Statistical Analysis
Professor Lynne Stokes
Department of Statistical Science
Lecture 12
Analysis of Factor-Effects:
Linear Combinations, Contrasts,
Polynomials
Analysis of Ballistic Limit Velocities
MGH Exercise #6.6
Nose Shape
Blunt
Conical
Hemispherical
Angle
0
938
942
943
889
890
892
876
877
881
45
1162
1167
1163
1151
1145
1152
1124
1125
1128
Model and Assumptions
yijk = m + ai + bj + (ab)ij + eijk
where
 ai  b j   (ab)ij   (ab)ij  0,
yij = ballistic velocity for the kth repeat of the ith nose
cone and the jth angle
m = overall mean ballistic velocity
ai = fixed effect of the ith nose cone on mean velocity
bi = fixed effect of the jth angle on mean velocity
(ab)ij = fixed effect of the interaction between the ith nose
cone and the jth angle on mean velocity
eij = random experimental error, NID(0,s2)
Analysis of Variance Table
Mason, Gunst, & Hess: Exercise 6.6
3
The ANOVA Procedure
Dependent Variable: velocity
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
5
275139.6111
55027.9222
7861.13
<.0001
Error
12
84.0000
7.0000
Corrected Total
17
275223.6111
Source
Source
shape
angle
shape*angle
R-Square
Coeff Var
Root MSE
velocity Mean
0.999695
0.258192
2.645751
1024.722
DF
Anova SS
Mean Square
F Value
Pr > F
2
1
2
7916.4444
266206.7222
1016.4444
3958.2222
266206.7222
508.2222
565.46
38029.5
72.60
<.0001
<.0001
<.0001
Interaction Null Hypothesis
H 0 : (ab)ij  0 for all i, j
H a : (ab)ij  0 for some i, j
The effect of differing nose cones on mean ballistic velocity
is the same for all angles of launch.
Test Procedure
Reject H0 if F = MSAB/MSE > F0.05(2,12)=3.89
Conclusion
There is sufficient evidence (p = 0.0001) to conclude at
a significance level of 0.05 that the effects of nose cone
design and launch angle differ on the mean ballistic
velocities of 10 mm rolled armor projectiles.
Interaction Averages
Level of
SHAPE
Level of
ANGLE
Blunt
0
Blunt
45
Conical
0
Conical
45
Hemispherical 0
Hemispherical 45
-----------VELOCITY---------N
Mean
SD
3
3
3
3
3
3
941.00000
1164.00000
890.33333
1149.33333
878.00000
1125.66667
2.64575131
2.64575131
1.52752523
3.78593890
2.64575131
2.08166600
Exercise 6.6: Interaction Averages for Ballistic Limit Velocities.
Average Velocity (m/sec)
1200
1100
Angle = 0
Angle = 45
Fisher’s
LSD = 5.53
1000
900
Statistical vs. Practical Significance for
the Interaction
800
Blunt
Conic
Shape
Hemispheric
Linear Combinations of Parameters
Estimable Functions of Parameters
   cim i
Example: m 1  m 2  2m  a 1  a 2
Estimator
Standard Error
t Statistic
ˆ   ci yi

seˆ  s 
ci2
ˆ  
t
se ˆ
/r

1/ 2
Not Usually
of Interest
Several individual
comparisons could be of interest
Interest
in comparing
Wheelchair
and crutches
vs.
amputee and
hearing loss
Contrasts of Effects
Estimable Factor Effects
   c i m i  m c i   c i a i
  cia i

