Design and Analysis of
Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
1/33
Factorial Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
2/33
Outline






Basic Definition and Principles
The Advantages of Factorials
The Two Factors Factorial Design
The General Factorial Design
Fitting Response Curve and Surfaces
Blocking in Factorial Design
3
Basic Definitions and Principles




Factorial Design—all of the possible
combinations of factors’ level are investigated
When factors are arranged in factorial design,
they are said to be crossed
Main effects – the effects of a factor is defined
to be changed
Interaction Effect – The effect that the
difference in response between the levels of
one factor is not the same at all levels of the
4
other factors.
Basic Definitions and Principles

Factorial Design without interaction
5
Basic Definitions and Principles

Factorial Design with interaction
6
Basic Definitions and Principles



Average response – the average value at one
factor’s level
Average response increase – the average value
change for a factor from low level to high level
No Interaction:
A  y A  y A 
B  yB  yB 
AB 
52  20
2

40  52

20  30
2
30  52
2

20  40
2
30  40
2
 21
 11
2
 1
7
Basic Definitions and Principles

With Interaction:
A  y A  y A 
B  yB  yB 
AB 
12  20
2

50  12

20  40
2
2
40  12
20  50
2
40  50

1
 9
2
  29
2
8
Basic Definitions and Principles


Another way to look at interaction:
When factors are quantitative
y   0   1 x 1   2 x 2   1 2 x1 x 2  
T he least squares fit is
yˆ  35.5  10.5 x1  5.5 x 2  0.5 x1 x 2  35.5  10.5 x1  5.5 x 2


In the above fitted regression model, factors are
coded in (-1, +1) for low and high levels
This is a least square estimates
9
Basic Definitions and Principles


Since the interaction is small, we can ignore it.
Next figure shows the response surface plot
10
Basic Definitions and Principles

The case with interaction
11
Advantages of Factorial design



Efficiency
Necessary if interaction effects are presented
The effects of a factor can be estimated at several
levels of the other factors
12
The Two-factor Factorial
Design




Two factors
a levels of factor A, b levels of factor B
n replicates
In total, nab combinations or experiments
13
The Two-factor Factorial
Design – An example


Two factors, each with three levels and four
replicates
32 factorial design
14
The Two-factor Factorial
Design – An example

Questions to be answered:


What effects do material type and temperature
have on the life the battery
Is there a choice of material that would give
uniformly long life regardless of temperature?
15
The Two-factor Factorial
Design
Statistical (effects) model:
y ijk     i   j  ( ) ij   ijk
 i  1, 2, ..., a

 j  1, 2, ..., b
 k  1, 2, ..., n

means model
y ijk   ij   ijk
 i  1 ,2 ,...,a

 j  1, 2 ,..., b
 k  1, 2 ,..., n

16
The Two-factor Factorial
Design

Hypothesis

Row effects:
H 0 :1   2     a  0
H a : at least one  i  0

Column effects:
H 0 : 1   2     a  0
H a : at least one  i  0

Interaction:
H 0 : ( ) ij  0
H a : at least one ( ) ij  0
The Two-factor Factorial
Design -- Statistical Analysis
a
b
n

i 1
j 1 k 1
a
b
( y ijk  y ... )  bn  ( y i ..  y ... )  an  ( y . j .  y ... )
2
2
i 1
a
 n
i 1
2
j 1
b

j 1
a
( y ij .  y i ..  y . j .  y ... )  
2
i 1
b
n

( y ijk  y ij . )
j 1 k 1
SS T  SS A  SS B  SS A B  SS E
df breakdow n:
abn  1  a  1  b  1  ( a  1)( b  1)  ab ( n  1)
18
2
The Two-factor Factorial
Design -- Statistical Analysis

Mean square:
 A:
 SS A 
E ( MS A )  E 

 a 1

a
bn   i
2
2

i 1
a 1
B:
a
an   i
2
 SS B 
E ( MS B )  E 

 b 1

2
i 1

b 1
Interaction:
a
E ( MS


SS AB
)

E

  
AB

 ( a  1)( b  1) 
n
2

i 1
b
 ( )
2
ij
j 1
( a  1)( b  1)
19
The Two-factor Factorial
Design -- Statistical Analysis

