Statistical Analysis
Professor Lynne Stokes
Department of Statistical Science
Lecture 14
Sequential Experimentation,
Screening Designs, Fold-Over
Designs
Don’t Risk all Experimental Resources
on a Single Comprehensive Experiment
•Usually many inert factors, few dominant ones
•Unexpected effects may be found early
•Experiment could be terminated early with
substantial cost savings
•Comprehensive evaluation of a few dominant
factors is usually more informative than
little information on many factors
Conduct a Screening Experiment
to Identify Dominant Factors
Augment the Screening
Experiment to Identify
Strong Two-Factor
Interactions
Conduct a RV Experiment
with the Dominant Factors
Comprehensive Experiment with
a Few Factors and Multiple Levels
or
Design to Quantitatively
Characterize the Response Surface
Figure 7.7 A simple strategy for a sequence of experiments.
Sequential Experimentation


Large experiments
Design so that key fractions can be run in
sequence



Key fractions : Resolution III, IV, or V
Analyze each sequence of data as it is
completed
Based on the results of the analysis



Continue experiment
Terminate
Redesign with dominant/new factors
Acid Plant Corrosion Study
Factor
Raw Material Feed Rate
Gas Temperature
Scrubber Water
Reactor Bed Acid
Exit Temperture
Reactant Distribution Point
Coded Level
-1
+1
3,000pph
6,000pph
100oC
220oC
5%
20%
20%
30%
300oC
360oC
East
West
Plant must cease commercial
production during experimentation -- test runs
must be minimized
MGH Table 7.1
Screening Experiments
Highly effective for isolating vital few strong effects
should be used ONLY under the proper circumstances




Very few test runs
Ability to assess main effects only
Generally leads to a comprehensive
evaluation of a few dominant factors
Potential for bias
Plackett-Burman Screening
Designs




Any number of factors, each having 2 levels
Interactions nonexistent or negligible Relative
to main effects
Number of test runs is a multiple of 4
At least 6 more test runs than factors should
be used
Construction


Determine the number of factors (k) to be included
in the design
Determine the experiments size : at least k + 6



6df for error
Select the design generator from Table 7A2
Generate the rows of the design




Design generator is the first row
Move all levels in the previous row one position to the left;
move the first level of the previous row to the last position
Continue the previous step until n - 1 rows are filled
The last row has all levels equal to -1
Construction (con’t)

Randomize


Randomly Assign Factors to Columns; Delete Unassigned
Columns
Randomly Permute the Rows
Acid Plant Corrosion Study
Factor
Raw Material Feed Rate
Gas Temperature
Scrubber Water
Reactor Bed Acid
Exit Temperture
Reactant Distribution Point
Coded Level
-1
+1
3,000pph
6,000pph
100oC
220oC
5%
20%
20%
30%
300oC
360oC
East
West
Seeking identification of dominant main effects
Plackett-Burman Design :
Corrosion Study


k = 6 Factors
n = 12 (Minimum Recommended)
Design Generator
Ru n No .
A
B
C
D
E
F
G
H
I
J
K
1
1
1
-1
1
1
1
-1
-1
-1
1
-1
2
1
-1
1
1
1
-1
-1
-1
1
-1
1
3
-1
1
1
1
-1
-1
-1
1
-1
1
1
4
1
1
1
-1
-1
-1
1
-1
1
1
-1
5
1
1
-1
-1
-1
1
-1
1
1
-1
1
6
1
-1
-1
-1
1
-1
1
1
-1
1
1
7
-1
-1
-1
1
-1
1
1
-1
1
1
1
8
-1
-1
1
-1
1
1
-1
1
1
1
-1
9
-1
1
-1
1
1
-1
1
1
1
-1
-1
10
1
-1
1
1
-1
1
1
1
-1
-1
-1
11
-1
1
1
-1
1
1
1
-1
-1
-1
1
12
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Plackett-Burman Design :
Corrosion Study