 ci  0
Contrasts
Elimination of the overall mean requires contrasts of main effect averages.
(Note: Want to compare factor effects.)
Elimination of main effects from interaction comparisons requires contrasts
of the interaction Aaverages.
(Note: Want interaction effects to measure variability that is unaccounted for
by or in addition to the main effects.)
Statistical Independence
y ~ N(Xb , s2I)
mˆ  c  y
1
1
mˆ 2  c 2 y
mˆ 1 and mˆ 2 are statistica lly independen t  c1c2  0
Orthogonal linear combinations
are statistically independent
Orthogonal contrasts
are statistically independent
Analysis of Water Pump Prototypes
Company A
Design 1 Design 2
31,189
24,944
31,416
24,712
30,643
24,576
30,321
25,488
30,661
24,403
30,756
24,625
31,316
24,953
30,826
25,930
30,924
24,215
31,168
24,858
Company B
Design 3 Design 4
24,356
27,077
24,036
26,030
24,544
26,573
26,233
25,804
23,075
25,906
25,264
27,190
25,667
26,539
21,613
27,724
21,752
26,384
26,135
26,712
Designs 1 & 3 are Nominally Identical
Designs 2 & 4 are Nominally Identical
MGH Ex 6.22
Analysis of Water Pump Prototypes
Model and Assumptions
yij = m + ai + eij
where
yij = jth mileage driven before failure using ith pump design
m = overall mean mileage
ai = fixed effect of the ith design on mileage
eij = random experimental error, NID(0,s2)
Prespecified Contrasts
Are these
statistics
optimal ?
Companies
t
sC
Company
A
A
B
B
Divisor
yA  yB
1/ n A 1/ n B
Designs 1 vs. 3
y1  y 3
t
s1&3 1 / n1  1 / n 3
Design #
1
2
3
4
Average
Miles Driven
y1
y2
y3
y4
Designs 1&3 vs. 2&4
y1&3  y 2&4
t
s D 1 / n1&3  1 / n 2&4
Designs 2 vs. 4
y2  y4
t
s 2& 4 1 / n 2  1 / n 4
t-test Comparison of Company Effects
The TTEST Procedure
Statistics
Variable
company
mileage
mileage
mileage
A
B
Diff (1-2)
N
20
20
Lower CL
Mean
Mean
Upper CL
Mean
Lower CL
Std Dev
Std Dev
26430
24630
849.45
27896
25431
2465.5
29363
26232
4081.6
2382.8
1301.4
2063.1
3133.2
1711.3
2524.4
Statistics
Variable
company
mileage
mileage
mileage
A
B
Diff (1-2)
Upper CL
Std Dev
Std Err
Minimum
Maximum
4576.3
2499.4
3253.4
700.6
382.65
798.29
24215
21613
31416
27724
DF
t Value
Pr > |t|
38
29.4
3.09
3.09
0.0037
0.0044
T-Tests
Variable
Method
Variances
mileage
mileage
Pooled
Satterthwaite
Equal
Unequal
t-test Comparison of Designs 1&3
with Designs 2&4
Designs Recoded: 1 & 3 = 1, 2 & 4 = 2
The TTEST Procedure
Statistics
Variable
mileage
mileage
mileage
design
N
2
4
10
10
Diff (1-2)
Lower CL
Mean
Mean
Upper CL
Mean
Lower CL
Std Dev
Std Dev
24506
26158
-2251
24870
26594
-1724
25235
27030
-1196
350.47
419.17
424.42
509.53
609.41
561.69
Statistics
Variable
mileage
mileage
mileage
design
2
4
Diff (1-2)
Upper CL
Std Dev
Std Err
Minimum
Maximum
930.2
1112.5
830.65
161.13
192.71
251.2
24215
25804
25930
27724
T-Tests
Variable
Method
Variances
mileage
mileage
Pooled
Satterthwaite
Equal
Unequal
DF
t Value
Pr > |t|
18
17.5
-6.86
-6.86
<.0001
<.0001
t-test Comparison of Design #1 with
Design #3
The TTEST Procedure
Statistics
Variable
mileage
mileage
mileage
design
N
1
3
10
10
Diff (1-2)
Lower CL