Mean square:
 Error:
 SS E

E ( MS A )  E 
  
 ab ( n  1) 
2
20
The Two-factor Factorial
Design -- Statistical Analysis

ANOVA table
21
The Two-factor Factorial
Design -- Statistical Analysis

Example
22
The Two-factor Factorial
Design -- Statistical Analysis

Example
23
The Two-factor Factorial
Design -- Statistical Analysis

Example
24
The Two-factor Factorial
Design -- Statistical Analysis

Example STATANOVA--GLM
General Linear Model: Life versus Material, Temp
Factor Type Levels Values
Material fixed
3 1, 2, 3
Temp
fixed
3 15, 70, 125
Analysis of Variance for Life, using Adjusted SS for Tests
Source
DF
Material
2
Temp
2
Material*Temp 4
Error
27
Total
35
Seq SS
10683.7
39118.7
9613.8
18230.7
77647.0
Adj SS
10683.7
39118.7
9613.8
18230.7
Adj MS
5341.9
19559.4
2403.4
675.2
F P
7.91 0.002
28.97 0.000
3.56 0.019
S = 25.9849 R-Sq = 76.52% R-Sq(adj) = 69.56%
Unusual Observations for Life
Obs Life
Fit
SE Fit Residual St Resid
2 74.000 134.750 12.992 -60.750 -2.70 R
8 180.000 134.750 12.992 45.250
2.01 R
R denotes an observation with a large standardized residual.
25
The Two-factor Factorial
Design -- Statistical Analysis

Example STATANOVA--GLM
26
The Two-factor Factorial
Design -- Statistical Analysis

Example STATANOVA--GLM
27
The Two-factor Factorial
Design -- Statistical Analysis

Estimating the model parameters
y ijk     i   j  ( ) ij   ijk
 i  1, 2, ..., a

 j  1, 2, ..., b
 k  1, 2, ..., n


  y ...

 i  y i ..  y ...

 j  y . j .  y ...
  
    y ij .  y i ..  y . j .  y ...

 ij
28
The Two-factor Factorial
Design -- Statistical Analysis

Choice of sample size
 Row effects
 
2

nbD
2
2a
2
Column effects
 
2

naD
2
2b
2
Interaction effects
 
nD
2

2
2  [( a  1)( b  1)  1]
2
D:difference, :standard deviation
29
The Two-factor Factorial
Design -- Statistical Analysis
30
The Two-factor Factorial
Design -- Statistical Analysis

Appendix Chart V

For n=4, giving D=40 on temperature, v1=2,
v2=27, Φ 2 =1.28n. β =0.06
n
Φ2
Φ
υ1
υ2
β
2
2.56
1.6
2
9
0.45
3
3.84
1.96
2
18
0.18
4
5.12
2.26
2
27
0.06
31
The Two-factor Factorial
Design -- Statistical Analysis – example
with no interaction
Analysis of Variance for Life, using Adjusted SS for Tests
Source
Material
Temp
Error
Total
DF Seq SS Adj SS Adj MS F P
2 10684 10684 5342
5.95 0.007
2 39119 39119 19559 21.78 0.000
31 27845 27845 898
35 77647
S = 29.9702 R-Sq = 64.14% R-Sq(adj) = 59.51%
32
The Two-factor Factorial
Design – One observation per cell


Single replicate
The effect model
y ij     i   j  ( ) ij   ij
 i  1, 2 ,..., a

 j  1, 2 ,..., b
33
The Two-factor Factorial
Design – One observation per cell

ANOVA table
34
The Two-factor Factorial
Design -- One observation per cell



The error variance is not estimable unless
interaction effect is zero
Needs Tuckey’s method to test if the
interaction exists.
Check page 183 for details.
35
The General Factorial Design



In general, there will be abc…n total
observations if there are n replicates of the
complete experiment.
There are a levels for factor A, b levels of
factor B, c levels of factor C,..so on.
We must have at least two replicate (n≧2)
to include all the possible interactions in
model.
36
The General Factorial Design