k = 6 Factors
n = 12 (Minimum Recommended)
Ru n No .
A
B
C
D
E
F
G
H
I
J
K
1
1
1
-1
1
1
1
-1
-1
-1
1
-1
2
1
-1
1
1
1
-1
-1
-1
1
-1
1
3
-1
1
1
1
-1
-1
-1
1
-1
1
1
4
1
1
1
-1
-1
-1
1
-1
1
1
-1
5
1
1
-1
-1
-1
1
-1
1
1
-1
1
6
1
-1
-1
-1
1
-1
1
1
-1
1
1
7
-1
-1
-1
1
-1
1
1
-1
1
1
1
8
-1
-1
1
-1
1
1
-1
1
1
1
-1
9
-1
1
-1
1
1
-1
1
1
1
-1
-1
10
1
-1
1
1
-1
1
1
1
-1
-1
-1
11
-1
1
1
-1
1
1
1
-1
-1
-1
1
12
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Plackett-Burman Design :
Corrosion Study


k = 6 Factors
n = 12 (Minimum Recommended)
Ru n No .
A
B
C
D
E
F
G
H
I
J
K
1
1
1
-1
1
1
1
-1
-1
-1
1
-1
2
1
-1
1
1
1
-1
-1
-1
1
-1
1
3
-1
1
1
1
-1
-1
-1
1
-1
1
1
4
1
1
1
-1
-1
-1
1
-1
1
1
-1
5
1
1
-1
-1
-1
1
-1
1
1
-1
1
6
1
-1
-1
-1
1
-1
1
1
-1
1
1
7
-1
-1
-1
1
-1
1
1
-1
1
1
1
8
-1
-1
1
-1
1
1
-1
1
1
1
-1
9
-1
1
-1
1
1
-1
1
1
1
-1
-1
10
1
-1
1
1
-1
1
1
1
-1
-1
-1
11
-1
1
1
-1
1
1
1
-1
-1
-1
1
12
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Plackett-Burman Design :
Corrosion Study


k = 6 Factors
n = 12 (Minimum Recommended)
Ru n No .
A
B
C
D
E
F
G
H
I
J
K
1
1
1
-1
1
1
1
-1
-1
-1
1
-1
2
1
-1
1
1
1
-1
-1
-1
1
-1
1
3
-1
1
1
1
-1
-1
-1
1
-1
1
1
4
1
1
1
-1
-1
-1
1
-1
1
1
-1
5
1
1
-1
-1
-1
1
-1
1
1
-1
1
6
1
-1
-1
-1
1
-1
1
1
-1
1
1
7
-1
-1
-1
1
-1
1
1
-1
1
1
1
8
-1
-1
1
-1
1
1
-1
1
1
1
-1
9
-1
1
-1
1
1
-1
1
1
1
-1
-1
10
1
-1
1
1
-1
1
1
1
-1
-1
-1
11
-1
1
1
-1
1
1
1
-1
-1
-1
1
12
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Plackett-Burman Design :
Corrosion Study
S cru b .
Distn .
Be d
Ex it
Gas
Fe e d
Ru n No .
A
W a te r
C
P o in t
E
Acid
Te m p.
Te m p.
I
J
Ra te
1
1
1
-1
1
1
1
-1
-1
