3667
7.8348
7.9628
7.9121
8.5040
8.5694
9.2767
8.5040
8.5040
9.9288
10.0241
10.0241
10.0465
7.7587
8.4652
8.3067
6.9067
6.9067
6.9067
7.4521
7.4207
7.4521
7.3401
7.9870
8.0027
7.9828
9.2414
6.7534
6.7534
7.1366
7.2674
8.1014
7.1366
7.2674
8.2759
8.3276
7.9370
7.9323
7.9561
7.9323
7.9561
7.9560
7.9560
8.9212
9.2687
6.6001
7.0681
7.1962
7.1454
7.1454
8.3914
329
Table of Criteria Values for UNFSDs (K = 4)
Dsgn
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
p
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
dv
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
l
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
c
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
q
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
D
78.1820
76.4720
78.4870
49.9110
80.0920
88.9510
88.9510
101.7060
71.2550
49.6810
71.2550
55.7070
55.4290
80.6390
80.2370
80.2360
80.6390
80.2360
80.2370
80.2370
64.3050
62.7760
90.8730
91.2400
71.4530
65.1350
65.7390
48.2140
72.6710
75.8430
73.0370
73.0370
73.0360
73.0360
48.2140
55.2990
55.2990
83.2150
54.5510
85.5390
99.6220
54.4790
76.2060
76.2060
76.2080
62.8710
62.8710
87.9440
87.9450
73.5090
46.7640
67.8920
68.6260
68.6260
46.7640
53.4460
53.1350
53.4460
53.1350
79.6420
79.6420
53.1350
61.4400
36.8000
73.0180
96.1720
62.7910
83.7210
75.3910
51.6060
51.6060
74.7300
51.6070
61.2860
61.2870
37.6750
37.6750
45.4500
63.4490
50.4350
77.6960
60.7990
40.5330
50.9440
47.5780
n0 = 1
A
G
58.8681 40.0925
58.6689 42.9931
58.4457 40.0925
37.2846 40.0925
76.4769 54.6944
76.8040 37.7462
76.8038 46.0240
78.1800 53.1591
66.9845 45.2098
42.6825 38.6573
66.9845 45.2098
45.0691 37.7462
45.3122 37.7462
67.7279 37.7462
68.1083 37.7462
67.7276 43.8445
67.7279 37.7462
67.7276 43.8445
68.1085 40.9507
67.7271 48.8806
48.4736 35.0809
47.9234 35.9827
67.7030 38.8829
68.1016 44.0370
43.6248 53.0547
60.8684 42.0238
60.8686 51.3528
38.7416 36.8931
61.7941 37.7462
63.6073 37.7462
61.4814 40.9507
61.4815 40.9507
61.4813 37.7462
61.4813 37.7462
38.7416 36.8931
40.8135 35.0809
40.8135 35.0809
62.4491 38.8829
40.3102 38.8829
82.1905 46.8809
88.1708 46.0943
47.3675 38.7525
71.8546 40.1506
71.8546 40.1506
71.8552 45.2988
52.0328 32.3539
52.0328 39.4492
74.4030 43.6093
74.4043 35.1006
58.2568 45.5650
39.9996 36.0214
63.8286 40.1505
63.8282 44.0167
63.8287 40.1506
39.9996 36.0214
42.4578 32.3539
42.7096 32.3539
42.4578 32.3539
42.7096 32.3539
67.0169 35.1006
67.0169 35.1006
42.7098 35.1006
45.4320 33.3282
25.7214 30.0694
60.2713 35.1006
96.1537 96.1493
55.9684 39.0674
79.9986 44.0633
67.0274 38.4119
44.3776 33.4588
44.3776 33.4588
70.8903 44.0633
44.3779 37.7490
49.3521 36.3411
49.3529 29.2505
27.0320 26.9616
27.0320 26.9616
35.3523 37.9708
59.8791 36.5099
39.2724 29.2505
76.9217 76.9194
53.0954 35.2506
29.4106 31.2540
42.8730 30.7295
40.5394 29.3445
IV
8.0462
8.0805
8.1159
8.4343
5.8542
6.5867
6.5868
7.5319
5.7801
6.4849
5.7801
6.9217
6.8865
6.5369
6.4986
6.5369
6.5369
6.5369
6.4986
6.5370
7.4014
7.4247
7.5488
7.5490
9.2352
5.6401
5.6401
6.9507
6.3587
6.2187
6.3969
6.3969
6.3969
6.3969
6.9507
7.5071
7.5071
7.4089
7.5291
5.3600
5.9313
5.7089
5.2201
5.2201
5.2201
6.0554
6.0554
5.8868
5.8867
6.4935
5.6766
5.0801
5.0801
5.0801
5.6766
6.1134
6.0782
6.1134
6.0782
5.7241
5.7241
6.0781
6.6164
6.7783
5.5186
4.8000
4.9330
4.6601
5.1780
4.9006
4.9006
4.4542
4.9006
5.2471
5.2471
5.7018
5.7018
5.4686
4.3801
5.3051
4.1570
4.1247
4.5822
4.3906
4.0923
D
73.7845
71.9513
74.1087
47.1231
74.0842
83.0792
83.0791
96.1050
65.9103
45.9540
65.9103
52.0587
51.7839
75.3580
74.9603
74.9594
75.3580
74.9594
74.9606
74.9606
60.9440
59.3189
85.8679
86.3335
69.2518
60.2493
60.8073
45.0702
67.8922
70.6777
68.2535
68.2535
68.2531
68.2531
45.0702
52.4333
52.4333
78.6319
51.5633
79.2944
93.3434
50.5017
70.6425
70.6425
70.6438
58.9425
58.9426
82.4491
82.4506
69.8596
43.3494
62.9354
63.6157
63.6162
43.3494
50.1398
49.8312
50.1398
49.8312
74.6885
74.6885
49.8314
58.3896
35.0944
68.2779
89.4213
58.3831
77.8447
71.0062
47.9837
47.9837
69.4848
47.9842
57.7614
57.7625
35.5086
35.5086
43.5405
58.9950
47.5719
72.5714
56.7895
37.8598
48.3548
44.4403
n0 = 3
A
G
57.8246 37.4044
57.3023 39.2718
57.5265 37.4044
35.9920 37.4044
70.4965 50.2800
72.7697 34.6376
72.7696 43.3499
78.8052 48.8353
61.6730 41.5117
39.1787 35.4661
61.6730 41.5117
42.2867 34.6376
42.4800 34.6376
64.0486 34.6376
64.3487 34.6376
64.0484 40.8501
64.0486 34.6376
64.0484 40.8501
64.3489 37.5944
64.0479 45.8545
47.8220 32.7289
46.9770 32.8657
67.4537 35.5156
68.1766 43.4544
50.8363 48.5809
55.9990 38.5709
55.9991 47.1884
36.2586 33.7762
58.1955 34.6376
59.7369 34.6376
57.9505 37.5944
57.9506 37.5944
57.9503 34.6376
57.9503 34.6376
36.2586 33.7762
39.9286 32.7289
39.9286 32.7289
62.1282 35.5156
39.2477 35.5156
75.9482 43.0971
84.2935 42.6421
43.5475 35.5825
66.2968 36.8738
66.2968 36.8738
66.2973 41.6331
49.0781 29.6894
49.1384 37.1571
70.8427 40.6933
70.8439 32.2237
58.7441 41.8589
36.7343 33.0616
58.8224 36.8737
58.8221 40.4472
58.8226 36.8738
36.7343 33.0616
39.9459 29.6894
40.1472 29.6894
39.9459 29.6894
40.1472 29.6894
63.6283 32.2237
63.6283 32.2237
40.1474 32.2237
44.8850 30.4420
25.0904 28.0533
56.7752 32.2237
89.2856 89.2814
51.6039 35.9143
74.0727 40.5424
64.1818 35.5351
40.8338 30.7281
40.8338 30.7281
65.5339 40.5424
40.8341 34.6943
46.8289 33.9111
46.7640 26.8531
25.3678 24.7412
25.3678 24.7412
35.6013 34.8824
55.2476 33.5482
37.0730 26.8531
71.4273 71.4251
49.0445 32.4339
27.0260 28.7314
41.1895 28.4281
37.3433 26.9648
IV
7.7373
7.7789
7.7837
8.3914
6.2212
6.7614
6.7614
7.2668
6.1401
6.9375
6.1401
7.1955
7.1655
6.6854
6.6540
6.6854
6.6854
6.6854
6.6539
6.6855
7.3035
7.3665
7.2360
7.2204
7.8984
5.9868
5.9868
7.2162
6.5007
6.3699
6.5321
6.5320
6.5321
6.5321
7.2162
7.3906
7.3906
7.0434
7.4422
5.6800
6.0632
6.0877
5.5268
5.5268
5.5267
6.2648
6.2648
5.9948
5.9947
6.3600
6.0522
5.3734
5.3734
5.3734
6.0522
6.3102
6.2802
6.3102
6.2802
5.8067
5.8067
6.2802
6.4812
6.9044
5.6032
5.0667
5.2378
4.9134
5.3209
5.2024
5.2024
4.6879
5.2024
5.3796
5.3795
5.9777
5.9777
5.5503
4.6067
5.4249
4.3879
4.3526
4.8839
4.5567
4.3171
330
Table of Criteria Values for UNFSDs (K = 4)
Dsgn
220
221
222
223
224
p
4
3
3
3
2
dv
2
2
2
1
1
l
1
2
1
1
1
c
1
0
1
0
0
q
1
0
0
1
0
D
35.5330
60.9786
38.4140
26.0520
48.7940
n0 = 1
A
G
24.3955 23.4004
57.6908 57.6897
26.0858 26.4380
15.4408 23.0471
38.4600 38.4600
IV
4.7118
3.3942
3.5922
1.8595
1.2000
D
33.7563
57.3903
36.1532
25.0499
46.6241
n0 = 3
A
G
22.9754 21.4825
53.5700 53.5690
23.9989 24.3254
14.9375 21.3211
35.7128 35.7128
IV
4.8934
3.5827
3.7996
1.9743
1.2667
Table of Criteria Values for Hybrid 416A (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
D
70.0185
70.9059
71.1361
70.5016
72.1955
72.1955
72.4479
72.4482
69.7462
69.3850
71.8574
71.8574
71.8577
73.7296
73.7299
73.7299
74.0089
74.0092
74.0092
69.2203
71.0237
69.5993
70.7392
70.7395
73.4934
73.4940
73.4937
73.4937
75.5847
75.5854
75.5851
75.5851
75.5851
75.5851
75.8975
75.8978
67.9119
70.5560
70.5560
72.5637
72.5640
71.1168
72.3992
72.3999
72.3995
75.5058
75.5065
78.0956
75.5062
78.0956
75.5062
75.5062
77.8730
77.8730
77.8734
77.8734
78.2275
67.9965
69.2225
69.2229
72.1929
72.1932
72.1932
74.4558
48.3607
74.4558
69.0253
73.0167
73.0170
74.4810
74.4818
74.4814
74.4822
n0 = 1
A
G
39.0902 74.3053
52.8684 69.5935
38.2617 69.3516
59.7966 64.7016
52.6354 64.6225
52.6354 64.6225
37.3483 64.3979
37.3484 64.3979
51.2969 64.6225
63.5224 59.7618
60.1272 59.7246
60.1272 59.7246
60.1275 59.7246
52.3663 59.6516
52.3665 59.6516
52.3665 59.6516
36.3364 59.4442
36.3365 59.4442
36.3365 59.4442
58.2463 59.7246
50.9338 59.6516
67.6962 68.9681
64.2963 54.7817
64.2966 54.7817
60.5227 54.7475
60.5233 54.7475
60.5230 54.7475
60.5230 54.7475
52.0517 54.6806
52.0521 54.6806
52.0519 54.6806
52.0519 54.6806
52.0519 54.6806
52.0519 54.6806
35.2090 54.4905
35.2091 54.4905
61.9621 54.7817
58.4500 54.7475
58.4500 54.7475
50.5112 54.6806
50.5114 54.6806
69.1258 67.7239
65.2502 49.8015
65.2509 49.8015
65.2506 49.8015
61.0042 49.7705
61.0048 49.7705
65.0555 61.2575
61.0045 49.7705
65.0555 61.2575
61.0045 49.7705
61.0045 49.7705
51.6794 49.7096
51.6794 49.7096
51.6796 49.7096
51.6796 49.7096
33.9452 49.5369
66.1775 66.1410
62.6169 49.8015
62.6173 49.8015
58.6964 49.7705
58.6967 49.7705
58.6967 49.7705
50.0134 49.7096
34.7756 49.7096
50.0134 49.7096
56.5569 49.7705
70.9572 60.9515
70.9577 60.9515
66.4552 44.8214
66.4561 44.8214
66.4557 44.8214
66.4565 44.8214
IV
23.9666
15.0467
23.4912
11.6155
14.5714
14.5714
23.0159
23.0158
14.2879
9.6356
11.1401
11.1401
11.1400
14.0960
14.0960
14.0960
22.5405
22.5405
22.5405
10.8567
13.8126
7.8875
9.1602
9.1602
10.6647
10.6646
10.6647
10.6647
13.6206
13.6205
13.6206
13.6206
13.6206
13.6206
22.0651
22.0650
8.8768
10.3813
10.3813
13.3372
13.3372
7.4121
8.6848
8.6848
8.6848
10.1893
10.1893
9.4868
10.1893
9.4868
10.1893