Mean
Mean
Upper CL
Mean
Lower CL
Std Dev
Std Dev
30675
23070
5518.5
30922
24268
6654.5
31169
25465
7790.5
237.72
1151.8
913.56
345.61
1674.5
1209
Statistics
Variable
mileage
mileage
mileage
design
1
3
Diff (1-2)
Upper CL
Std Dev
Std Err
Minimum
Maximum
630.95
3057
1787.9
109.29
529.53
540.69
30321
21613
31416
26233
T-Tests
Variable
Method
Variances
mileage
mileage
Pooled
Satterthwaite
Equal
Unequal
DF
t Value
Pr > |t|
18
9.77
12.31
12.31
<.0001
<.0001
t-test Comparison of Design #2 with
Design #4
The TTEST Procedure
Statistics
Variable
mileage
mileage
mileage
design
N
2
4
10
10
Diff (1-2)
Lower CL
Mean
Mean
Upper CL
Mean
Lower CL
Std Dev
Std Dev
24506
26158
-2251
24870
26594
-1724
25235
27030
-1196
350.47
419.17
424.42
509.53
609.41
561.69
Statistics
Variable
mileage
mileage
mileage
design
2
4
Diff (1-2)
Upper CL
Std Dev
Std Err
Minimum
Maximum
930.2
1112.5
830.65
161.13
192.71
251.2
24215
25804
25930
27724
T-Tests
Variable
Method
Variances
mileage
mileage
Pooled
Satterthwaite
Equal
Unequal
DF
t Value
Pr > |t|
18
17.5
-6.86
-6.86
<.0001
<.0001
t-test Comparisons
Incorrect Analyses:
Standard deviations are inefficient and
possibly biased
Linear Model Analysis
The ANOVA Procedure
Dependent Variable: mileage
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
3
270956900.1
90318966.7
101.64
<.0001
Error
36
31990423.8
888622.9
Corrected Total
39
302947323.9
Source
R-Square
Coeff Var
Root MSE
mileage Mean
0.894403
3.535431
942.6680
26663.45
Source
DF
Anova SS
Mean Square
F Value
Pr > F
design
3
270956900.1
90318966.7
101.64
<.0001
Analysis of Water Pump Prototypes
Comparison of All 4 Designs
H0: ai = 0 for all i
vs
Ha: ai  0 for some i
Reject H0 and accept Ha if F > F0.05(3,36) = 2.87
From the ANOVA Table, F = 101.64.
Conclusion
On the basis of this analysis, there is sufficient
evidence (p = 0.001) to conclude that the mean
mileage before failure of these prototype pumps
differs by design type.
Analysis of Water Pump Prototypes
Multiple Comparisons of Pump Design Average Mileages
Design
Average
3
24,268
2
24,870
4
26,594
1
30,922
Using Fisher’s Least Significant Difference procedure, average mileages
for two designs are significantly different from each other if their
difference exceeds 1,135.4 miles. As indicated by the line in the above
table, the average mileages for Designs 2 and 3 are not significantly
different. Mileages for all other pairwise comparisons of designs are
significantly different; in particular, Design 1 has a significantly greater
average mileage measurement than the other designs.
Prespecified Contrasts
Company
A
A
B
B
Divisor
Design #
1
2
3
4
Possible Comparisons
Designs for
Designs for
Companies
Company A
Company B
1
1
0
1
-1
0
-1
0
1
-1
0
-1
2
1
1
Companies
t
Company A
t
Company B
t
se
se
se
yA  yB
1/ nA 1/ n B
y1  y 2
1 / n1  1 / n 2
y3  y 4
1/ n3 1/ n 4
Average
Miles Driven
y1
y2
y3
y4
Contrasts: the Details
Companies
t
yA  yB
1/ nA 1/ n B
se
1
1
1
y A  y B  ( y1  y 2 )  ( y3  y 4 )  ( y1  y 2  y3  y 4 )
2
2
2
s yA  yB
1/ 2
 1  s s s s 
   