-1
1
-1
2
1
-1
1
1
1
-1
-1
-1
1
-1
1
3
-1
1
1
1
-1
-1
-1
1
-1
1
1
4
1
1
1
-1
-1
-1
1
-1
1
1
-1
5
1
1
-1
-1
-1
1
-1
1
1
-1
1
6
1
-1
-1
-1
1
-1
1
1
-1
1
1
7
-1
-1
-1
1
-1
1
1
-1
1
1
1
8
-1
-1
1
-1
1
1
-1
1
1
1
-1
9
-1
1
-1
1
1
-1
1
1
1
-1
-1
10
1
-1
1
1
-1
1
1
1
-1
-1
-1
11
-1
1
1
-1
1
1
1
-1
-1
-1
1
12
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Randomly assign factors to columns
Plackett-Burman Design :
Corrosion Study
O ri g i n a l
S c ru b .
D i stn .
Be d
Ex i t
Gas
Fe e d
Run No.
W a te r
P o in t
Acid
Te m p.
Te m p.
R a te
11
1
-1
1
1
-1
1
3
1
1
-1
-1
1
1
8
-1
-1
1
-1
1
-1
1
1
1
1
-1
-1
-1
4
1
-1
-1
1
-1
-1
12
-1
-1
-1
-1
-1
-1
6
-1
-1
-1
1
1
1
10
-1
1
1
1
1
-1
7
-1
1
1
1
-1
1
5
1
-1
1
-1
1
1
2
-1
1
-1
-1
-1
1
9
1
1
-1
1
1
-1
Eliminate unassigned columns
randomly permute rows
Plackett-Burman Design :
Corrosion Study
O ri g i n a l
S c ru b .
D i stn .
Be d
Ex i t
Gas
Fe e d
Run No.
W a te r
P o in t
Acid
Te m p.
Te m p.
R a te
11
20
E as t
30
360
100
6000
3
20
W es t
20
300
220
6000
8
5
E as t
30
300
220
3000
1
20
W es t
30
300
100
3000
4
20
E as t
20
360
100
3000
12
5
E as t
20
300
100
3000
6
5
E as t
20
360
220
6000
10
5
W es t
30
360
220
3000
7
5
W es t
30
360
100
6000
5
20
E as t
30
300
220
6000
2
5
W es t
20
300
100
6000
9
20
W es t
20
360
220
3000
Resolution III
Human Performance Testing
Response
Eye Focus Time (ms)
Predictors
(A) Acuity or Sharpness of Vision
(B) Distance from Eye to Target
(C) Target Shape
2 Levels Each
(D) Illumination Level
(E) Target Size
(F) Target Density
(G) Subject
Only a few effects anticipated,
no interactions
Design Considerations


