Stat 470-11
•
Today: More Chapter 3
Analysis of Location and Dispersion Effects
• The epitaxial growth layer experiment is a 24 factorial design
• Have looked at ways to analyze response of a factorial experiment
– Plotting effects on a normal probability plot
– Regression
• May wish to model mean and also the variance
Analysis of Location and Dispersion Effects
•
Recall, from Section 3.2, the quadratic loss function
L( y , t )  c ( y  t ) 2
•
The expected loss E(y,t)=cVar(y)+c(E(y)-t)2 suggested
1. Selecting levels of some factors to minimize V(y)
2. Selecting levels of other factors to adjust the mean as close as possible to
the target, t.
•
Need a model for the variance (dispersion)
Analysis of Location and Dispersion Effects
1
yi 
ni
ni
•
 yij be the sample mean of observations taken at the ith
Let
j 1
treatment of the experiment
•
Let si2 be the sample variance of observations taken at the ith treatment
of the experiment
n
i
1

( yij  yi ) 2

(ni  1) j 1
•
That is,
•
Can model both the mean and variance using regression
si2
Analysis of Location and Dispersion Effects
•
Would like to model the variance as a function of the factors
•
Regression assumes that quantities measured at each treatment be
normally distributed
•
•
Is it likely that
si2
n
i
1

