Chapter 5 Introduction to Factorial
Designs
1
5.1 Basic Definitions and Principles
•
•
•
•
Study the effects of two or more factors.
Factorial designs
Crossed: factors are arranged in a factorial design
Main effect: the change in response produced by a
change in the level of the factor
2
Definition of a factor effect: The change in the mean response when
the factor is changed from low to high
40  52 20  30

 21
2
2
30  52 20  40
B  yB  yB 

 11
2
2
52  20 30  40
AB 

 1
2
2
A  y A  y A 
3
50  12 20  40
A  y A  y A 

1
2
2
40  12 20  50
B  yB  yB 

 9
2
2
12  20 40  50
AB 

 29
2
2
4
Regression Model &
The Associated
Response Surface
y   0  1 x1   2 x2
 12 x1 x2  
The least squares fit is
yˆ  35.5  10.5 x1  5.5 x2
0.5 x1 x2
 35.5  10.5 x1  5.5 x2
5
The Effect of
Interaction on the
Response Surface
Suppose that we add an
interaction term to the
model:
yˆ  35.5  10.5 x1  5.5 x2
8x1 x2
Interaction is actually
a form of curvature
6
• When an interaction is large, the corresponding
main effects have little practical meaning.
• A significant interaction will often mask the
significance of main effects.
7
5.2 The Advantage of Factorials
• One-factor-at-a-time desgin
• Compute the main effects of factors
A: A+B- - A-BB: A-B- - A-B+
Total number of experiments: 6
• Interaction effects
A+B-, A-B+ > A-B- => A+B+ is
better???
8
5.3 The Two-Factor Factorial Design
5.3.1 An Example
• a levels for factor A, b levels for factor B and n
replicates
• Design a battery: the plate materials (3 levels) v.s.
temperatures (3 levels), and n = 4: 32 factorial design
• Two questions:
– What effects do material type and temperature have
on the life of the battery?
– Is there a choice of material that would give
uniformly long life regardless of temperature? 9
• The data for the Battery Design:
10
• Completely randomized design: a levels of factor
A, b levels of factor B, n replicates
11
• Statistical (effects) model:
 i  1, 2,..., a

yijk     i   j  ( )ij   ijk  j  1, 2,..., b
k  1, 2,..., n

 is an overall mean, i is the effect of the ith level
of the row factor A, j is the effect of the jth
column of column factor B and ( )ij is the
interaction between i and j .
• Testing hypotheses:
H 0 :  1     a  0 v.s. H 1 : at least one i  0
H 0 : 1     b  0 v.s. H 1 : at least one  j  0
12
H 0 : ( ) ij  0 i, j v.s. H 1 : at least one ( ) ij  0
• 5.3.2 Statistical Analysis of the Fixed Effects
Model
b
n
yi..   yijk yi..
j 1 k 1
a
yi..

bn
n
y. j .   yijk y. j . 
i 1 k 1
n
yij.   yijk
yij. 
k 1
a
b
n
y...   yijk
i 1 j 1 k 1
y. j .
an
yij.
n
y...
y... 
abn
13
a
b
n
a
b
2
(
y

y
)

bn
(
y

y
)