10.1893
13.1452
13.1452
13.1452
13.1452
21.5897
7.1287
8.4014
8.4014
9.9059
9.9059
9.9059
12.8618
12.2765
12.8618
9.6225
6.9367
6.9367
8.2094
8.2094
8.2094
8.2094
D
67.1423
66.1560
68.5525
64.8639
67.5765
67.5765
70.2164
70.2166
65.2840
63.2594
66.2651
66.2651
66.2654
69.2724
69.2727
69.2727
72.2087
72.2090
72.2090
63.8332
66.7301
62.9059
64.6044
64.6047
67.9601
67.9607
67.9604
67.9604
71.3316
71.3322
71.3319
71.3319
71.3319
71.3319
74.6367
74.6370
62.0222
65.2438
65.2438
68.4805
68.4808
64.3425
66.2561
66.2567
66.2564
70.0515
70.0521
72.1293
70.0518
72.1293
70.0518
70.0518
73.8839
73.8839
73.8842
73.8842
77.6582
61.5194
63.3490
63.3493
66.9779
66.9782
66.9782
70.6418
45.8834
70.6418
64.0391
66.1431
66.1434
68.3323
68.3331
68.3327
68.3334
n0 = 3
A
G
52.7264 69.1018
55.6347 64.8579
52.9174 64.4950
57.5395 60.4597
56.1028 60.2252
56.1028 60.2252
53.1394 59.8882
53.1396 59.8882
54.4113 60.2252
58.6858 55.9682
58.2503 55.8089
58.2503 55.8089
58.2506 55.8089
56.6589 55.5925
56.6592 55.5925
56.6592 55.5925
53.4009 55.2814
53.4011 55.2814
53.4011 55.2814
56.2824 55.8089
54.7953 55.5925
60.9653 62.1182
59.6029 51.3042
59.6032 51.3042
59.1132 51.1582
59.1138 51.1582
59.1135 51.1582
59.1135 51.1582
57.3306 50.9598
57.3312 50.9598
57.3309 50.9598
57.3309 50.9598
57.3309 50.9598
57.3309 50.9598
53.7134 50.6747
53.7137 50.6747
57.3640 51.3042
56.9103 51.1582
56.9103 51.1582
55.2562 50.9598
55.2565 50.9598
62.3027 61.0301
60.7420 46.6402
60.7427 46.6402
60.7424 46.6402
60.1831 46.5074
60.1838 46.5074
62.8130 54.8106
60.1835 46.5074
62.8130 54.8106
60.1835 46.5074
60.1835 46.5074
58.1583 46.3271
58.1583 46.3271
58.1586 46.3271
58.1586 46.3271
54.0934 46.0679
59.6268 59.5937
58.1957 46.6402
58.1961 46.6402
57.6826 46.5074
57.6829 46.5074
57.6829 46.5074
55.8196 46.3271
37.2106 46.3271
55.8196 46.3271
55.3815 46.5074
64.0193 54.9271
64.0198 54.9271
62.1948 41.9762
62.1956 41.9762
62.1952 41.9762
62.1960 41.9762
IV
15.1685
13.0278
14.6372
11.3885
12.4965
12.4965
14.1059
14.1058
12.1797
10.0484
10.8572
10.8572
10.8572
11.9652
11.9651
11.9651
13.5746
13.5745
13.5745
10.5404
11.6484
8.5802
9.5171
9.5170
10.3259
10.3258
10.3259
10.3259
11.4339
11.4338
11.4338
11.4338
11.4338
11.4338
13.0432
13.0432
9.2003
10.0091
10.0091
11.1171
11.1171
8.0488
8.9857
8.9857
8.9857
9.7946
9.7945
9.3621
9.7945
9.3621
9.7945
9.7945
10.9025
10.9025
10.9025
10.9025
12.5119
7.7321
8.6690
8.6690
9.4778
9.4778
9.4778
10.5858
10.9595
10.5858
9.1611
7.5175
7.5175
8.4544
8.4544
8.4544
8.4544
331
Table of Criteria Values for Hybrid 416A (K = 4)
Dsgn
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
p
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
l
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
q
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
D
74.4814
78.0409
78.0413
78.0409
78.0413
81.0206
78.0413
80.7640
80.7640
81.1727
69.4658
70.8589
70.8592
73.3939
74.2452
74.2460
49.0009
74.2452
74.2452
50.7104
76.8360
76.8360
76.8364
67.4129
70.6346
70.6350
75.4629
75.4634
75.4634
77.1681
77.1685
77.1690
80.2808
81.3301
81.3297
81.3297
84.5292
71.3471
71.3475
72.9588
72.9597
49.1235
72.9588
72.9597
51.7724
51.7727
76.8934
76.8934
76.8938
76.8934
76.8938
76.8938
76.8938
53.8091
79.9184
67.4557
68.9795
68.9799
68.9799
71.7619
47.2197
72.6991
72.6999
72.6995
47.2197
78.7301
80.7659
80.7664
85.7626
73.8410
51.0111
73.8410
52.3302
52.3305
75.7509
75.7514
75.7519
75.7509
79.2524
79.2524
75.7519
55.5677
55.5677
80.4374
80.4379
n0 = 1
A
G
66.4557 44.8214
61.6035 44.7934
61.6038 44.7934
61.6035 44.7934
61.6038 44.7934
66.2313 55.4076
61.6038 44.7934
51.2315 44.7387
51.2315 44.7387
32.5186 44.5832
67.5260 60.9515
63.4364 44.8214
63.4368 44.8214
66.8106 55.6543
59.0004 44.7934
59.0010 44.7934
39.6256 44.7934
59.0004 44.7934
59.0004 44.7934
35.1747 44.7387
49.4183 44.7387
49.4183 44.7387
49.4185 44.7387
60.6799 44.8214
56.6086 44.7934
56.6089 44.7934
73.3876 63.2516
73.3882 54.1791
73.3882 54.1791
68.0261 39.8412
68.0266 39.8412
68.0271 39.8412
72.4398 49.4704
62.3698 39.8164
62.3694 39.8164
62.3694 39.8164
50.6825 39.7677
69.2910 54.1791
69.2915 54.1791
64.4914 39.8412
64.4922 39.8412
42.4129 39.8412
64.4914 39.8412
64.4922 39.8412
40.9345 39.8164
40.9349 39.8164
59.3852 39.8164
59.3852 39.8164
59.3855 39.8164
59.3852 39.8164
59.3855 39.8164
59.3855 39.8164
59.3855 39.8164
35.6869 39.7677
48.6940 39.7677
65.6275 60.2131
61.3062 39.8412
61.3066 39.8412
61.3066 39.8412
64.8681 49.4710
37.7447 39.8164
56.6735 39.8164
56.6741 39.8164
56.6738 39.8164
37.7447 39.8164
76.7698 58.3195
70.1584 34.8611
70.1590 34.8611
63.3826 34.8393
71.7005 55.8326
46.4787 47.4067
71.7005 55.8326
44.6059 34.8611
44.6063 34.8611
65.9010 34.8611
65.9015 34.8611
65.9020 34.8611
65.9010 34.8611
70.6675 43.3702
70.6674 43.2871
65.9020 34.8611
42.7506 34.8393
42.7506 34.8393
59.8873 34.8393
59.8878 34.8393
IV
8.2094
9.7139
9.7139
9.7139
9.7139
9.0114
9.7139
12.6698
12.6698
21.1143
6.6533
7.9260
7.9260
7.4519
9.4305
9.4305
9.6335
9.4305
9.4305
11.6178
12.3864
12.3864
12.3864
7.6426
9.1471
9.1471
6.4614
6.4613
6.4613
7.7340
7.7340
7.7340
7.2599
9.2385
9.2385
9.2385
12.1944
6.1780
6.1779
7.4507
7.4506
7.9662
7.4507
7.4506
8.9748
8.9748
8.9551
8.9551
8.9551
8.9551
8.9551
8.9551
8.9551
10.9591
11.9110
5.8945
7.1672
7.1672
7.1672
6.6931
8.7573
8.6717
8.6717
8.6717
8.7573
5.9859
7.2586
7.2586
8.7631
5.7026
6.3368
5.7026
7.3075
7.3075
6.9752
6.9752
6.9752
6.9752
6.5012
6.5012
6.9752
8.3160
8.3160
8.4797
8.4797
D
68.3327
72.6956
72.6959
72.6956
72.6959
75.0953
72.6959
77.1277
77.1277
81.5178
62.9265
65.0093
65.0096
67.1128
69.1599
69.1607
45.6446
69.1599
69.1599
48.4273
73.3765
73.3765
73.3769
61.8478
65.7966
65.7969
68.4648
68.4652
68.4652
71.0198
71.0202
71.0206
73.6103
76.1419
76.1414
76.1414
81.3836
64.7306
64.7310
67.1459
67.1467
45.2096
67.1459
67.1467
48.4697
48.4700
71.9882
71.9882
71.9886
71.9882
71.9886
71.9886
71.9886
51.8067
76.9443
61.2001
63.4836
63.4840
63.4840
65.7993
44.2074
68.0614
68.0622
68.0618
44.2074
71.5709
74.6310
74.6315
80.8130
67.1264
46.3726
67.1264
48.3552
48.3556
69.9969
69.9974
69.9979
69.9969
72.9219
72.9219
69.9979
52.3607
52.3607
75.7951
75.7956
n0 = 3
A
G
62.1952 41.9762
61.5450 41.8567
61.5454 41.8567
61.5450 41.8567
61.5454 41.8567
64.6191 50.0245
61.5454 41.8567
59.2029 41.6944
59.2029 41.6944
54.5652 41.4611
60.8990 54.9271
59.2457 41.9762
59.2460 41.9762
61.6791 50.5855
58.6554 41.8567
58.6561 41.8567
38.5684 41.8567
58.6554 41.8567
58.6554 41.8567
38.5736 41.6944
56.5243 41.6944
56.5243 41.6944
56.5246 41.6944
56.5636 41.9762
56.0253 41.8567
56.0256 41.8567
66.3028 57.0685
66.3034 48.8241
66.3034 48.8241
64.1120 37.3121
64.1125 37.3121
64.1130 37.3121
67.3466 44.9649
63.3370 37.2059
63.3365 37.2059
63.3365 37.2059
60.5627 37.0617
62.5676 48.8241
62.5681 48.8241
60.6126 37.3121
60.6135 37.3121
39.4562 37.3121
60.6126 37.3121
60.6135 37.3121
40.4181 37.2059
40.4184 37.2059
59.9190 37.2059
59.9190 37.2059
59.9194 37.2059
59.9190 37.2059
59.9194 37.2059
59.9194 37.2059
59.9194 37.2059
40.4249 37.0617
57.4306 37.0617
59.2308 54.3051
57.4758 37.3121
57.4762 37.3121
57.4762 37.3121
60.0620 44.9654
36.9702 37.2059
56.8514 37.2059
56.8522 37.2059
56.8518 37.2059
36.9702 37.2059
69.4910 52.6423
66.7578 32.6481
66.7584 32.6481
65.7991 32.5552
64.8523 50.3784
41.8789 42.7211
64.8523 50.3784
41.8320 32.6481
41.8324 32.6481
62.4661 32.6481
62.4666 32.6481
62.4672 32.6481
62.4661 32.6481
65.9951 39.7386
65.9951 39.3443
62.4672 32.6481
43.0745 32.5552
43.0745 32.5552
61.6259 32.5552
61.6264 32.5552
IV
8.4544
9.2632
9.2632
9.2632
9.2632
8.8308
9.2632
10.3712
10.3712
11.9806
7.2008
8.1377
8.1377
7.7718
8.9465
8.9465
9.5470
8.9465
8.9465
10.2233
10.0545
10.0545
10.0544
7.8209
8.6298
8.6297
6.9862
6.9862
6.9862
7.9231
7.9231
7.9231
7.5572
8.7319
8.7319
8.7319
9.8399
6.6695
6.6695
7.6064
7.6063
8.3181
7.6064
7.6063
8.8108
8.8108
8.4152
8.4152
8.4152
8.4152
8.4152
8.4152
8.4152
9.4871
9.5231
6.3527
7.2896
7.2896
7.2896
6.9237
8.5678
8.0985
8.0984
8.0984
8.5678
6.4549
7.3918
7.3917
8.2006
6.1382
6.8785
6.1382
7.5819
7.5819
7.0750
7.0750
7.0750
7.0750
6.7092
6.7092
7.0750
8.0746
8.0746
7.8838
7.8838
332
Table of Criteria Values for Hybrid 416A (K = 4)
Dsgn
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
58.0730
69.2565
69.2569
47.1050
71.0483
71.0473
71.0483
74.3319
71.0478
71.0478
47.1050
50.0189
50.0189
75.4433
50.0192
83.3060
85.8252
55.2630
77.3029
77.3029
77.3029
56.9341
56.9346
79.6413
79.6406
61.0644
48.8804
71.7325
71.7330
71.7331
48.8803
50.3581
50.3585
50.3581
50.3585
77.9028
73.9023
50.3585
54.0113
31.4555
68.5764
90.1634
61.8180
82.4240
64.0681
53.3521
53.3521
75.3497
53.3520
55.2945
55.2940
36.2112
33.9913
36.9714
68.8814
47.7215
73.1369
60.8389
40.5593
42.4130
50.6089
31.8802
57.7881
39.4943
20.9596
46.8666
n0 = 1
A
G
36.3677 34.7968
67.2603 53.0588
67.2608 53.0589