Complete factorial : 27 + repeats = 128 +
repeats
Very few effects expected, no interactions
Solution
Fractional Factorial RIII
Human Performance Testing
Design:
7 4
2 III
n=8
Defining Equation
I = ABD = ACE = BCF = ABCG
Added Factors
D = AB , E = AC , F = BC , G = ABC
Implicit Equations
24 - 4 - 1 = 11
Human Performance Testing
Complete Defining Relation
I = ABD = ACE = BCF = ABCG = BCDE = ACDF
= CDG = ABEF = BEG = AFG = DEF = ADEG
Implicit Contrasts
= CEFG = BDFG = ABCDEFG
Human Performance Testing
Complete Defining Relation
I = ABD = ACE = BCF = ABCG = BCDE = ACDF
= CDG = ABEF = BEG = AFG = DEF = ADEG
Implicit Contrasts
= CEFG = BDFG = ABCDEFG
Main-Effect Aliases
A = BD = CE = FG
B = AD = CF = EG
C = AE = BF = DG
D = AB = CG = EF
E = AC = BG = DF
F = BC = AG = DE
G = CD = BE = AF
Assuming No 3fi
Human Performance Testing
Complete Defining Relation
I = ABD = ACE = BCF = ABCG = BCDE = ACDF
= CDG = ABEF = BEG = AFG = DEF = ADEG
Implicit Contrasts
= CEFG = BDFG = ABCDEFG
Main-Effect Aliases
A + BD + CE + FG
B + AD + CF + EG
C + AE + BF + DG
D + AB + CG + EF
E + AC + BG + DF
F + BC + AG + DE
G + CD + BE + AF
Alternative Interpretation
of Aliasing:
Each Measured Effect is
the Sum of Four Effects
Human Performance Testing
Run
A
B
C
1
-1
-1
-1
+1
+1
+1
-1
85.5
2
+1
-1
-1
-1
-1
+1
+1
75.1
3
-1
+1
-1
-1
+1
-1
+1
93.2
4
+1
+1
-1
+1
-1
-1
-1
145.4
5
-1
-1
+1
+1
-1
-1
+1
83.7
6
+1
-1
+1
-1
+1
-1
-1
77.6
7
-1
+1
+1
-1
-1
+1
-1
95.0
8
+1
+1
+1
+1
+1
+1
+1
141.8
-.28
28.88
-.28
-.63
-2.43
Effects 20.63 38.38
D=AB E=AC
F=BC
G=ABC Time
Human Performance Testing
Conclusions
A, B, and D are the primary factors that affect
eye focus times
Human Performance Testing
Conclusions
A, B, and D are the primary factors that affect
eye focus times
Key Main-Effect Aliases
A = BD
B = AD
D = AB
Could the primary effects be only
two factors and their interaction ?
Human Performance Testing
Fold-Over Design:
7 4
2 III
Reverse the signs on all levels
of all factors in the design
Defining Equation
I = -ABD = -ACE = -BCF = -ABCG
Added Factors
-D = AB , -E = AC , -F = BC , -G = ABC
Human Performance Testing:
Fold-Over Design
Run
A
B
C
1
+1
+1
+1
-1
-1
-1
+1
91.3
2
-1
+1
+1
+1
+1
-1
-1
136.7
3
+1
-1
+1
+1
-1
+1
-1
82.4
4
-1
-1
+1
-1
+1
+1
+1
73.4
5
+1
+1
-1
-1
+1
+1
-1
94.1
6
-1
+1
-1
+1
-1
+1
+1
143.8
7
+1
-1
-1
+1
+1
-1
+1
87.3
8
-1
-1
-1
-1
-1
-1
-1
71.9
29.88
.53
1.63
2.68
Effects -17.68 37.73 -3.33
D=-AB E=-AC F=-BC G=-ABC Time
Human Performance Testing:
Fold-Over Design
Combined Effects:
Original Design
A + BD + CE + FG
Fold-Over Design A - BD - CE - FG
Average
A
Difference/2
BD + CE + FG
Conclusion:
Reversing ALL signs in a second fraction
unaliases ALL main effects from two-factor interactions
(still assumes higher-order interactions are negligible)
Human Performance Testing:
Fold-Over Design
n = 16
Contrasts
Average
Difference / 2
A
A= 1.48
BD+CE+FG= 19.15
B
B= 38.05
AD+CF+EG=
C
C= -1.80
AE+BF+DG= 1.53
D
D= 29.38
AB+CG+EF=
E
E=
.13
AC+BG+DF= -.40
F
F=
.50
BC+AG+DE= -1.53
G
G= .13
Original Design:
D = AB
.33
-.50
CD+BE+AF= -2.55
Conclusions ?
Fold-Over Designs



Reverse the signs on one or more factors
Run a second fraction with the sign reversals
Use the confounding pattern of the original
and the fold-over design to determine the
alias structure


Averages
Half-Differences
Human Performance Testing
Complete Defining Relation Reversing the Signs on B
I = -ABD = ACE = -BCF = -ABCG = -BCDE = ACDF
= CDG = -ABEF = -BEG = AFG = DEF = ADEG
= CEFG = -BDFG = -ABCDEFG
Main-Effect Aliases
A - BD + CE + FG
-B + AD + CF + EG
C + AE - BF + DG
D - AB + CG + EF
E + AC - BG + DF
F - BC + AG + DE
G + CD - BE + AF
Human Performance Testing:
Fold-Over Design
Combined Effects:
Original Design
A + BD + CE + FG
Fold-Over Design A - BD + CE + FG
Average
A
+ CE + FG
Difference/2
BD
Similar
With All
Main Effects
Except B
Original Design
B + AD + CF + EG
Fold-Over Design -B + AD + CF + EG
Average
AD + CE + FG
Difference/2
B
Conclusion:
Reversing the signs on ONE factor in a second fraction
unaliases its main effect and ALL its two-factor interactions