an
(
y

y
)
 ijk ...
 i.. ...
 . j. ...
2
i 1 j 1 k 1
2
i 1
a
j 1
b
a
b
n
 n ( yij .  yi..  y. j .  y... )   ( yijk  yij . ) 2
2
i 1 j 1
i 1 j 1 k 1
SST  SS A  SS B  SS AB  SS E
df breakdown:
abn  1  a  1  b  1  (a  1)(b  1)  ab(n  1)
14
• Mean squares
a
E ( MS A )  E ( SS A /(a  1))   2 
bn i2
i 1
a 1
b
E ( MS B )  E ( SSB /(b  1))   
2
an  j2
j 1
b 1
a
b
n ( ) ij2
SS AB
i 1 j 1
E ( MS AB )  E (
) 2 
(a  1)(b  1)
(a  1)(b  1)
SSE
E ( MS E )  E (
) 2
ab(n  1)
15
• The ANOVA table:
16
17
Example 5.1
Response:
Life
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Source
Model
A
B
AB
Pure E
C Total
Sum of
Squares
59416.22
10683.72
39118.72
9613.78
18230.75
77646.97
DF
8
2
2
4
27
35
Mean
F
Square Value
7427.03 11.00
5341.86
7.91
19559.36 28.97
2403.44
3.56
675.21
Std. Dev. 25.98
Mean
105.53
C.V.
24.62
R-Squared
Adj R-Squared
Pred R-Squared
0.7652
0.6956
0.5826
PRESS
Adeq Precision
8.178
32410.22
Prob > F
< 0.0001
0.0020
< 0.0001
0.0186
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DESIGN-EXPERT Plot
Life
Interaction Graph
A: Material
188
X = B: Temperature
Y = A: Material
A1 A1
A2 A2
A3 A3
Life
146
104
2
2
62
2
20
15
70
125
B: Tem perature
19
• Multiple Comparisons:
– Use the methods in Chapter 3.
– Since the interaction is significant, fix the factor
B at a specific level and apply Turkey’s test to
the means of factor A at this level.
– See Page 174
– Compare all ab cells means to determine which
one differ significantly
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5.3.3 Model Adequacy Checking
• Residual analysis: eijk  yijk  yˆ ijk  yijk  yij
DESIGN-EXPERT Plot
Life
Normal plot of residuals
DESIGN-EXPERT Plot
Life
Residuals vs. Predicted
45.25
99
95
18.75
80
70
Res iduals
Norm al % probability
90
50
30
20
10
-7.75
-34.25
5
1
-60.75
49.50
-60.75
-34.25
-7.75
18.75
76.06
102.62
129.19
155.75
45.25
Predicted
Res idual
21
DESIGN-EXPERT Plot
Life
Residuals vs. Run
45.25
Res iduals
18.75
-7.75
-34.25
-60.75
1
6
11
16
21
26
31
36
Run Num ber
22
DESIGN-EXPERT Plot
Life
Residuals vs. Material
Residuals vs. Temperature
45.25
45.25
18.75
18.75
Res iduals
Res iduals
DESIGN-EXPERT Plot
Life
-7.75
-7.75
-34.25
-34.25
-60.75
-60.75
1
2
Material
3
1
2
3
Tem perature
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5.3.4 Estimating the Model Parameters
• The model is
yijk     i   j  ( ) ij   ijk
• The normal equations:
a
b
i 1
j 1
a
b
 : abn  bn i  an  j  n ( ) ij  y
i 1 j 1
b
b
j 1
j 1
 i : bn  bn i  n  j  n ( ) ij  yi
a
a
i 1
i 1
 j : an  n i  an j  n ( ) ij  y j
( ) ij : n  n i  n j  n( ) ij  y ij
• Constraints:
a

i 1
i
b
a
b
j 1
i 1
j 1
 0,   j  0,   ij    ij  0
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• Estimations:
ˆ  y
ˆi  y i  y
ˆ j  y j  y
 ij
 y ij  y i  y j  y
• The fitted value:
yˆ ijk  ˆ  ˆi  ˆ j   ij  yij
• Choice of sample size: Use OC curves to choose
the proper sample size.
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• Consider a two-factor model without interaction:
– Table 5.8
– The fitted values: yˆ ijk  yi  y j  y
26
• One observation per cell:
– The error variance is not estimable because the
two-factor interaction and the error can not be
separated.
– Assume no interaction. (Table 5.9)
– Tukey (1949): assume ()ij = rij (Page 183)
– Example 5.2
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5.4 The General Factorial Design
• More than two factors: a levels of factor A, b
levels of factor B, c levels of factor C, …, and n
replicates.
• Total abc … n observations.
• For a fixed effects model, test statistics for each
main effect and interaction may be constructed by
dividing the corresponding mean square for effect
or interaction by the mean square error.
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• Degree of freedom:
– Main effect: # of levels – 1
– Interaction: the product of the # of degrees of
freedom associated with the individual
components of the interaction.
• The three factor analysis of variance model:
– yijkl     i   j   k  ( ) ij
 ( ) ik  (  ) jk  ( ) ijk   ijkl
– The ANOVA table (see Table 5.12)
– Computing formulas for the sums of squares
(see Page 186)
– Example 5.3
29
30
• Example 5.3: Three factors: the percent
carbonation (A), the operating pressure (B); the
line speed (C)
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32
5.5 Fitting Response Curves and
Surfaces
• An equation relates the response (y) to the factor
(x).
• Useful for interpolation.
• Linear regression methods
• Example 5.4
– Study how temperatures affects the battery life
– Hierarchy principle
33
– Involve both quantitative and qualitative factors
– This can be accounted for in the analysis to produce
regression models for the quantitative factors at each
level (or combination of levels) of the qualitative
factors
A = Material type
B = Linear effect of Temperature
B2 = Quadratic effect of
Temperature
AB = Material type – TempLinear
AB2 = Material type - TempQuad
B3 = Cubic effect of
Temperature (Aliased)
34
35
36
37
5.6 Blocking in a Factorial Design
• A nuisance factor: blocking
• A single replicate of a complete factorial
experiment is run within each block.
• Model:
yijk     i   j  ( ) ij   k   ijk
– No interaction between blocks and treatments
• ANOVA table (Table 5.20)
38
39
• Example 5.6:
– Two factors: ground clutter and filter type
– Nuisance factor: operator
40
• Two randomization restrictions: Latin square
design
• An example in Page 200.
• Model:
yijkl    i  j  k  ( ) jk  l   ijkl
• Tables 5.23 and 5.24
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