40.3583 34.8611
62.1316 34.8611
62.1307 34.8611
62.1316 34.8611
66.3504 43.2871
62.1311 34.8611
62.1311 34.8611
40.3583 34.8611
38.8332 34.8393
38.8332 34.8393
56.7578 34.8393
38.8335 34.8393
81.7940 49.9882
73.2184 29.8809
50.4564 47.8565
75.1873 49.9685
75.1873 49.9685
75.1872 49.9688
47.9094 29.8809
47.9100 29.8809
67.8798 29.8809
67.8791 29.8809
45.4384 29.8623
44.3031 45.4790
69.5680 49.9685
69.5687 49.9695
69.5687 49.9685
44.3031 45.4790
42.3268 29.8809
42.3273 29.8809
42.3268 29.8809
42.3273 29.8809
68.4363 37.1745
63.2662 29.8809
42.3273 29.8809
40.3865 29.8623
22.6338 29.8623
59.2394 29.8809
90.0457 90.0455
57.3257 41.6568
80.6801 47.9390
53.4514 24.9008
48.1980 41.6404
48.1980 41.6404
73.0801 47.9390
48.1980 41.6415
45.4301 24.9008
45.4295 24.9008
27.9811 30.9190
25.2544 24.9008
25.8025 24.8852
66.7872 47.2373
39.5012 24.9008
72.0365 72.0364
55.5207 38.3512
31.5363 33.3254
31.5769 19.9206
45.1657 37.7899
22.8282 19.9206
54.0274 54.0273
28.8256 28.7634
9.8171
14.9405
36.0183 36.0182
IV
10.3003
5.4192
5.4191
7.0900
6.6918
6.6919
6.6918
6.2178
6.6918
6.6918
7.0900
8.0986
8.0986
8.1963
8.0986
5.5106
6.7832
5.6780
5.2272
5.2272
5.2272
6.6488
6.6487
6.4998
6.4998
7.6573
5.4606
4.9438
4.9438
4.9438
5.4606
6.4313
6.4313
6.4313
6.4313
5.7424
6.2164
6.4313
7.4399
7.7142
5.9331
5.0352
5.0193
4.7518
5.9900
4.8019
4.8019
4.4684
4.8019
5.7725
5.7726
5.7990
6.2567
6.7057
4.1850
5.5551
4.3606
4.1431
4.5689
5.2482
3.9257
5.1836
3.5604
3.4958
2.4731
1.2588
D
56.5004
62.9588
62.9592
43.5269
65.6515
65.6506
65.6515
68.3944
65.6511
65.6511
43.5269
47.1321
47.1321
71.0892
47.1324
75.9316
79.7331
50.3710
70.4599
70.4599
70.4599
52.8928
52.8932
73.9881
73.9875
58.0389
44.5534
65.3826
65.3831
65.3831
44.5533
46.7836
46.7839
46.7836
46.7839
72.0152
68.6565
46.7839
51.3352
29.8970
63.7086
82.4872
56.5551
75.4067
59.9696
48.8099
48.8099
68.9347
48.8098
51.7573
51.7568
33.6937
31.8169
35.5668
63.0170
44.6687
67.2834
55.9697
37.3132
40.1501
46.5584
30.1793
53.6581
36.6717
20.2178
44.3314
n0 = 3
A
G
43.0834 32.4289
60.7952 47.8555
60.7956 47.8556
37.6759 32.6481
58.6938 32.6481
58.6929 32.6481
58.6938 32.6481
61.7983 39.3447
58.6933 32.6481
58.6933 32.6481
37.6759 32.6481
38.6804 32.5552
38.6804 32.5552
57.9510 32.5552
38.6808 32.5552
74.2496 45.1220
70.6450 27.9841
45.5484 43.1815
68.1721 45.1040
68.1721 45.1040
68.1720 45.1043
45.4842 27.9841
45.4848 27.9841
65.1219 27.9841
65.1213 27.9841
47.2118 27.9045
39.9501 41.0190
63.0141 45.1040
63.0148 45.1040
63.0148 45.1040
39.9501 41.0190
39.9003 27.9841
39.9007 27.9841
39.9003 27.9841
39.9007 27.9841
64.2765 34.0617
60.3993 27.9841
39.9007 27.9841
41.2236 27.9045
22.5618 27.9045
56.3148 27.9841
82.1241 82.1239
51.9180 37.6016
73.4348 43.3301
51.8180 23.3201
43.5666 37.5867
43.5666 37.5867
66.4092 43.3301
43.5666 37.5875
43.4968 23.3201
43.4962 23.3201
26.0257 28.1030
23.8983 23.3201
26.8476 23.2537
60.6091 42.6895
37.4774 23.3201
65.6992 65.6991
50.4130 34.6641
28.4529 30.0813
30.8832 18.6561
40.8975 34.1516
21.7654 18.6561
49.2745 49.2743
26.0548 25.9981
9.7443
13.9921
32.8496 32.8495
IV
8.7508
5.8214
5.8214
7.3388
6.7583
6.7583
6.7583
6.3924
6.7583
6.7583
7.3388
7.8316
7.8316
7.5671
7.8316
5.9236
6.8604
6.1423
5.6068
5.6068
5.6068
6.8457
6.8456
6.5437
6.5437
7.3384
5.8992
5.2901
5.2901
5.2901
5.8992
6.6026
6.6026
6.6026
6.6026
5.8611
6.2270
6.6026
7.0954
7.8210
5.9103
5.3922
5.4060
5.0755
6.1094
5.1630
5.1630
4.7588
5.1630
5.8664
5.8664
6.1549
6.5581
6.6939
4.4421
5.6234
4.6698
4.4268
4.9400
5.4310
4.1838
5.3588
3.8129
3.7407
2.6274
1.3481
Table of Criteria Values for Hybrid 416B (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
p
15
14
14
13
13
13
13
13
13
12
12
12
dv
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
q
4
3
4
2
3
3
4
4
3
1
2
2
D
73.5228
73.9403
74.1787
74.1640
74.6834
74.6834
74.9427
74.9430
73.0142
74.2344
74.9907
74.9907
n0 = 1
A
G
52.3632 70.0683
58.6489 66.4910
51.6493 65.3971
63.8618 62.4215
58.2159 61.7416
58.2159 61.7416
50.8495 60.7258
50.8496 60.7258
57.2112 61.7416
68.4109 58.0693
63.7747 57.6198
63.7747 57.6198
IV
17.0991
13.6652
16.6884
11.2132
13.2544
13.2544
16.2777
16.2777
12.8998
9.3055
10.8024
10.8024
D
68.9424
68.5893
69.7909
68.1762
69.4713
69.4713
70.7829
70.7831
67.9186
67.6897
69.0917
69.0917
n0 = 3
A
G
58.0290 62.8626
59.6687 60.3379
58.0246 58.6717
61.4443 57.5410
59.7935 56.0281
59.7935 56.0281
58.0194 54.4809
58.0195 54.4809
58.6120 56.0281
63.4118 54.4806
61.7412 53.1147
61.7412 53.1147
IV
13.9233
12.4500
13.4643
11.0345
11.9909
11.9909
13.0052
13.0052
11.5945
9.6708
10.5755
10.5755
333
Table of Criteria Values for Hybrid 416B (K = 4)
Dsgn
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
p
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
l
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
c
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
q
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
D
74.9909
75.5599
75.5601
75.5601
75.8441
75.8444
75.8444
73.1766
73.7320
75.1405
75.1440
75.1443
75.9796
75.9801
75.9798
75.9798
76.6089
76.6094
76.6092
76.6092
76.6092
76.6092
76.9236
76.9239
73.1632
73.9768
73.9768
74.5895
74.5897
76.2464
76.2504
76.2509
76.2506
77.1835
77.1841
78.3453
77.1838
78.3453
77.1838
77.1838
77.8873
77.8873
77.8876
77.8876
78.2394
74.0384
74.0423
74.0426
74.9485
74.9487
74.9487
75.6316
49.1243
75.6316
72.7781
77.6201
77.6205
77.6247
77.6253
77.6250
77.6256
77.6250
78.6812
78.6815
78.6812
78.6815
79.9982
78.6815
79.4788
79.4788
79.8781
75.1268
75.1311
75.1315
76.3598
76.1534
76.1541
50.2602
76.1534
76.1534
50.7695
76.9254
76.9254
76.9257
72.7177
n0 = 1
A
G
63.7748 57.6198
57.7188 56.9923
57.7190 56.9923
57.7190 56.9923
49.9470 56.0546
49.9471 56.0546
49.9471 56.0546
62.4726 57.6198
56.6503 56.9923
74.1556 75.4719
68.7463 53.2302
68.7465 53.2302
63.6720 52.8182
63.6724 52.8182
63.6722 52.8182
63.6722 52.8182
57.1421 52.2429
57.1425 52.2429
57.1423 52.2429
57.1423 52.2429
57.1423 52.2429
57.1423 52.2429
48.9211 51.3834
48.9212 51.3834
67.1017 53.2302
62.2587 52.8182
62.2587 52.8182
56.0013 52.2429
56.0015 52.2429
75.2249 73.7877
69.1531 48.3911
69.1537 48.3911
69.1534 48.3911
63.5492 48.0165
63.5497 48.0165
65.3846 53.0468
63.5494 48.0165
65.3846 53.0468
63.5494 48.0165
63.5494 48.0165
56.4653 47.4935
56.4653 47.4935
56.4655 47.4935
56.4655 47.4935
47.7443 46.7122
73.0694 72.4913
67.3273 48.3911
67.3275 48.3911
62.0040 48.0165
62.0042 48.0165
62.0042 48.0165
55.2419 47.4935
38.0193 47.4935
55.2419 47.4935
60.5321 48.0165
76.5745 66.4090
76.5749 66.4090
69.6570 43.5520
69.6576 43.5520
69.6573 43.5520
69.6579 43.5520
69.6573 43.5520
63.4000 43.2149
63.4003 43.2149
63.4000 43.2149
63.4003 43.2149
65.4362 47.7421
63.4003 43.2149
55.6596 42.7442
55.6596 42.7442
46.3806 42.0410
74.1021 66.4090
67.6051 43.5520
67.6054 43.5520
69.5257 48.3436
61.6954 43.2149
61.6959 43.2149
41.6322 43.2149
61.6954 43.2149
61.6954 43.2149
38.1934 42.7442
54.3415 42.7442
54.3415 42.7442
54.3417 42.7442
65.6706 43.5520
IV
10.8024
12.8437
12.8437
12.8437
15.8670
15.8670
15.8670
10.4478
12.4891
7.5259
8.8948
8.8948
10.3917
10.3917
10.3917
10.3917
12.4330
12.4330
12.4330
12.4330
12.4330
12.4330
15.4562
15.4562
8.5402
10.0371
10.0371
12.0784
12.0783
7.1152
8.4841
8.4841
8.4841
9.9810
9.9810
9.6827
9.9810
9.6827
9.9810
9.9810
12.0223
12.0223
12.0223
12.0223
15.0455
6.7605
8.1295
8.1294
9.6263
9.6263
9.6263
11.6676
11.4496
11.6676
9.2717
6.7044
6.7044
8.0734
8.0733
8.0734
8.0733
8.0734
9.5702
9.5702
9.5702
9.5702
9.2720
9.5702
11.6115
11.6115
14.6348
6.3498
7.7187
7.7187
7.4770
9.2156
9.2156
9.4302
9.2156
9.2156
10.8805
11.2569
11.2569
11.2569
7.3641
D
69.0919
70.5147
70.5149
70.5149
71.9580
71.9582
71.9582
67.4203
68.8089
67.9142
68.6372
68.6374
70.1895
70.1900
70.1897
70.1897
71.7680
71.7684
71.7682
71.7682
71.7682
71.7682
73.3722
73.3725
66.8279
68.3393
68.3393
69.8761
69.8764
68.9834
69.7917
69.7922
69.7919
71.5299
71.5304
72.4745
71.5302
72.4745
71.5302
71.5302
73.3016
73.3016
73.3019
73.3019
75.1061
66.9859
67.7707
67.7709
69.4586
69.4588
69.4588
71.1787
46.2321
71.1787
67.4472
70.3132
70.3135
71.2292
71.2297
71.2295
71.2300
71.2295
73.2033
73.2036
73.2033
73.2036
74.2782
73.2036
75.2207
75.2207
77.2809
68.0546
68.9411
68.9414
69.9519
70.8515
70.8521
46.7610
70.8515
70.8515
48.0495
72.8041
72.8041
72.8044
66.7265
n0 = 3
A
G
61.7414 53.1147
59.9398 51.7182
59.9400 51.7182
59.9400 51.7182
58.0133 50.2900
58.0135 50.2900
58.0135 50.2900
60.3796 53.1147
58.6557 51.7182
66.8240 68.0188
63.9441 49.9406
63.9443 49.9406
62.0958 48.6885
62.0963 48.6885
62.0960 48.6885
62.0960 48.6885
60.1137 47.4084
60.1141 47.4084
60.1139 47.4084
60.1139 47.4084
60.1139 47.4084
60.1139 47.4084
58.0064 46.0992
58.0066 46.0992
62.3554 49.9406
60.5965 48.6885
60.5965 48.6885
58.7075 47.4084
58.7077 47.4084
67.8437 66.5375
64.5948 45.4005
64.5954 45.4005
64.5951 45.4005