 
r
r
r 
4  r
2
2
2
2
1/ 2
 s2 s2 
  
n A n B 
1/ 2
 1
1 
 s  
n A n B 
, n A  n B  2r
Start with difference of averages
Contrasts: the Details
Companies
t
yA  yB
1/ nA 1/ n B
se
y1  y 2  y3  y 4
s y1  y2  y3  y4
2 1 / 2
s s s s
 

 
r
r
r 
 r
2
2
2
1/ 2
 1
1 
 2s  
n A n B 
Start with a contrast
Orthogonal Sets of Contrasts
Company
A
A
B
B
Design #
1
2
3
4
Possible Comparisons
Designs for
Designs for
Companies
Company A
Company B
1
1
0
1
-1
0
-1
0
1
-1
0
-1
Average
Miles Driven
y1
y2
y3
y4
Possible Comparisons
Company
A
A
B
B
Design #
1
2
3
4
Companies
1
1
-1
-1
(1&3) vs. (2&4) (1&4) vs. (2&3)
1
1
-1
-1
1
-1
-1
1
Many other sets of orthogonal contrasts
Average
Miles Driven
y1
y2
y3
y4
General Method for Forming
Orthogonal Contrasts
1 
 1
 1
 1

 
 
 


1 
 1
 1
 1

 y1 
 2

1 


*  0
*
*  1
*
c1    , c 2    , c 3    , . . . , c k-1  
0
0
3
1  y =  ... 
y 
 
 
 


 k 
 . . .
 . . .
 . . .
 ... 
 
 
 


 0
 0
 0
  (k  1)
Orthonormal : c j  {j( j  1)}1 c *j
j = 1, 2, . . . , k - 1
k-1
ANY Contrast Vector c =
 a jc j
j=1
Some Contrasts for Comparing
Factor Level Averages
a  (a 1 a 2
... a a )

Comparing Two Factor Level Effects
a i  a j  c a
c  (0 ... 0 1 0 ... 0 - 1 0 ... ) / 2
Comparing the Mean of Two Factor Level Effects
with a Third Factor Effect
ai  a j
 ak
or
a i  a j  2a k
2
c  (0 . .. 0 1 0 . .. 0 1 0 . .. 0 - 2 0 . . . 0) / 6
One Set of Main Effects Contrasts :
Qualitative Factor Levels
 1
 1
 1
 
 
 
1  1
1  1
1  1
c1 
, c2 
, c3 




2 0
6 2
12  1 
 
 
 
 0
 0
 3
 1  c1 a  a 1  a 2
a1  a 2

 2  c2 a 
 a3
2
a1  a 2  a 3

 3  c3 a 
 a4
3
Main Effects Contrasts :
Qualitative Factor Levels
 1
 1
 1
 
 
 
1  1
1  1
1  1
c1 
, c2 
, c3 




2 0
6 2
12  1 
 
 
 
 0
 0
 3


Three statistically independent contrasts of the
response averages
A partitioning of the main effects degrees of freedom
into single degree-of-freedom contrasts (a = 4: df = 3)
Sums of Squares and Contrasts
C a x (a-1) = c 1 c 2 ... c a-1 

1/ 2
Pa x a  a
Orthonormal
Basis Set
1a : C

PP   I a  a 1J a  CC 
 CC   I a  a 1J a
SS A  r  (y i - y )
2
 ry ( I a  a 1 J a ) y
 ry CC y