If all the factors are fixed, we may easily
formulate and test hypotheses about the
main effects and interaction effects using
ANOVA.
For example, the three factor analysis of
variance model:
y ijkl     i   j   k  ( ) ij  ( ) ik  (  ) jk  ( ) ijk   ijkl
 i  1, 2 ,..., a

j  1, 2 ,..., b

 k  1, 2 ,..., c
 l  1, 2 ,..., n
37

The General Factorial Design

ANOVA.
38
The General Factorial Design

where
a
SS T 
b
2
n
 y
i 1
SS A 
c
1
bcn
SS AB 
SS BC 
j 1 k 1 l 1
a
2
i ...
i 1
a
1
cn
1
an
SS ABC 


2
ijkl
abcn
2
y ....

y
i 1
a
2
ij ..

abcn
y . jk . 
a
b
SS E  SS T 
1
abcn
y
j 1
b
 SS A  SS B
2
. j ..
y ....

SS C 
abcn
SS AC 
1
bn
a
i 1 k 1

abn
y
k 1
2
.. k .

y ....
abcn
2
c

2
c
1
y
2
i.k .

y ....
abcn
 SS A  SS C
 SS B  SS C
y ....

abcn
j 1 k 1
 SS A  SS B  SS C  SS AB  SS BC  SS AC
2
c
y



n
i 1
acn
2
b
2
2
ijk .
j 1 k 1
a
2
....
c
y



n
i 1
y
2
j 1 k 1
1
1
2
y ....
j 1
c

SS B 
abcn
b
 y
y ....
2
ijk .

y ....
abcn
39
The General Factorial Design -example

Three
factors:
pressure,
percent of
carbonation,
and line
speed.
40
The General Factorial Design -example

ANOVA
41
Fitting Response Curve and
Surfaces



When factors are quantitative, one can fit a
response curve (surface) to the levels of the
factor so the experimenter can relate the
response to the factors.
These surface could be linear or quadratic.
Linear regression model is generally used
42
Fitting Response Curve and
Surfaces -- example


Battery life data
Factor temperature is quantitative
43
Fitting Response Curve and
Surfaces -- example




Example STATANOVA—GLM
Response  life
Model  temp, material temp*temp,
material*temp, material*temp*temp
Covariates  temp
44
Fitting Response Curve and
Surfaces -- example
coding method: -1, 0, +1
General Linear Model: Life versus Material
Factor Type Levels Values
Material fixed
3
1, 2, 3
Analysis of Variance for Life, using Sequential SS for Tests
Source
DF Seq SS Adj SS Seq MS
Temp
1 39042.7 1239.2 39042.7
Material
2 10683.7 1147.9 5341.9
Temp*Temp
1 76.1
76.1
76.1
Material*Temp
2 2315.1 7170.7 1157.5
Material*Temp*Temp
2 7298.7 7298.7 3649.3
Error
27 18230.8 18230.8 675.2
Total
35 77647.0
S = 25.9849 R-Sq = 76.52% R-Sq(adj) = 69.56%
Term
Coef
SE Coef T
P
Constant
153.92
11.87
12.96 0.000
Temp
-0.5906
0.4360
-1.35 0.187
Temp*Temp
-0.001019 0.003037 -0.34 0.740
Temp*Material
1
-1.9108 0.6166
-3.10
0.005
2
0.4173
0.6166
0.68
0.504
Temp*Temp*Material
1
0.013871 0.004295 3.23 0.003
2
-0.004642 0.004295 -1.08 0.289
Two kinds of coding methods:
1. 1, 0, -1
2. 0, 1, -1
F
57.82
7.91
0.11
1.71
5.40
P
0.000
0.002
0.740
0.199
0.011
45
Fitting Response Curve and
Surfaces -- example

Final regression equation:
Life  153 . 92  0 . 5906 * A  15 . 46 * B [1]
 5 . 7 * B [ 2 ]  0 . 001019 * A  1 . 9108 * AB [1]
2
 0 . 4173 * AB [ 2 ]  0 . 013871 * A B [1]  0 . 004642 * A B [ 2 ]
2
2
46
Fitting Response Curve and
Surfaces – example –3 factorial design
2