62.5268 44.2623
62.5273 44.2623
63.7699 48.3832
62.5270 44.2623
63.7699 48.3832
62.5270 44.2623
62.5270 44.2623
60.3238 43.0985
60.3238 43.0985
60.3241 43.0985
60.3241 43.0985
57.9980 41.9084
65.8846 65.3594
62.8164 45.4005
62.8167 45.4005
60.8589 44.2623
60.8592 44.2623
60.8592 44.2623
58.7698 43.0985
39.0515 43.0985
58.7698 43.0985
59.2777 44.2623
69.1332 59.8837
69.1335 59.8837
65.4084 40.8605
65.4090 40.8605
65.4087 40.8605
65.4093 40.8605
65.4087 40.8605
63.0619 39.8360
63.0622 39.8360
63.0619 39.8360
63.0622 39.8360
64.4703 43.5449
63.0622 39.8360
60.5827 38.7887
60.5827 38.7887
57.9879 37.7175
66.8815 59.8837
63.3892 40.8605
63.3895 40.8605
64.7687 44.8348
61.1827 39.8360
61.1832 39.8360
40.4493 39.8360
61.1827 39.8360
61.1827 39.8360
39.9167 38.7887
58.8463 38.7887
58.8463 38.7887
58.8465 38.7887
61.4910 40.8605
IV
10.5755
11.5319
11.5319
11.5319
12.5462
12.5462
12.5462
10.1791
11.1355
8.1760
9.2118
9.2118
10.1165
10.1164
10.1164
10.1164
11.0728
11.0728
11.0728
11.0728
11.0728
11.0728
12.0871
12.0871
8.8154
9.7201
9.7201
10.6765
10.6764
7.7169
8.7528
8.7527
8.7527
9.6574
9.6574
9.4672
9.6574
9.4672
9.6574
9.6574
10.6138
10.6138
10.6138
10.6138
11.6281
7.3206
8.3564
8.3564
9.2610
9.2610
9.2610
10.2174
10.7419
10.2174
8.8647
7.2579
7.2579
8.2937
8.2937
8.2937
8.2937
8.2937
9.1984
9.1983
9.1984
9.1983
9.0081
9.1983
10.1547
10.1547
11.1690
6.8615
7.8974
7.8973
7.7138
8.8020
8.8020
9.3948
8.8020
8.8020
10.1059
9.7584
9.7584
9.7583
7.5010
334
Table of Criteria Values for Hybrid 416B (K = 4)
Dsgn
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
p
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
l
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
c
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
q
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
D
73.7072
73.7075
79.3722
79.3726
79.3726
79.3778
79.3782
79.3785
80.8397
80.5947
80.5943
80.5943
81.5140
76.5097
76.5100
76.5147
76.5154
51.5177
76.5147
76.5154
52.3069
52.3072
77.6873
77.6873
77.6877
77.6873
77.6877
77.6877
77.6877
52.9038
78.5738
73.7504
73.7552
73.7555
73.7555
75.1135
48.6396
74.8852
74.8859
74.8855
48.6396
81.6840
81.6902
81.6906
83.1225
78.3252
54.1088
78.3252
54.1129
54.1132
78.3315
78.3319
78.3323
78.3315
79.9819
79.9823
78.3323
55.0617
55.0617
79.7049
79.7053
55.7804
75.1052
75.1056
49.7991
75.1117
75.1109
75.1117
76.6942
75.1113
75.1113
49.7991
50.6720
50.6720
76.4283
50.6722
84.8704
84.8784
57.7726
80.8135
80.8135
80.8140
57.7781
57.7784
80.8216
n0 = 1
A
G
60.0803 43.2149
60.0806 43.2149
78.3312 68.2446
78.3316 59.0302
78.3316 59.0302
70.2976 38.7129
70.2979 38.7129
70.2983 38.7129
72.6451 42.9721
63.2148 38.4132
63.2146 38.4132
63.2146 38.4132
54.6842 37.9948
75.4346 59.0302
75.4350 71.6509
67.9555 38.7129
67.9562 38.7129
45.1681 38.7129
67.9555 38.7129
67.9562 38.7129
42.3735 38.4132
42.3737 38.4132
61.3143 38.4132
61.3143 38.4132
61.3146 38.4132
61.3143 38.4132
61.3146 38.4132
61.3146 38.4132
61.3146 38.4132
38.4134 37.9948
53.2564 37.9948
72.7447 65.8211
65.7648 38.7129
65.7651 38.7129
65.7651 38.7129
67.8150 42.9721
39.8265 38.4132
59.5250 38.4132
59.5255 38.4132
59.5252 38.4132
39.8265 38.4132
80.7128 62.6945
71.1387 33.8738
71.1391 33.8738
62.9777 33.6116
77.2201 60.2040
50.6992 51.6514
77.2201 60.2040
46.7493 33.8738
46.7496 33.8738
68.4119 33.8738
68.4123 33.8738
68.4126 33.8738
68.4119 33.8738
70.9619 37.6006
70.9623 37.6006
68.4126 33.8738
43.3665 33.6116
43.3665 33.6116
60.8311 33.6116
60.8314 33.6116
38.7001 33.2455
74.0180 58.5003
74.0185 58.5009
43.2609 33.8738
65.8871 33.8738
65.8864 33.8738
65.8871 33.8738
68.2491 37.6006
65.8867 33.8738
65.8867 33.8738
43.2609 33.8738
40.3482 33.6116
40.3482 33.6116
58.8264 33.6116
40.3484 33.6116
84.1217 53.7382
72.2921 29.0346
54.2066 51.6035
79.7370 53.7382
79.7370 53.7382
79.7376 53.7382
49.0387 29.0346
49.0391 29.0346
69.0304 29.0346
IV
8.8610
8.8610
6.2937
6.2937
6.2937
7.6626
7.6626
7.6626
7.4209
9.1595
9.1595
9.1595
11.2008
5.9391
5.9391
7.3080
7.3080
7.7558
7.3080
7.3080
8.8611
8.8611
8.8049
8.8049
8.8049
8.8049
8.8049
8.8049
8.8049
10.3114
10.8462
5.5844
6.9534
6.9533
6.9533
6.7116
8.5465
8.4503
8.4502
8.4502
8.5465
5.8830
7.2519
7.2519
8.7488
5.5284
6.0908
5.5284
7.1867
7.1867
6.8973
6.8973
6.8972
6.8973
6.6556
6.6555
6.8972
8.2920
8.2920
8.3941
8.3941
9.7422
5.1737
5.1737
6.8720
6.5426
6.5426
6.5426
6.3009
6.5426
6.5426
6.8720
7.9774
7.9774
8.0395
7.9773
5.4722
6.8412
5.5217
5.1176
5.1176
5.1176
6.6176
6.6175
6.4865
D
68.5756
68.5759
72.0115
72.0119
72.0119
73.0680
73.0684
73.0687
74.2745
75.3506
75.3502
75.3502
77.6903
69.4144
69.4148
70.4325
70.4332
47.4225
70.4325
70.4332
48.9035
48.9037
72.6324
72.6324
72.6327
72.6324
72.6327
72.6327
72.6327
50.4222
74.8881
66.9110
67.8924
67.8927
67.8927
69.0133
45.4747
70.0126
70.0132
70.0129
45.4747
74.2563
75.5023
75.5027
78.2033
71.2029
49.1886
71.2029
50.0139
50.0142
72.3980
72.3984
72.3988
72.3980
73.7653
73.7657
72.3988
51.8032
51.8032
74.9880
74.9884
53.6458
68.2757
68.2761
46.0269
69.4221
69.4214
69.4221
70.7332
69.4217
69.4217
46.0269
47.6732
47.6732
71.9052
47.6735
77.3575
78.8745
52.6585
73.6597
73.6597
73.6602
53.6911
53.6914
75.1047
n0 = 3
A
G
59.4125 39.8360
59.4127 39.8360
70.8156 61.6143
70.8160 53.2300
70.8160 53.2300
66.4549 36.3204
66.4553 36.3204
66.4556 36.3204
68.1673 39.8532
63.7443 35.4098
63.7439 35.4098
63.7439 35.4098
60.9094 34.4788
68.1708 53.2300
68.1712 64.7189
64.1201 36.3204
64.1207 36.3204
42.1762 36.3204
64.1201 36.3204
64.1207 36.3204
41.6888 35.4098
41.6890 35.4098
61.5926 35.4098
61.5926 35.4098
61.5929 35.4098
61.5926 35.4098
61.5929 35.4098
61.5929 35.4098
61.5929 35.4098
41.0539 34.4788
58.9422 34.4788
65.7164 59.4072
61.9440 36.3204
61.9443 36.3204
61.9443 36.3204
63.4292 39.8532
38.9498 35.4098
59.5818 35.4098
59.5823 35.4098
59.5821 35.4098
38.9498 35.4098
73.1040 56.6290
67.8507 31.7804
67.8511 31.7804
64.6428 30.9836
69.9034 54.3589
45.7110 46.5762
69.9034 54.3589
44.0037 31.7804
44.0040 31.7804
65.0853 31.7804
65.0856 31.7804
65.0860 31.7804
65.0853 31.7804
66.9678 34.8715
66.9682 34.8715
65.0860 31.7804
43.3990 30.9836
43.3990 30.9836
62.1278 30.9836
62.1282 30.9836
42.6149 30.1690
66.9721 52.8069
66.9725 52.8074
40.5627 31.7804
62.5371 31.7804
62.5364 31.7804
62.5371 31.7804
64.2732 34.8715
62.5368 31.7804
62.5368 31.7804
40.5627 31.7804
40.0481 30.9836
40.0481 30.9836
59.8015 30.9836
40.0483 30.9836
76.3942 48.5392
69.8056 27.2403
48.9663 46.5934
72.3558 48.5392
72.3558 48.5392
72.3564 48.5392
46.7022 27.2403
46.7026 27.2403
66.4188 27.2403
IV
8.4056
8.4056
6.7989
6.7988
6.7988
7.8347
7.8346
7.8346
7.6511
8.7393
8.7393
8.7393
9.6957
6.4025
6.4025
7.4383
7.4383
8.0754
7.4383
7.4383
8.7588
8.7587
8.3429
8.3429
8.3429
8.3429
8.3429
8.3429
8.3429
9.4698
9.2993
6.0061
7.0419
7.0419
7.0419
6.8584
8.4071
7.9466
7.9466
7.9466
8.4071
6.3398
7.3756
7.3756
8.2802
5.9435
6.6036
5.9435
7.4394
7.4394
6.9792
6.9792
6.9792
6.9792
6.7957
6.7957
6.9792
8.1227
8.1227
7.8839
7.8839
8.8337
5.5471
5.5471
7.0877
6.5829
6.5829
6.5829
6.3993
6.5829
6.5829
7.0877
7.7710
7.7710
7.4875
7.7710
5.8807
6.9165
5.9675
5.4844
5.4844
5.4844
6.8033
6.8033
6.5202
335
Table of Criteria Values for Hybrid 416B (K = 4)
Dsgn
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
80.8211
58.9617
52.4360
76.9506
76.9515
76.9510
52.4360
52.4406
52.4409
52.4406
52.4409
78.8533
76.9582
52.4409
53.5149
31.1664
73.2791
89.5420
63.3230
84.4306
63.3301
56.3698
56.3698
79.6116
56.3702
56.3766
56.3762
35.6832
34.6566
35.5103
75.0665
50.1858
72.6640
62.8314
41.8876
41.8935
54.3292
32.7326
57.4555
41.3470
20.6774
46.6636
n0 = 1
A
G
69.0300 29.0346
44.7653 28.8099
48.8751 50.1431
75.7868 53.7382
75.7878 53.7407
75.7873 53.7382
48.8751 50.1431
44.6337 29.0346
44.6341 29.0346
44.6337 29.0346
44.6341 29.0346
68.8363 32.2291
66.0499 29.0346
44.6341 29.0346
41.0657 28.8099
22.9868 28.8099
63.3158 29.0346
89.4093 89.4047
60.0202 44.7818
83.5495 52.1997
52.6482 24.1955
52.4217 44.7818
52.4217 44.7818
78.4113 52.1997
52.4221 44.7839
46.7097 24.1955
46.7093 24.1955
27.6486 26.8576
26.2069 24.1955
25.0741 24.0083
73.8673 51.6995
41.9744 24.1955
71.5272 71.5237
58.8253 41.7597
34.0328 35.8254
30.9647 19.3564
49.9544 41.3596
23.8338 19.3564
53.6450 53.6428
31.7216 31.3198
9.5569
14.5173
35.7626 35.7619
IV
6.4865
7.7229
5.2070
4.7630
4.7630
4.7630
5.2070
6.3029
6.3029
6.3029
6.3029
5.8902
6.1319
6.3029
7.4082
7.7659
5.7773
5.0615
4.9525
4.7069
6.0484
4.6379
4.6379
4.3522
4.6379
5.7338
5.7338
5.9504
6.1862
6.8946
3.9976
5.4192
4.3834
4.0688
4.4504
5.3149
3.7541
5.1038
3.5791
3.3680
2.5247
1.2654
D
75.1042
55.9386
47.7942
70.1387
70.1396
70.1392
47.7942
48.7312
48.7315
48.7312
48.7315
73.0928
71.5146
48.7315
50.7711
29.5684
68.0957
81.9187
57.9319
77.2425
59.2978
51.5707
51.5707
72.8337
51.5711
52.7870
52.7866
33.3112
32.4500
34.0867
68.6756
46.9904