 r  (ci y ) 2
a-1 mutually
orthonormal contrast
vectors
ANY set of
orthonormal contrast
vectors
Simultaneous
Test
Single degree-offreedom contrasts
Possible Analyses of Water Pump
Prototypes
Source
Four Distinct Designs
Design
df
S.S.
3
270,956,900
Single Degree of Freedom Contrasts
Companies
1
Designs for Company #1
1
Designs for Company #2
1
Total
3
60,786,902
183,109,313
27,060,685
270,956,900
Two Fixed Designs for Each Company
Companies
1
Designs
1
Companies x Designs
1
Total
3
60,786,902
34,692,788
175,477,210
270,956,900
Warping of Copper Plates:
Quantitative Factor Levels
Temperature
50
75
100
125
Average
40
17,20
12,9
16,12
21,17
15.50
Copper Content
60
80
16,21
24,22
18,13
17,12
18,21
25,23
23,21
23,22
18.87
21.00
100
28,27
27,31
30,23
29,31
28.25
Average
21.88
17.38
21.00
23.38
20.91
MGH Table 6.7
Model and Assumptions
yijk = m + ai + bj + (ab)ij + eijk
where
 ai   b j   (ab)ij   (ab)ij  0,
i
j
i
j
yijk = warping measurement for the kth repeat at the ith temp.
using a plate having the jth amount of copper
m = overall mean warping measurement
ai = fixed effect of the ith temperature on the mean warping
bi = fixed effect of the jth copper content on the mean warping
(ab)ij = fixed effect of the interaction between the ith temperature
and the jth copper content on the mean warping
eij = random experimental error, NID(0,s2)
Warping of Copper Plates
Source
Copper Content
Temperature
Cx T
Error
Total
df
3
3
9
16
31
SS
698.34
156.09
113.78
108.50
1076.71
MS
232.78
52.03
12.64
6.78
F
34.33
7.67
1.86
p-Value
0.000
0.002
0.134
Quantitative factor levels
HOW does mean warping change with the factor levels ?
MGH Table 6.7
Warping of Copper Plates
Are there contrast vectors
that quantify curvature ?
35
30
Average 25
Warping 20
15
10
0
50
75
100
125
Temperature (deg F)
150
Warping of Copper Plates
Are there contrast vectors
that quantify curvature ?
35
30
Average 25
Warping 20
15
10
0
20
40
60
80
Copper Content (%)
100
Main Effects Contrasts :
Quantitative Factor Levels
Assumption
Mean response can be well approximated by a
low-order polynomial function of the factor levels
yij  m  ai  eij
 b0  b1x i  b2 x i2  ... + eij
Assumption : m  ai  b0  b1x i  b2 x i2  ...
Orthogonal Polynomials
Coded Factor Levels
xi = a0 + a1i
Equally Spaced
e.g., Temp = 50, 75, 100, 125 >>>> xi = 25 + 25i
Orthogonal Polynomials
y ij  b 0  b 1x  b 2 x 2  . . . + e ij
  0   1c1   2 c 2  .. . + e ij
With coded factor levels,
orthogonal polynomials are only
a function of n
Not Orthongonal
Orthogonal
Orthogonal Polynomials
Linear Effect
c1  a 0 1  x
Orthogonality Constraint
c1’1 = 0
Orthogonal Polynomials
Linear Effect
c1  a 0 1  x
Orthogonality Constraint
c1’1 = 0
Solution
a 0 n   xi  0  a 0   x
c1  x  x1n  {x i  x}
Centered Data Values
Linear Orthogonal Polynomial
(Coded Factor Levels)
n  1

1

2

n  1


c1   2  2 
 ... 
 n  n  1

2 
n=4
 1.5  3

  
0.5  1

c1 

 0.5  1

  
 1.5  3
Orthogonal Polynomials
Quadratic Effect
c 2  a 0 1  a 1x  x 2
Orthogonality Constraints
c2’1 = 0 and c2’c1 = 0
Solution
a 0 n  a1  x i   x i2  0
a 0  ( x i  x )  a1  x i ( x i  x )   x i2 ( x i  x )  0
Main Effects Contrasts :
Equally Spaced Quantitative Factor Levels
Recurrence Relation for Coded Levels
c i,0  1
n1
c i,1  i 
2
c i, j1  c i,1c i, j 
j2 (n 2  j2 )
4(4 j  1)
2
c i, j1
j = 2, 3, ... , n - 1
Main Effects Contrasts :
Equally Spaced Quantitative Factor Levels
n=4
 3
 1
 1
 