Tool life
Factors: cutting speed, total angle
Data are coded
47
Fitting Response Curve and
Surfaces – example –3 factorial design
2
48
Fitting Response Curve and
Surfaces – example –3 factorial design
2
Regression Analysis: Life versus Speed, Angle, ...
The regression equation is
Life = - 1068 + 14.5 Speed + 136 Angle - 4.08 Angle*Angle - 0.0496 Speed*Speed
- 1.86 Angle*Speed + 0.00640 Angle*Speed*Speed + 0.0560 Angle*Angle*Speed
- 0.000192 Angle*Angle*Speed*Speed
Predictor
Constant
Speed
Angle
Angle*Angle
Speed*Speed
Angle*Speed
Angle*Speed*Speed
Angle*Angle*Speed
Angle*Angle*Speed*Speed
Coef
SE Coef
-1068.0
702.2
14.480
9.503
136.30
72.61
-4.080
1.810
-0.04960 0.03164
-1.8640
0.9827
0.006400 0.003272
0.05600
0.02450
-0.00019200 0.00008158 -
T
-1.52
1.52
1.88
-2.25
-1.57
-1.90
1.96
2.29
2.35
P
0.163
0.162
0.093
0.051
0.151
0.090
0.082
0.048
0.043
S = 1.20185 R-Sq = 89.5% R-Sq(adj) = 80.2%
49
Fitting Response Curve and
Surfaces – example –3 factorial design
2
Analysis of Variance
Source
Regression
Residual Error
Total
DF
8
9
17
Source
Speed
Angle
Angle*Angle
Speed*Speed
Angle*Speed
Angle*Speed*Speed
Angle*Angle*Speed
Angle*Angle*Speed*Speed
SS
111.000
13.000
124.000
MS
13.875
1.444
DF
1
1
1
1
1
1
1
1
Seq SS
21.333
8.333
16.000
4.000
8.000
42.667
2.667
8.000
F
9.61
P
0.001
50
Fitting Response Curve and
Surfaces – example –3 factorial design
2
51
Blocking in a Factorial Design


We may have a nuisance factor presented in a
factorial design
Original two factor factorial model:
y ij     i   j  ( ) ij   ij

 i  1, 2 ,..., a

 j  1, 2 ,..., b
Two factor factorial design with a block factor
model:
y ijk     i   j  ( ) ij   k   ijk
 i  1, 2 ,... a

 j  1, 2 ,..., b
 k  1, 2 ,..., n

52
Blocking in a Factorial Design
53
Blocking in a Factorial Design - example



Response: intensity level
Factors: Ground cutter and filter type
Block factor: Operator
54
Blocking in a Factorial Design - example
General Linear Model: Intensity versus Clutter, Filter, Blocks
Factor Type Levels Values
Clutter fixed
3 High, Low, Medium
Filter fixed
2 1, 2
Blocks fixed 4 1, 2, 3, 4
Analysis of Variance for Intensity, using Sequential SS for Tests
Source
Clutter
Filter
Clutter*Filter
Blocks
Error
Total
DF
2
1
2
3
15
23
Seq SS
335.58
1066.67
77.08
402.17
166.33
2047.83
Adj SS
335.58
1066.67
77.08
402.17
166.33
Seq MS
167.79
1066.67
38.54
134.06
11.09
F
15.13
96.19
3.48
12.09
P
0.000
0.000
0.058
0.000
S = 3.33000 R-Sq = 91.88% R-Sq(adj) = 87.55%
55
Blocking in a Factorial Design - example
General Linear Model: Intensity versus Clutter, Filter, Blocks
Term
Coef
Constant
94.9167
Clutter
High
4.3333
Low
-4.7917
Filter
1
6.6667
Clutter*Filter
High 1
2.0833
Low 1
-2.2917
Blocks
1
0.417
2
1.583
3
4.583
SE Coef
0.6797
T
139.64
P
0.000
0.9613
0.9613
4.51
-4.98
0.000
0.000
0.6797
9.81
0.000
0.9613
0.9613
2.17
-2.38
0.047
0.031
1.177
1.177
1.177
0.35
1.34
3.89
0.728
0.199
0.001
56