66.8484
57.8027
38.5351
39.6741
49.9810
30.9986
53.3493
38.3920
19.9562
44.1393
n0 = 3
A
G
66.4183 27.2403
45.9101 26.5574
44.1086 45.2631
68.7229 48.5392
68.7239 48.5415
68.7234 48.5392
44.1086 45.2631
42.2626 27.2403
42.2630 27.2403
42.2626 27.2403
42.2630 27.2403
65.4336 29.8899
63.3450 27.2403
42.2630 27.2403
41.6129 26.5574
22.7655 26.5574
60.5427 27.2403
81.5325 81.5284
54.3896 40.4493
76.0933 47.2240
51.0883 22.7003
47.4270 40.4493
47.4270 40.4493
71.3351 47.2240
47.4274 40.4512
44.8977 22.7003
44.8972 22.7003
26.0267 24.9082
24.9033 22.7003
25.6808 22.1311
67.1358 46.7665
40.0445 22.7003
65.2257 65.2227
53.4608 37.7792
30.7256 32.3594
30.3220 18.1602
45.2914 37.4132
22.8423 18.1602
48.9189 48.9170
28.7020 28.3344
9.5005
13.6202
32.6119 32.6113
IV
6.5202
7.4866
5.6158
5.0880
5.0880
5.0880
5.6158
6.4516
6.4516
6.4516
6.4516
5.9403
6.1238
6.4516
7.1349
7.9321
5.7275
5.4217
5.3314
5.0253
6.1672
4.9798
4.9798
4.6290
4.9797
5.8155
5.8155
6.2678
6.4722
6.9583
4.2326
5.4639
4.6953
4.3437
4.8076
5.4984
3.9920
5.2624
3.8337
3.5978
2.6802
1.3555
Table of Criteria Values for Hybrid 416C (K = 4)
Dsgn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
p
15
14
14
13
13
13
13
13
13
12
12
12
12
12
12
12
12
12
12
12
12
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
l
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c
6
6
5
6
5
5
4
4
6
6
5
5
5
4
4
4
3
3
3
6
5
6
5
5
4
4
4
4
3
3
3
3
3
3
2
2
q
4
3
4
2
3
3
4
4
3
1
2
2
2
3
3
3
4
4
4
2
3
0
1
1
2
2
2
2
3
3
3
3
3
3
4
4
D
74.9411
76.9314
75.1191
78.1361
77.2838
77.2838
75.3251
75.8200
75.8350
78.9947
78.8219
78.6258
78.6258
77.6971
77.8929
78.2504
75.5660
76.1041
76.1041
77.0302
76.1204
79.7561
79.6140
79.6140
79.4240
79.2084
79.8239
79.8239
78.1884
78.4034
78.7959
78.7959
78.7959
78.7959
76.4412
77.0351
n0 = 1
A
G
40.8478 77.4937
54.6071 73.4870
39.6126 72.3275
64.7804 69.2542
53.5900 68.2379
53.5900 68.2379
38.2772 67.1612
38.4158 67.1612
52.9335 68.2379
72.3986 64.5238
64.2111 63.9269
64.2111 63.9269
64.6351 63.9269
52.4503 62.9888
52.7328 62.9888
52.7328 62.9888
36.8286 61.9950
36.9677 61.9950
36.9677 61.9950
63.1938 63.9269
51.7695 62.9888
78.8031 79.6941
72.3900 59.1468
72.3900 59.1468
63.5511 58.5997
64.0044 58.5997
64.0044 58.5997
64.0044 58.5997
51.1643 57.7398
51.7545 57.7398
51.4577 57.7398
51.4577 57.7398
51.4577 57.7398
51.4577 57.7398
35.3910 56.8287
35.5312 56.8287
IV
24.4033
15.7172
24.0355
11.5171
15.3493
15.3493
23.6676
23.6349
14.9981
8.9864
11.1492
11.1492
11.1166
14.9815
14.9488
14.9488
23.2998
23.2671
23.2671
10.7980
14.6303
7.1814
8.6185
8.6185
10.7814
10.7487
10.7487
10.7487
14.6136
14.5482
14.5809
14.5809
14.5809
14.5809
22.8992
22.8665
D
73.8686
74.4311
74.2889
74.8084
74.9962
74.9309
74.7768
75.2681
73.5263
75.1537
75.6344
75.3846
75.3846
75.5896
75.5183
76.0561
75.3500
75.8866
75.8866
73.8548
74.0557
75.4794
75.8171
75.8171
76.3463
76.0712
76.6623
76.6623
76.2970
76.2185
76.8898
76.8107
76.8898
76.8898
76.6240
77.2194
n0 = 2
A
G
52.9251 72.9368
59.0324 69.5527
52.0905 68.0743
64.7032 66.2428
58.4266 64.5847
58.4266 64.5847
51.1597 63.2119
51.4233 63.2119
57.5991 64.5847
69.6590 62.3952
64.4257 61.1472
64.4257 61.1472
64.8035 61.1472
57.7354 59.6166
58.0995 59.6166
58.0995 59.6166
50.1149 58.3494
50.3890 58.3494
50.3890 58.3494
63.3387 61.1472
56.8609 59.6166
74.4814 75.3273
69.7920 57.1956
69.7920 57.1956
64.1010 56.0516
64.5091 56.0516
64.5912 56.0516
64.5912 56.0516
56.9393 54.6486
57.7176 54.6486
57.3258 54.6486
57.3258 54.6486
57.3258 54.6486
57.3258 54.6486
49.2191 53.4870
49.5077 53.4870
IV
16.9110
13.5990
16.5201
11.0727
13.2082
13.2082
16.1293
16.0945
12.8350
9.1232
10.6819
10.6819
10.6471
12.8173
12.7826
12.7826
15.7384
15.7037
15.7037
10.3087
12.4442
7.5053
8.7323
8.7323
10.2910
10.2563
10.2563
10.2563
12.4265
12.3570
12.3917
12.3917
12.3917
12.3917
15.3129
15.2781
336
Table of Criteria Values for Hybrid 416C (K = 4)
Dsgn
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
p
11
11
11
11
11
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
dv
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
3
3
4
4
l
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
c
6
5
5
4
4
5
4
4
4
3
3
3
3
3
3
3
2
2
2
2
1
6
5
5
4
4
4
3
3
3
5
4
4
3
3
3
3
3
2
2
2
2
2
2
1
1
0
5
4
4
4
3
3
3
3
3
2
2
2
2
5
4
4
3
3
3
2
2
2
2
1
1
1
0
4
4
3
3
3
3
3
2
2
2
2
q
1
2
2
3
3
0
1
1
1
2
2
2
2
2
2
2
3
3
3
3
4
0
1
1
2
2
2
3
3
3
2
0
0
1
1
1
1
1
2
2
2
2
2
2
3
3
4
0
1
1
1
2
2
2
2
2
3
3
3
3
1
2
2
0
0
0
1
1
1
1
2
2
2
3
0
0
1
1
1
1
1
2
2
2
2
D
77.8531
77.6673
77.4565
76.4590
77.0531
80.5213
80.8915
80.3635
81.0506
80.1526
79.9132
80.5966
80.5966
80.5966
80.5966
80.5966
79.4556
79.4556
79.6960
80.1350
77.5047
78.5645
78.4104
78.4104
78.2047
77.9712
78.6379
76.8674
50.3537
76.8674
76.3041
81.4665
82.2409
81.8828
81.2891
82.0618
82.0618
82.0618
81.8225
81.5511
81.8225
81.5511
82.3263
82.3263
81.0324
81.0324
78.8248
79.2697
79.6748
79.0971
79.8489
78.8665
78.6049
52.3710
78.8665
78.8665
51.5479
78.1049
78.1049
78.3675
76.9642
76.7398
77.2123
82.6637
83.5482
83.5482
84.0286
83.3435
83.3435
84.2353
83.6458
83.6458
83.9590
83.0474
80.1602
81.0179
80.6212
79.9638
54.4158
80.6212
79.9638
54.2374
54.0350
80.5544
80.5544
n0 = 1
A
G
70.9846 59.1468
62.4654 58.5997
62.4654 58.5997
50.4582 57.7398
50.7436 57.7398
79.4950 77.3774
73.0277 53.7698
72.3797 53.7698
73.0277 53.7698
62.7768 53.2725
63.6021 53.2725
63.2637 53.2725
63.2637 53.2725
63.2637 53.2725
63.2637 53.2725
63.2637 53.2725
50.0067 52.4907
50.0067 52.4907
50.3152 52.4907
50.3152 52.4907
33.8074 51.6625
77.6382 76.2965
70.8372 53.7698
70.8372 53.7698
61.6131 53.2725
62.0821 53.2725
62.0821 53.2725
48.9697 52.4907
34.9978 52.4907
48.9697 52.4907
60.4918 53.2725
80.3575 69.6397
81.2468 80.3617
73.3457 48.3928
72.3672 48.3928
73.0877 48.3928
73.0877 48.3928
73.0877 48.3928
62.3812 47.9452
62.9159 47.9452
62.3812 47.9452
62.9159 47.9452
62.9159 47.9452
62.9159 47.9452
48.6611 47.2416
48.6611 47.2416
32.0544 46.4962
78.2554 69.6397
71.3446 48.3928
70.6579 48.3928
71.3446 48.3928
60.6025 47.9452
61.4580 47.9452
41.8726 47.9452
60.6025 47.9452
60.6025 47.9452
34.4552 47.2416
47.5719 47.2416
47.5719 47.2416
47.8822 47.2416
69.0275 48.3928
59.3992 47.9452
59.8837 47.9452
81.4622 70.8442
82.4920 61.9019
82.4920 71.4327
73.9923 43.0159
73.1627 43.0159
73.1627 43.0159
73.9923 43.0159
62.4865 42.6180
61.8938 42.6180
61.8938 42.6180
47.0775 41.9926
79.0407 61.9019
80.0099 74.4260
71.4792 43.0159
70.4350 43.0159
47.5693 43.0159
71.4792 43.0159
70.4350 43.0159
42.0137 42.6180
42.4921 42.6180
59.9304 42.6180
59.9304 42.6180
IV
8.2673
10.4302
10.4302
14.2624
14.2297
6.8136
8.2180
8.2507
8.2180
10.4135
10.3482
10.3809
10.3809
10.3809
10.3809
10.3809
14.2131
14.2131
14.1804
14.1804
22.4987
6.4624
7.8995
7.8995
10.0623
10.0296
10.0296
13.8946
12.8549
13.8946
9.7111
6.4457
6.4130
7.8174
7.8828
7.8501
7.8501
7.8501
10.0130
9.9803
10.0130
9.9803
9.9803
9.9803
13.8125
13.8125
22.0981
6.0945
7.4989
7.5316
7.4989
9.6945
9.6291
9.6012
9.6945
9.6945
12.3452
13.4940
13.4940
13.4613
7.1804
9.3433
9.3106
6.0779
6.0452
6.0452
7.4496
7.4823
7.4823
7.4496
9.5798
9.6125
9.6125
13.4120
5.7267
5.6940
7.0984
7.1638
7.4886
7.0984
7.1638
9.0915
9.0462
9.2939
9.2939
D
74.1402
74.6577
74.3887
74.6095
75.1119
76.2456
77.0918
76.6209
77.2760
77.2094
76.9035
77.5610
77.5610
77.5610
77.5610
77.5610
77.8142
77.8142
77.7262
78.3908
78.1812
74.3927
74.7588
74.7588
75.3330
75.0345
75.6761
75.2795
49.3136
75.2795
73.5023
77.1926
77.9264
78.1451
77.6149
78.3526
78.3526
78.3526
79.0216
78.6738
79.0216
78.6738
79.4216
79.4216
79.6095
79.7097
80.1275
75.1111
76.0378
75.5219
76.2398
76.1667
75.8315
50.5232
76.1667
76.1667
50.7065
76.8300
76.8300
76.7335
73.4854
74.1128
74.4880
78.3930
79.2317
79.2317
80.3324
79.7195
79.7195
80.5724
80.9441
80.9441
81.3468
82.0280
76.0188
76.8322
77.0748
76.4868
52.0496
77.0748
76.4868
52.5499
52.2897
78.0481
78.0481
n0 = 2
A
G
68.4046 57.1956
62.9287 56.0516
62.9287 56.0516
56.0125 54.6486
56.3865 54.6486
75.1704 73.1588
70.5955 51.9960
69.9523 51.9960
70.5955 51.9960
63.7155 50.9560
64.1593 50.9560
64.2487 50.9560
64.2487 50.9560
64.2487 50.9560
64.2487 50.9560
64.2487 50.9560
56.4242 49.6805
56.4242 49.6805
56.8420 49.6805
56.8420 49.6805
48.1855 48.6245
73.4065 72.1322
68.4223 51.9960
68.4223 51.9960
62.4437 50.9560
62.8699 50.9560
62.9558 50.9560
55.0273 49.6805
38.1524 49.6805
55.0273 49.6805
61.2217 50.9560
76.0299 65.8429
76.8758 76.0340
70.9575 46.7964
70.1492 46.7964
70.8687 46.7964
70.8687 46.7964
70.8687 46.7964
63.8350 45.8604
64.3304 45.8604
63.8350 45.8604
64.3304 45.8604
64.4303 45.8604
64.4303 45.8604
55.8071 44.7125
55.8071 44.7125
46.9797 43.7621
74.0307 65.8429
69.1287 46.7964
68.4439 46.7964
69.1287 46.7964
61.8609 45.8604
62.3260 45.8604
42.1236 45.8604
61.8609 45.8604
61.8609 45.8604
38.1275 44.7125
54.2924 44.7125
54.2924 44.7125