 
 
1  1
1  1
1  3
c1 
, c2 
, c3 
2  1
20  1 
20  3
 
 
 
 3
 1
 1
1 = Linear
2 = Quadratic
3 = Cubic
Linear Combinations of Parameters
Estimable Functions of Parameters
   ci m i
Estimator
Standard Error
ˆ   ci yi
Same for Contrasts

seˆ  s 
t Statistic
ci2
/r

1/ 2
ˆ  
t
, F  t2
se ˆ
Average warping measurements.
24
29
Changes in Average Warping
Possible Trends
in Average Warping
28
23
27
26
25
Average warping
Average warping
22
21
20
19
24
23
22
21
20
19
18
18
17
16
17
15
50
75
100
T emperature (deg C)
125
40
60
80
Copper content (%)
100
Warping of Copper Plates
Temperature
50
75
100
125
Normalized Contrast
Single df S.S.
Linear
-3
-1
1
3
1.82
26.41
Quadratic
1
-1
-1
1
3.44
94.53
Cubic
-1
3
-3
1
-2.09
35.16
1/ 2


2
ˆ
Normalized Contrast :  j   cij yi /   cij 
 i1 
i 1
Single df Sum of Squares : br ˆ 2j
4
4
Average
21.88
17.38
21.00
23.38
156.10
SAS Output
The GLM Procedure
Dependent Variable: warping
Warping Measurement
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
15
968.218750
64.547917
9.52
<.0001
Error
16
108.500000
6.781250
Corrected Total
31
1076.718750
R-Square
Coeff Var
Root MSE
warping Mean
0.899231
12.45600
2.604083
20.90625
Source
temp
content
temp*content
Source
temp
content
temp*content
Contrast
Linear Temperature
Quadratic Temperature
Cubic Temperature
Linear Copper Content
Quadratic Copper Content
Cubic Copper Content
DF
Type I SS
Mean Square
F Value
Pr > F
3
3
9
156.0937500
698.3437500
113.7812500
52.0312500
232.7812500
12.6423611
7.67
34.33
1.86
0.0021
<.0001
0.1327
DF
Type III SS
Mean Square
F Value
Pr > F
3
3
9
156.0937500
698.3437500
113.7812500
52.0312500
232.7812500
12.6423611
7.67
34.33
1.86
0.0021
<.0001
0.1327
DF
Contrast SS
Mean Square
F Value
Pr > F
1
1
1
1
1
1
26.4062500
94.5312500
35.1562500
652.0562500
30.0312500
16.2562500
26.4062500
94.5312500
35.1562500
652.0562500
30.0312500
16.2562500
3.89
13.94
5.18
96.16
4.43
2.40
0.0660
0.0018
0.0369
<.0001
0.0515
0.1411
Scaled Contrasts
Note: Need scaling to make polynomial contrasts comparable
ˆ 
k
 a i y i
i 1
ˆ s 
k
 a i y i
i 1
, var (ˆ )  se2  a i2 / n
k
i 1
1/ 2
 k a 2 /n 
 i 
 i1

, var (ˆ s )  se2
Warping of Copper Plates
Copper Content
Linear
Quadratic
Cubic
40
-3
1
-1
60
-1
-1
3
80
1
-1
-3
100
3
1
1
Scaled Effect
25.51
5.47
4.04
95% C.I.
(19.99,31.03) (-0.05,10.99) (-1.48,9.56)
Average
15.50
18.88
21.00
28.25
Note: Quadratic Effect is Forced to be Orthogonal to Linear
Cubic Effect is Forced to be Orthogonal to Linear and Quadratic
Single Degree of Freedom
Contrasts for Interactions
Linear x Linear
Linear
Linear
-3
-1
1
3
-3
9
3
-3
-9
-1
3
1
-1
-3
1
-3
-1
1
3
3
-9
-3
3
9
2(xi  x )