54.7224 44.7125
66.8195 46.7964
60.5310 45.8604
61.0660 45.8604
77.1323 67.0260
78.1133 58.5271
78.1133 67.5858
72.0486 41.5968
71.2132 41.5968
71.2132 41.5968
72.0486 41.5968
64.5455 40.7648
63.9850 40.7648
63.9850 40.7648
55.0545 39.7444
74.8262 58.5271
75.7490 70.4335
69.3383 41.5968
68.4709 41.5968
45.9575 41.5968
69.3383 41.5968
68.4709 41.5968
42.6437 40.7648
42.8487 40.7648
61.7625 40.7648
61.7625 40.7648
IV
8.3592
9.9179
9.9179
12.0533
12.0186
7.1144
8.3068
8.3415
8.3068
9.9002
9.8307
9.8654
9.8654
9.8654
9.8654
9.8654
12.0009
12.0009
11.9662
11.9662
14.8873
6.7413
7.9683
7.9683
9.5270
9.4923
9.4923
11.6625
11.4025
11.6625
9.1539
6.7236
6.6889
7.8812
7.9507
7.9159
7.9159
7.9159
9.4746
9.4399
9.4746
9.4399
9.4399
9.4399
11.5753
11.5753
14.4617
6.3504
7.5428
7.5775
7.5428
9.1362
9.0667
9.3119
9.1362
9.1362
10.8609
11.2369
11.2369
11.2022
7.2043
8.7630
8.7283
6.3328
6.2980
6.2980
7.4903
7.5251
7.5251
7.4903
9.0143
9.0490
9.0490
11.1497
5.9596
5.9249
7.1172
7.1867
7.6121
7.1172
7.1867
8.7704
8.7281
8.7106
8.7106
337
Table of Criteria Values for Hybrid 416C (K = 4)
Dsgn
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
p
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
dv
4
4
4
4
4
3
4
4
4
4
4
4
3
4
4
4
3
4
4
4
4
4
3
4
3
3
4
4
4
4
4
4
4
3
3
4
4
3
4
4
3
4
4
4
4
4
4
3
3
3
4
3
4
4
3
4
4
4
3
3
4
4
3
3
4
4
4
3
3
3
3
3
4
4
3
3
2
4
4
3
4
3
3
3
4
l
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
4
3
3
3
2
2
2
c
2
2
2
2
2
1
1
5
4
4
4
4
3
3
3
3
3
2
1
1
0
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
0
4
4
3
3
3
3
3
3
3
3
2
2
2
2
1
0
2
2
2
2
1
1
1
1
0
3
3
3
3
3
2
2
2
2
2
2
2
1
1
3
0
1
1
0
2
2
2
q
2
2
2
2
2
3
3
0
1
1
1
1
2
2
2
2
2
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
3
0
0
1
1
1
1
1
1
1
1
2
2
2
2
0
1
0
0
0
0
1
1
1
1
2
0
0
0
0
0
1
1
1
1
1
1
1
2
2
1
0
0
0
1
0
0
0
D
80.2539
80.5544
80.2539
80.2540
80.2540
53.6485
79.6798
77.7326
78.1796
77.5421
78.3718
78.3718
50.5478
77.2878
77.8235
78.1148
50.5478
86.3029
86.0613
86.0613
86.4182
81.3199
56.8648
81.3199
57.2387
56.7056
82.8562
82.0846
82.0846
82.8562
82.0846
82.0847
82.0847
57.1845
56.9407
82.4250
82.4249
56.4755
78.5114
79.4722
52.5429
78.2916
79.0276
79.2497
79.2498
79.2497
79.2497
52.5429
52.3460
52.3460
79.5784
52.1229
88.8479
89.8235
60.1056
84.0769
84.0769
84.0770
60.7655
60.7655
85.0000
85.0001
61.0595
54.9900
79.5620
80.6991
80.6991
54.9900
55.4120
54.8104
55.4120
54.8104
81.5851
81.5851
54.8105
55.0757
32.2356
76.9517
94.1268
66.0708
88.0944
66.9423
58.3790
58.3790
83.8646
n0 = 1
A
G
60.4859 42.6180
59.9304 42.6180
60.4859 42.6180
60.4859 42.6180
60.4859 42.6180
34.1143 41.9926
45.9330 41.9926
76.7591 68.7147
69.3467 43.0159
68.6175 43.0159
69.3467 43.0159
69.3467 43.0159
39.8890 42.6180
58.0877 42.6180
58.6095 42.6180
58.6095 42.6180
39.8890 42.6180
85.4089 65.1228
74.2116 37.6389
74.2116 37.6389
61.2781 37.2907
80.0740 62.5036
53.4052 54.1642
80.0740 62.5036
49.4795 37.6389
48.7288 37.6389
71.9177 37.6389
71.0232 37.6389
71.0232 37.6389
71.9177 37.6389
71.0232 37.6389
71.0232 37.6389
71.0232 37.6389
42.7582 37.2907
42.7582 37.2907
59.0878 37.2907
59.7057 37.2907
33.6857 36.7435
77.4099 61.2258
78.4738 74.7945
45.5153 37.6389
68.0974 37.6389
69.2146 37.6389
68.9194 37.6389
68.9195 37.6389
68.9194 37.6389
68.9194 37.6389
45.5153 37.6389
39.7653 37.2907
39.7653 37.2907
57.6244 37.2907
40.2556 37.2907
88.0227 55.8195
75.6578 32.2619
56.3117 53.5745
82.8744 55.8195
82.8744 55.8195
82.8745 64.8064
51.3044 32.2619
51.3044 32.2619
71.8228 32.2619
71.8229 32.2619
43.7930 31.9635
51.4174 52.4793
78.2950 55.8195
79.5679 64.1096
79.5679 55.8195
51.4174 52.4793
47.2101 32.2619
46.4142 32.2619
47.2101 32.2619
46.4142 32.2619
69.3261 32.2619
69.3261 32.2619
46.4142 32.2619
40.1795 31.9635
22.8867 31.9635
66.4093 32.2619
94.0843 94.0819
62.6199 46.5163
87.1409 54.0053
55.4040 26.8849
54.2484 46.5163
54.2484 46.5163
82.7995 54.0053
IV
9.2612
9.2939
9.2612
9.2612
9.2612
11.7901
13.0935
5.3755
6.7798
6.8125
6.7798
6.7798
8.7709
8.9754
8.9427
8.9427
8.7709
5.6447
7.0817
7.0817
9.2119
5.3588
5.8427
5.3588
6.9336
6.9789
6.7305
6.7632
6.7632
6.7305
6.7632
6.7632
6.7632
8.5365
8.5365
8.8934
8.8607
11.2351
5.0076
4.9749
6.6583
6.4447
6.3793
6.4120
6.4120
6.4120
6.4120
6.6583
8.2612
8.2612
8.5422
8.2159
5.2768
6.6812
5.3330
4.9583
4.9583
4.9583
6.4238
6.4238
6.3627
6.3627
7.9815
5.0124
4.6398
4.6071
4.6071
5.0124
6.1033
6.1486
6.1033
6.1486
6.0115
6.0115
6.1485
7.7062
7.7331
5.6602
4.8763
4.7780
4.5577
5.8688
4.5027
4.5027
4.2065
D
77.6617
78.0481
77.6617
77.6618
77.6618
53.0650
78.7018
73.7166
74.7406
74.1704
74.9640
74.9640
48.9152
74.8832
75.3098
75.6844
48.9152
81.9328
82.5095
82.5095
83.9596
77.2021
53.9854
77.2021
54.8433
54.3653
79.3887
78.6969
78.6969
79.3887
78.6969
78.6970
78.6970
55.6355
55.3208
80.0801
80.0800
56.1683
74.5358
75.4480
50.3744
75.0604
75.7203
75.9790
75.9791
75.9790
75.9790
50.3744
50.9280
50.9280
77.3144
50.6401
84.4708
86.3819
57.1444
79.9348
79.9348
79.9349
58.4373
58.4373
81.7433
81.7434
59.6373
52.2809
75.6423
76.7234
76.7234
52.2809
53.2514
52.7104
53.2514
52.7104
78.4593
78.4593
52.7105
53.7929
31.5362
73.9512
89.6706
62.9428
83.9238
64.6556
55.6152
55.6152
79.8943
n0 = 2
A
G
62.2845 40.7648
61.7625 40.7648
62.2845 40.7648
62.2846 40.7648
62.2846 40.7648
38.5211 39.7444
53.4011 39.7444
72.6539 65.0011
67.3787 41.5968
66.6475 41.5968
67.3787 41.5968
67.3787 41.5968
40.3273 40.7648
59.6892 40.7648
60.2750 40.7648
60.2750 40.7648
40.3273 40.7648
80.9659 61.6293
72.6295 36.3972
72.6295 36.3972
64.1789 35.6692
75.8743 59.1375
50.4903 51.2112
75.8743 59.1375
47.9744 36.3972
47.2916 36.3972
70.2980 36.3972
69.3902 36.3972
69.3902 36.3972
70.2980 36.3972
69.3902 36.3972
69.3903 36.3972
69.3903 36.3972
43.9619 35.6692
43.9619 35.6692
61.6364 35.6692
62.2313 35.6692
39.0393 34.7764
73.3334 57.9223
74.3480 70.8401
44.0823 36.3972
66.4276 36.3972
67.3619 36.3972
67.2590 36.3972
67.2591 36.3972
67.2590 36.3972
67.2590 36.3972
44.0823 36.3972
40.6219 35.6692
40.6219 35.6692
59.9491 35.6692
40.8346 35.6692
83.5660 52.8251
74.6080 31.1976
53.2935 50.6893
78.6383 52.8251
78.6383 52.8251
78.6384 61.3843
50.1480 31.1976
50.1480 31.1976
70.6551 31.1976
70.6552 31.1976
45.8516 30.5736
48.6380 49.6477
74.2594 52.8251
75.4762 60.7201
75.4762 52.8251
48.6380 49.6477
45.9309 31.1976
45.2020 31.1976
45.9309 31.1976
45.2020 31.1976
68.0920 31.1976
68.0920 31.1976
45.2020 31.1976
41.6815 30.5736
23.3515 30.5736
64.8955 31.1976
89.5411 89.5388
59.3738 44.0210
82.8645 51.1536
54.7802 25.9980
51.3853 44.0210
51.3853 44.0210
78.6955 51.1536
IV
8.6759
8.7106
8.6759
8.6759
8.6759
10.2712
10.8113
5.5864
6.7788
6.8135
6.7788
6.7788
8.4297
8.3722
8.3374
8.3374
8.4297
5.8724
7.0995
7.0995
8.6234
5.5688
6.0996
5.5688
7.0224
7.0706
6.7263
6.7611
6.7611
6.7263
6.7611
6.7611
6.7611
8.1807
8.1807
8.2850
8.2503
9.6815
5.1956
5.1609
6.7300
6.4227
6.3532
6.3879
6.3879
6.3879
6.3879
6.7300
7.8882
7.8882
7.9119
7.8459
5.4816
6.6739
5.5580
5.1432
5.1432
5.1432
6.4809
6.4809
6.3355
6.3355
7.5910
5.2174
4.8048
4.7700
4.7700
5.2174
6.1403
6.1884
6.1403
6.1884
5.9623
5.9623
6.1884
7.2985
7.6376
5.5892
5.0560
4.9683
4.7176
5.8912
4.6758
4.6758
4.3444
338
Table of Criteria Values for Hybrid 416C (K = 4)
Dsgn
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
p
5
5
5
5
5
5
5
5
4
4
4
4
4
4
3
3
3
2
dv
3
3
3
2
2
2
4
3
3
3
2
2
3
2
2
2
1
1
l
2
2
2
2
2
2
1
1
3
2
2
2
1
1
2
1
1
1
c
2
1
1
1
1
0
3
2
0
1
1
0
2
1
0
1
0
0
q
0
1
1
1
1
2
0
1
0
0
0
1
0
1
0
0
1
0
D
58.3791
59.1491
59.1492
36.3612
36.3612
36.5725
77.1646
52.9522
76.1468
65.2320
43.4880
44.2062
55.8818
34.2193
59.8973
42.5718
21.7560
48.1437
n0 = 1
A
G
54.2485 54.0053
48.7482 26.8849
48.7483 26.8849
27.5346 26.8849
27.5346 26.8849
24.7427 26.6362
75.9332 53.4247
44.3616 26.8849
75.2674 75.2655
61.1008 43.2043
35.3277 37.2130
32.9196 21.5079
51.4223 42.7397
24.9972 21.5079
56.4505 56.4491
32.7319 32.4032
10.3334 16.1309
37.6335 37.6328
IV
4.5027
5.5935
5.5935
5.9359
5.9359
6.8834
3.9207
5.2729
4.2230
3.9477
4.2977
5.0862
3.6724
4.9190
3.4480
3.2808
2.3355
1.2191
D
55.6153
57.1286
57.1287
35.1191
35.1191
35.9864
73.5115
51.1002
72.7621
62.3325
41.5550
42.9732
53.3978
33.2648
57.5247
40.8855
21.3783
46.7062
n0 = 2
A
G
51.3854 51.1536
47.9085 25.9980
47.9085 25.9980
26.8643 25.9980
26.8643 25.9980
25.8926 25.4780
72.1108 50.6001
43.3460 25.9980
71.6328 71.6311
58.0280 40.9229
33.4232 35.2168
32.7108 20.7984
48.7662 40.4801
24.5098 20.7984
53.7245 53.7233
31.0055 30.6921
10.3542 15.5988
35.8162 35.8155
IV
4.6758
5.5987
5.5987
6.0527
6.0527
6.7627
4.0407
5.2581
4.3786
4.0861
4.4779
5.1499
3.7937
4.9722
3.5752
3.3975
2.4044
1.2640
339
APPENDIX B
D, A, G, and IV Criteria Plots for Small Composite, Uniform Shell, and Hybrid
Designs for 3 Factors
340
Figure 69. The D, A, G, and IV -Criteria Plots for 3 Factor SCDs (Plotting Symbol
= Q-Path).
341
Figure 70. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 3 Factor SCDs.
342
Figure 71. The D, A, G, and IV -Criteria Plots for 3 Factor SCDs (Plotting Symbol
= C-Path).
343
Figure 72. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 3 Factor SCDs.
344
Figure 73. The D, A, G, and IV -Criteria Plots for 3 Factor UNFSDs (Plotting
Symbol = Q-Path).
345
Figure 74. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 3 Factor UNFSDs.
346
Figure 75. The D, A, G, and IV -Criteria Plots for 3 Factor UNFSDs (Plotting
Symbol = C-Path).
347
Figure 76. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 3 Factor UNFSDs.
348
Figure 77. The D, A, G, and IV -Criteria Plots for 3 Factor 310 Designs (Plotting
Symbol = Q-Path).
349
Figure 78. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 3 Factor 310 Designs.
350
Figure 79. The D, A, G, and IV -Criteria Plots for 3 Factor 310 Designs (Plotting
Symbol = C-Path).
351
Figure 80. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 3 Factor 310 Designs.
352
Figure 81. The D, A, G, and IV -Criteria Plots for 3 Factor 311A Designs (Plotting
Symbol = Q-Path).
353
Figure 82. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 3 Factor 311A Designs.
354
Figure 83. The D, A, G, and IV -Criteria Plots for 3 Factor 311A Designs (Plotting
Symbol = C-Path).
355
Figure 84. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 3 Factor 311A Designs.
356
Figure 85. The D, A, G, and IV -Criteria Plots for 3 Factor 311B Designs (Plotting
Symbol = Q-Path).
357
Figure 86. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 3 Factor 311B Designs.
358
Figure 87. The D, A, G, and IV -Criteria Plots for 3 Factor 311B Designs (Plotting
Symbol = C-Path).
359
Figure 88. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 3 Factor 311B Designs.
360
APPENDIX C
D, A, G, snd IV Criteria Plots for Small Composite, Plackett-Burman Composite,
Uniform Shell, and Hybrid Designs for 4 Factors
361
Figure 89. The D, A, G, and IV -Criteria Plots for 4 Factor SCDs for dv = 4 (Plotting
Symbol = Q-Path).
362
Figure 90. The D, A, G, and IV -Criteria Plots for 4 Factor SCDs for dv = 1, 2, and
3 (Plotting Symbol = Q-Path).
363
Figure 91. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 4 Factor SCDs.
364
Figure 92. The D, A, G, and IV -Criteria Plots for 4 Factor SCDs for dv = 4 (Plotting
Symbol = C-Path).
365
Figure 93. The D, A, G, and IV -Criteria Plots for 4 Factor SCDs for dv = 4 → 3
and 4 → 3 → 2 (Plotting Symbol = C-Path).
366
Figure 94. The D, A, G, and IV -Criteria Plots for 4 Factor SCDs for dv = 3, 4 → 3,
and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path).
367
Figure 95. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 4 Factor SCDs.
368
Figure 96. The D, A, G, and IV -Criteria Plots for 4 Factor PBCDs for dv = 4
(Plotting Symbol = Q-Path).
369
Figure 97. The D, A, G, and IV -Criteria Plots for 4 Factor PBCDs for dv = 1, 2,
and 3 (Plotting Symbol = Q-Path).
370
Figure 98. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 4 Factor PBCDs.
371
Figure 99. The D, A, G, and IV -Criteria Plots for 4 Factor PBCDs for dv = 4
(Plotting Symbol = C-Path).
372
Figure 100. The D, A, G, and IV -Criteria Plots for 4 Factor PBCDs for dv = 4 → 3
and 4 → 3 → 2 (Plotting Symbol = C-Path).
373
Figure 101. The D, A, G, and IV -Criteria Plots for 4 Factor PBCDs for dv = 3,
4 → 3, and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path).
374
Figure 102. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 4 Factor PBCDs.
375
Figure 103. The D, A, G, and IV -Criteria Plots for 4 Factor UNFSDs for dv = 4
(Plotting Symbol = Q-Path).
376
Figure 104. The D, A, G, and IV -Criteria Plots for 4 Factor UNFSDs for dv = 1, 2,
and 3 (Plotting Symbol = Q-Path).
377
Figure 105. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 4 Factor UNFSDs.
378
Figure 106. The D, A, G, and IV -Criteria Plots for 4 Factor UNFSDs for dv = 4
(Plotting Symbol = C-Path).
379
Figure 107. The D, A, G, and IV -Criteria Plots for 4 Factor UNFSDs for dv = 4 → 3
and 4 → 3 → 2 (Plotting Symbol = C-Path).
380
Figure 108. The D, A, G, and IV -Criteria Plots for 4 Factor UNFSDs for dv = 3,
4 → 3, and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path).
381
Figure 109. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 4 Factor UNFSDs.
382
Figure 110. The D, A, G, and IV -Criteria Plots for 4 Factor 416A Designs for dv =
4 (Plotting Symbol = Q-Path).
383
Figure 111. The D, A, G, and IV -Criteria Plots for 4 Factor 416A Designs for dv =
1, 2, and 3 (Plotting Symbol = Q-Path).
384
Figure 112. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 4 Factor 416A Designs.
385
Figure 113. The D, A, G, and IV -Criteria Plots for 4 Factor 416A Designs for dv =
4 (Plotting Symbol = C-Path).
386
Figure 114. The D, A, G, and IV -Criteria Plots for 4 Factor 416A Designs for dv =
4 → 3 and 4 → 3 → 2 (Plotting Symbol = C-Path).
387
Figure 115. The D, A, G, and IV -Criteria Plots for 4 Factor 416A Designs for dv =
3, 4 → 3, and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path).
388
Figure 116. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 4 Factor 416A Designs.
389
Figure 117. The D, A, G, and IV -Criteria Plots for 4 Factor 416B Designs for dv =
4 (Plotting Symbol = Q-Path).
390
Figure 118. The D, A, G, and IV -Criteria Plots for 4 Factor 416B Designs for dv =
1, 2, and 3 (Plotting Symbol = Q-Path).
391
Figure 119. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 4 Factor 416B Designs.
392
Figure 120. The D, A, G, and IV -Criteria Plots for 4 Factor 416B Designs for dv =
4 (Plotting Symbol = C-Path).
393
Figure 121. The D, A, G, and IV -Criteria Plots for 4 Factor 416B Designs for dv =
4 → 3 and 4 → 3 → 2 (Plotting Symbol = C-Path).
394
Figure 122. The D, A, G, and IV -Criteria Plots for 4 Factor 416B Designs for dv =
3, 4 → 3, and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path).
395
Figure 123. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 4 Factor 416B Designs.
396
Figure 124. The D, A, G, and IV -Criteria Plots for 4 Factor 416C Designs for dv =
4 (Plotting Symbol = Q-Path).
397
Figure 125. The D, A, G, and IV -Criteria Plots for 4 Factor 416C Designs for dv =
1, 2, and 3 (Plotting Symbol = Q-Path).
398
Figure 126. The Change in D, A, G, and IV -Criteria Plots by Reduction of Squared
Terms in Models for 4 Factor 416C Designs.
399
Figure 127. The D, A, G, and IV -Criteria Plots for 4 Factor 416C Designs for dv =
4 (Plotting Symbol = C-Path).
400
Figure 128. The D, A, G, and IV -Criteria Plots for 4 Factor 416C Designs for dv =
4 → 3 and 4 → 3 → 2 (Plotting Symbol = C-Path).
401
Figure 129. The D, A, G, and IV -Criteria Plots for 4 Factor 416C Designs for dv =
3, 4 → 3, and 4 → 3 → 2 → 1 (Plotting Symbol = C-Path).
402
Figure 130. The Change in D, A, G, and IV -Criteria Plots by Reduction of CrossProduct Terms in Models for 4 Factor 416C Designs.
403
APPENDIX D
Programming Codes
404
In this section, the program for calculating D, A, G, and IV criteria for 3 and 4
factor response surface designs in a sphical region using matlab software version 5.3
and 6.1 (Mathworks [40]) are given.
Matlab Code for 3 Design Variables:
%%
%% Load the Design Matrix (X) for the Central Composite Design %%
%%
format compact; format short;
load ccdr1cp1.dat;
xtmp = ccd3r1cp1(:,2:10); % the design matrix for the CCD;
n = length(xtmp);
% n = # of design pointts;
k = 3;
prmax = (k+2)*(k+1)/2;
%%
%% Load the Set of Reduced
%%
load Red3.dat;
vec = Red3(:,3:11);
vlast = length(vec);
out=[];
for rw = 1:vlast
xd = ones(n,1);
dsgn = Red3(rw,1);
prm = Red3(rw,2);
dv = Red3(rw,12);
l = Red3(rw,13);
c = Red3(rw,14);
q = Red3(rw,15);
x1 = vec(rw,1);
x12 = vec(rw,4);
x11 = vec(rw,7);
%
%
%
%
%
%
% k = # of design variable;
% prmax = # of full model parameters;
Models Data for 3 Factor %%
% vec = the matrix indicating model terms;
% vlast = the # of row in vec(# of Models);
dsgn = # of design;
prm = # of parameters in model;
dv = # of design variables in model;
l = # of linear term in model;
c = # of cross product term in model;
q = # of quadratic term in model;
x2 = vec(rw,2);
x13 = vec(rw,5);
x22 = vec(rw,8);
xdl = sqrt(dv/k)*xtmp(:,1:3);
xdcq = (dv/k)*xtmp(:,4:9);
x = [xdl xdcq];
for cl = 1:prmax-1
if vec(rw,cl) == 1
xd = [xd x(:,cl)];
end;
end;
%%
%% Calculate D-Criterion %%
%%
d = det(xd’*xd);
x3 = vec(rw,3);
x23 = vec(rw,6);
x33 = vec(rw,9);
% Transform design matrix if dv < k;
405
deff = 100*(d^(1/prm))/n;
xtx = xd’*xd;
xtxin = inv(xtx);
%%
%% Calculate A-Criterion %%
%%
aeff = 100*prm/(trace(n*xtxin));
%%
%% Calculate G-Criterion %%
%% By Searching for the Maximum Value of G in the Spherical Region %%
%%
i1 = .1; i2 = .1; i3 = .1; gmax = 0;
for r = 0:i1:1;
for ang1 = 0 :i2:1.9;
for ang2 = 0:i3:1.9;
rho = r*sqrt(dv);
X1 = rho*cos(ang1*pi);
X2 = rho*sin(ang1*pi)*cos(ang2*pi);
X3 = rho*sin(ang1*pi)*sin(ang2*pi);
fx = zeros(prm,1);
fx(1,1) =1;
fct = 2;
if vec(rw,1)== 1; fx(fct,1)=X1; fct=fct+1; end;
if vec(rw,2)== 1; fx(fct,1)=X2; fct=fct+1; end;
if vec(rw,3)== 1; fx(fct,1)=X3; fct=fct+1; end;
if vec(rw,4)== 1; fx(fct,1)=X1*X2; fct=fct+1; end;
if vec(rw,5)== 1; fx(fct,1)=X1*X3; fct=fct+1; end;
if vec(rw,6)== 1; fx(fct,1)=X2*X3; fct=fct+1; end;
if vec(rw,7)== 1; fx(fct,1)=X1^2; fct=fct+1; end;
if vec(rw,8)== 1; fx(fct,1)=X2^2; fct=fct+1; end;
if vec(rw,9)== 1; fx(fct,1)=X3^2; fct=fct+1; end;
gcrit = n*fx’*xtxin*fx;
if gcrit > gmax
gmax = gcrit;
fxbest = [X1 X2 X3];
end;
end;
end;
end;
geff = 100*prm/gmax;
%%
%% Calculate IV-Criterion %%
%% Involves Integration Over the Spherical Region %%
%%
syms X1 X2 X3 Fx Fxt;
Fx(1,1) = 1; Fxt(1,1) = 1;
fct = 2;
if vec(rw,1) == 1; Fx(fct,1)= sym(’X1’);
Fxt(1,fct)= sym(’X1’); fct= fct+1; end;
if vec(rw,2) == 1; Fx(fct,1)= sym(’X2’);
Fxt(1,fct)= sym(’X2’); fct= fct+1; end;
406
if vec(rw,3) == 1; Fx(fct,1)= sym(’X3’);
Fxt(1,fct)= sym(’X3’); fct= fct+1; end;
if vec(rw,4) == 1; Fx(fct,1)= sym(’X1*X2’);
Fxt(1,fct)= sym(’X1*X2’); fct= fct+1; end;
if vec(rw,5) == 1; Fx(fct,1)= sym(’X1*X3’);
Fxt(1,fct)= sym(’X1*X3’); fct= fct+1; end;
if vec(rw,6) == 1; Fx(fct,1)= sym(’X2*X3’);
Fxt(1,fct)= sym(’X2*X3’); fct= fct+1; end;
if vec(rw,7) == 1; Fx(fct,1)= sym(’X1^2’);
Fxt(1,fct)= sym(’X1^2’); fct= fct+1; end;
if vec(rw,8) == 1; Fx(fct,1)= sym(’X2^2’);
Fxt(1,fct)= sym(’X2^2’); fct= fct+1; end;
if vec(rw,9) == 1; Fx(fct,1)= sym(’X3^2’);
Fxt(1,fct)= sym(’X3^2’); fct= fct+1; end;
syms f fp f1 f2 w j iveff rr theta theta1 theta2 X1 X2 X3;
f = n*Fxt*xtxin*Fx; collect(f);
if dv == 0;
iveff = int(f,0,1); end;
if dv == 1;
w = 2;
iveff = int(f,0,1)/w; end;
if dv == 2;
w = 2*rr*pi; j = rr;
if x1==1 & x2==1 & x3==0;
f = subs(f,X1,rr*cos(theta));
f = subs(f,X2,rr*sin(theta)); end;
if x1==1 & x2==0 & x3==0 & x12==1;
f = subs(f,X1,rr*cos(theta));
f = subs(f,X2,rr*sin(theta)); end;
if x1==0 & x2==1 & x3==0 & x12==1;
f = subs(f,X1,rr*cos(theta));
f = subs(f,X2,rr*sin(theta)); end;
if x1==1 & x2==0 & x3==1;
f = subs(f,X1,rr*cos(theta));
f = subs(f,X3,rr*sin(theta)); end;
if x1==1 & x2==0 & x3==0 & x13==1;
f = subs(f,X1,rr*cos(theta));
f = subs(f,X3,rr*sin(theta)); end;
if x1==0 & x2==0 & x3==1 & x13==1;
f = subs(f,X1,rr*cos(theta));
f = subs(f,X3,rr*sin(theta)); end;
if x1==0 & x2==1 & x3==1;
f = subs(f,X2,rr*cos(theta));
f = subs(f,X3,rr*sin(theta)); end;
if x1==0 & x2==1 & x3==0 & x23==1;
f = subs(f,X2,rr*cos(theta));
f = subs(f,X3,rr*sin(theta)); end;
if x1==0 & x2==0 & x3==1 & x23==1;
f = subs(f,X2,rr*cos(theta));
f = subs(f,X3,rr*sin(theta)); end;
407
fp = (f*j)/w;
f1 = int(fp,theta,0,2*pi);
collect(f1);
iveff = int(f1,rr,0,sqrt(dv));
end;
if dv == 3;
w = 4*rr^2*pi; j = rr^2*sin(theta1);
f = subs(f,X1,rr*cos(theta1));
f = subs(f,X2,rr*sin(theta1)*cos(theta2));
f = subs(f,X3,rr*sin(theta1)*sin(theta2));
fp = (f*j)/w;
f1 = int(fp,theta2,0,2*pi);
collect(f1);
f2 = int(f1,theta1,0,pi);
collect(f2);
iveff = int(f2,rr,0,sqrt(dv));
end;
collect(iveff);
iveff = double(iveff);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fprintf(1,’%3d %1d %2d %1d %1d %1d %1d %3.4f %3.4f %3.4f %3.4f \n’,
dsgn,k,prm,dv,l,c,q,deff,aeff,geff,iveff);
out = [out;[dsgn,k,prm,dv,l,c,q,deff aeff geff iveff]];
end;
% end of 44 models;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Matlab Code for 4 Design Variables:
%%
%% Load the Design Matrix (X) for the Box-Behnken Design %%
%%
format compact; format short;
load bbd4cp1.dat;
xtmp = bbd4cp1(:,2:15); % the design matrix for the BBD;
n = length(xtmp);
% n = # of design pointts;
k = 4;
% k = # of design variable;
prmax = (k+2)*(k+1)/2; % prmax = # of full model parameters;
%%
%% Load the Set of Reduced Models Data for 4 Factor %%
%%
load Red4.dat;
vec = Red4(:,3:16);
% vec = the matrix indicating model terms;
vlast = length(vec);
% vlast = the # of row in vec(# of Models);
out=[];
for rw = 1:vlast
xd = ones(n,1);
dsgn = Red4(rw,1);
prm = Red4(rw,2);
% dsgn = # of design;
% prm = # of parameters in model;
408
dv = Red4(rw,17);
l = Red4(rw,18);
c = Red4(rw,19);
q = Red4(rw,20);
%
%
%
%
dv = # of design variables in model;
l = # of linear term in model;
c = # of cross product term in model;
q = # of quadratic term in model;
xdl = sqrt(dv/k)*xtmp(:,1:4);
xdcq = (dv/k)*xtmp(:,5:14);
x = [xdl xdcq];
for cl = 1:prmax-1
if vec(rw,cl) == 1
xd = [xd x(:,cl)];
end;
end;
% Transform design matrix if dv < k;
%%
%% Calculate D-Criterion %%
%%
d = det(xd’*xd);
deff = 100*(d^(1/prm))/n;
xtx = xd’*xd;
xtxin = inv(xtx);
%%
%% Calculate A-Criterion %%
%%
aeff = 100*prm/(trace(n*xtxin));
%%
%% Calculate G-Criterion %%
%% By Searching for the Maximum Value of G in the Spherical Region %%
%%
i1 = .1; i2 = .1; i3 = .1; i4 = .1; gmax = 0;
for r = 0:i1:1;
for ang1 = 0 :i2:1.9;
for ang2 = 0:i3:1.9;
for ang3 = 0:i4:1.9;
rho = r*sqrt(dv);
X1 = rho*cos(ang1*pi);
X2 = rho*sin(ang1*pi)*cos(ang2*pi);
X3 = rho*sin(ang1*pi)*sin(ang2*pi)*cos(ang3*pi);
X4 = rho*sin(ang1*pi)*sin(ang2*pi)*sin(ang3*pi);
fx = zeros(prm,1);
fx(1,1) =1;
fct = 2;
if vec(rw,1)== 1; fx(fct,1)=X1; fct=fct+1; end;
if vec(rw,2)== 1; fx(fct,1)=X2; fct=fct+1; end;
if vec(rw,3)== 1; fx(fct,1)=X3; fct=fct+1; end;
if vec(rw,4)== 1; fx(fct,1)=X4; fct=fct+1; end;
if vec(rw,5)== 1; fx(fct,1)=X1*X2; fct=fct+1; end;
if vec(rw,6)== 1; fx(fct,1)=X1*X3; fct=fct+1; end;
if vec(rw,7)== 1; fx(fct,1)=X1*X4; fct=fct+1; end;
if vec(rw,8)== 1; fx(fct,1)=X2*X3; fct=fct+1; end;
if vec(rw,9)== 1; fx(fct,1)=X2*X4; fct=fct+1; end;
if vec(rw,10)== 1; fx(fct,1)=X3*X4; fct=fct+1; end;
if vec(rw,11)== 1; fx(fct,1)=X1^2; fct=fct+1; end;
409
if vec(rw,12)== 1; fx(fct,1)=X2^2; fct=fct+1; end;
if vec(rw,13)== 1; fx(fct,1)=X3^2; fct=fct+1; end;
if vec(rw,14)== 1; fx(fct,1)=X4^2; fct=fct+1; end;
gcrit = n*fx’*xtxin*fx;
if gcrit > gmax
gmax = gcrit;
fxbest = [X1 X2 X3 X4];
end;
end;
end;
end;
end;
geff = 100*prm/gmax;
%%
%% Calculate IV-Criterion %%
%% Involves Integration Over the Spherical Region %%
%%
syms X1 X2 X3 X4 Fx Fxt;
Fx(1,1) = 1; Fxt(1,1) = 1;
fct = 2;
if vec(rw,1) == 1; Fx(fct,1)= sym(’X1’);
Fxt(1,fct)= sym(’X1’); fct= fct+1; end;
if vec(rw,2) == 1; Fx(fct,1)= sym(’X2’);
Fxt(1,fct)= sym(’X2’); fct= fct+1; end;
if vec(rw,3) == 1; Fx(fct,1)= sym(’X3’);
Fxt(1,fct)= sym(’X3’); fct= fct+1; end;
if vec(rw,4) == 1; Fx(fct,1)= sym(’X4’);
Fxt(1,fct)= sym(’X4’); fct= fct+1; end;
if vec(rw,5) == 1; Fx(fct,1)= sym(’X1*X2’);
Fxt(1,fct)= sym(’X1*X2’); fct= fct+1; end;
if vec(rw,6) == 1; Fx(fct,1)= sym(’X1*X3’);
Fxt(1,fct)= sym(’X1*X3’); fct= fct+1; end;
if vec(rw,7) == 1; Fx(fct,1)= sym(’X1*X4’);
Fxt(1,fct)= sym(’X1*X4’); fct= fct+1; end;
if vec(rw,8) == 1; Fx(fct,1)= sym(’X2*X3’);
Fxt(1,fct)= sym(’X2*X3’); fct= fct+1; end;
if vec(rw,9) == 1; Fx(fct,1)= sym(’X2*X4’);
Fxt(1,fct)= sym(’X2*X4’); fct= fct+1; end;
if vec(rw,10) == 1; Fx(fct,1)= sym(’X3*X4’);
Fxt(1,fct)= sym(’X3*X4’); fct= fct+1; end;
if vec(rw,11) == 1; Fx(fct,1)= sym(’X1^2’);
Fxt(1,fct)= sym(’X1^2’); fct= fct+1; end;
if vec(rw,12) == 1; Fx(fct,1)= sym(’X2^2’);
Fxt(1,fct)= sym(’X2^2’); fct= fct+1; end;
if vec(rw,13) == 1; Fx(fct,1)= sym(’X3^2’);
Fxt(1,fct)= sym(’X3^2’); fct= fct+1; end;
if vec(rw,14) == 1; Fx(fct,1)= sym(’X4^2’);
Fxt(1,fct)= sym(’X4^2’); fct= fct+1; end;
syms f fp f1 f2 f3 w j rr theta theta1 theta2 theta3 X1 X2 X3 X4 iveff;
f = n*Fxt*xtxin*Fx; collect(f);
410
if dv == 0;
iveff = int(f,0,1); end;
if dv == 1;
w = 2;
iveff = int(f,0,1)/w; end;
if dv == 2;
w = 2*rr*pi; j = rr;
f = subs(f,X3,rr*cos(theta));
f = subs(f,X4,rr*sin(theta));
fp = (f*j)/w;
f1 = int(fp,theta,0,2*pi); collect(f1);
iveff = int(f1,rr,0,sqrt(dv)); end;
if dv == 3;
w = 4*rr^2*pi; j = rr^2*sin(theta1);
f = subs(f,X1,rr*cos(theta1));
f = subs(f,X2,rr*cos(theta1));
f = subs(f,X3,rr*sin(theta1)*cos(theta2));
f = subs(f,X4,rr*sin(theta1)*sin(theta2));
fp = (f*j)/w;
f1 = int(fp,theta2,0,2*pi); collect(f1);
f2 = int(f1,theta1,0,pi); collect(f2);
iveff = int(f2,rr,0,sqrt(dv)); end;
if dv == 4;
w = 2*rr^3*pi^2; j = rr^3*(sin(theta1))^2*sin(theta2);
f = subs(f,X1,rr*cos(theta1));
f = subs(f,X2,rr*sin(theta1)*cos(theta2));
f = subs(f,X3,rr*sin(theta1)*sin(theta2)*cos(theta3));
f = subs(f,X4,rr*sin(theta1)*sin(theta2)*sin(theta3));
fp = (f*j)/w;
f1 = int(fp,theta3,0,2*pi); collect(f1);
f2 = int(f1,theta2,0,pi); collect(f2);
f3 = int(f2,theta1,0,pi); collect(f3);
iveff = int(f3,rr,0,sqrt(dv)); end;
collect(iveff);
iveff = double(iveff);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fprintf(1,’%3d %1d %2d %1d %1d %1d %1d %3.4f %3.4f %3.4f
%3.4f\n’,dsgn,k,prm,dv,l,c,q,deff,aeff,geff,iveff);
out = [out;[dsgn k prm dv l c q deff aeff geff iveff]];
end;
% end of 224 models;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%