( yij  yi ) 2

(ni  1) j 1
is normally distributed?
Example: Original Growth Layer Experiment
A
-1
-1
-1
-1
-1
-1
-1
-1
+1
+1
+1
+1
+1
+1
+1
+1
B
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
C
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
D
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
14.812
13.768
14.722
13.860
14.886
14.182
14.758
13.996
15.272
14.324
13.918
13.614
14.648
13.970
14.184
13.866
14.774
13.778
14.736
13.876
14.810
14.172
14.784
13.988
14.656
14.092
14.044
13.202
14.350
14.448
14.402
14.130
Thickness
14.772
14.794
13.870
13.896
14.774
14.778
13.932
13.846
14.868
14.876
14.126
14.274
15.054
15.058
14.044
14.028
14.258
14.718
13.536
13.588
14.926
14.962
13.704
14.264
14.682
15.034
14.326
13.970
15.544
15.424
14.256
14.000
14.860
13.932
14.682
13.896
14.958
14.154
14.938
14.108
15.198
13.964
14.504
14.432
15.384
13.738
15.036
13.640
14.914
13.914
14.850
13.870
14.932
14.082
14.936
14.060
15.490
14.328
14.136
14.228
15.170
13.738
14.470
13.592
Example: Original Growth Layer Experiment
A
B
C
D
-1
-1
-1
-1
-1
-1
-1
-1
+1
+1
+1
+1
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
y
14.82
13.86
14.76
13.88
14.89
14.17
14.92
14.04
14.93
13.97
14.42
13.91
14.88
14.03
14.84
13.91
s2
.0031
.0049
.0033
.0009
.0027
.0041
.0165
.0020
.2148
.1205
.2061
.2260
.1471
.0880
.3268
.0704
ln(s2)
-5.771
-5.311
-5.704
-6.984
-5.917
-5.485
-4.107
-6.237
-1.538
-2.116
-1.579
-1.487
-1.916
-2.430
-1.118
-2.653
Example: Original Growth Layer Experiment
•
Model Matrix for a single replicate:
A
-1
-1
-1
-1
-1
-1
-1
-1
+1
+1
+1
+1
+1
+1
+1
+1
B
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
C
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
D
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
AB
+1
+1
+1
+1
-1
-1
-1
-1
-1
-1
-1
-1
+1
+1
+1
+1
AC
+1
+1
-1
-1
+1
+1
-1
-1
-1
-1
+1
+1
-1
-1
+1
+1
AD
-1
+1
-1
+1
-1
+1
-1
+1
+1
-1
+1
-1
+1
-1
+1
-1
BC
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
BD
-1
+1
-1
+1
+1
-1
+1
-1
-1
+1
-1
+1
+1
-1
+1
-1
CD
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
ABC
-1
-1
+1
+1
+1
+1
-1
-1
+1
+1
-1
-1
-1
-1
+1
+1
ABD
+1
-1
+1
-1
-1
+1
-1
+1
-1
+1
-1
+1
+1
-1
+1
-1
ACD
+1
-1
-1
+1
+1
-1
-1
+1
-1
+1
+1
-1
-1
+1
+1
-1
BCD
+1
-1
-1
+1
-1
+1
+1
-1
+1
-1
-1
+1
-1
+1
+1
-1
ABCD
-1
+1
+1
-1
+1
-1
-1
+1
+1
-1
-1
+1
-1
+1
+1
-1
Example: Original Growth Layer Experiment
Effect Estimates and QQ-Plot:
Effect
A
B
C
D
AB
AC
AD
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD
Estimate
0.055
0.142
-0.109
0.836
-0.032
-0.074
-0.025
0.047
0.010
-0.037
0.060
0.067
-0.056
0.098
0.036
y
0. 0.2 0.4 0.6 0.8
•
-1
0
1
Qu a n ti l e s
of
Example: Original Growth Layer Experiment
• Regression equation for the mean response:
Example: Original Growth Layer Experiment
Dispersion analysis:
Effect
A
B
C
D
AB
AC
AD
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD
Estimate
3.834
0.078
0.077
0.632
-0.428
0.214
0.002
0.331
0.305
0.582
-0.335
0.086
-0.494
0.314
0.109
l
n
(
s
2
)
0 1 2 3 4
•
-1
0
1
Qu a n ti l e s
o
Example: Original Growth Layer Experiment
• Regression equation for the ln(s2) response:
Example: Original Growth Layer Experiment
• Suggested settings for the process:
Example: Original Growth Layer Experiment
• Suggested settings for the process in the original units of the factors:
Location-Dispersion Modeling
• Steps:
Example
•
An experiment was conducted to improve a heat treatment process on
truck leaf springs
•
The heat treatment process, which forms the curvature of the leaf
spring, consists of
1. Heating in a furnace
2. Processing by machine forming
3. Quenching in an oil bath
•
The height of an unloaded spring, known as the free height, is the
quality characteristic of interest and has a target of 8 inches
Example
•
The experiment goals are to
1. Minimize the variability about the target
2. Keep the process mean as close to the target of 8 inches as possible
•
A 24 factorial experiment was conducted with factors:
•
•
•
•
•
A. Furnace Temperature
B. Heating Time
C. Transfer Time
Q. Quench Oil Temperature
There were 3 replicates of the experiment
Example
•
Data
A
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
B
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
C
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
Q
-1
-1
-1
-1
-1
-1
-1
-1
+1
+1
+1
+1
+1
+1
+1
+1
Free Height (inches)
7.78 7.78
7.81
8.15 8.18
7.88
7.50 7.56
7.50
7.59 7.56
7.75
7.94 8.00
7.88
7.69 8.09
8.06
7.56 7.62
7.44
7.56 7.81
7.69
7.50 7.25
7.12
7.88 7.88
7.44
7.50 7.56
7.50
7.63 7.75
7.56
7.32 7.44
7.44
7.56 7.69
7.62
7.18 7.18
7.25
7.81 7.50
7.59
Example
•
Data
A
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
-1
+1
B
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
+1
+1
-1
-1
C
+1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
+1
-1
-1
-1
-1
Q
-1
-1
-1
-1
-1
-1
-1
-1
+1
+1
+1
+1
+1
+1
+1
+1
y
7.7900
8.0700
7.5200
7.6333
7.9400
7.9467
7.5400
7.6867
7.2900
7.7333
7.5200
7.6467
7.4000
7.6233
7.2033
7.6333
s2
.0003
.0273
.0012
.0104
.0036
.0496
.0084
.0156
.0373
.0645
.0012
.0092
.0048
.0042
.0016
.0254
ln(s2)
-8.1117
-3.6009
-6.7254
-4.5627
-5.6268
-3.0031
-4.7795
-4.1583
-3.2888
-2.7406
-6.7254
-4.6849
-5.3391
-5.4648
-6.4171
-3.6717
Example: Location Model
Term
A
B
C
Q
AB
AC
AQ
BC
BQ
CQ
ABC
ABQ
ACQ
BCQ
ABCQ
Estimated
Effect
Regression Estimates
Coefficient
.111
.222
.088
.176
.014
.029
-.130
-.260
.009
.017
.010
.020
.042
.085
-.018
-.035
-.083
-.165
.027
.054
.052
.104
.005
.010
-.020
-.040
-.024
-.047
.014
.027
-0.10
-0.05
x
0.0
0.05
0.10
Example: Location Model
-1
0
Quantiles of Standard Normal
1
Example
• Regression equation for the mean response:
Example: Dispersion Model
Term
A
B
C
Q
AB
AC
AQ
BC
BQ
CQ
ABC
ABQ
ACQ
BCQ
ABCQ
Estimated
Effect
Regression Estimates
Coefficient
.945
1.890
.284
.568
-.124
-.248
.140
.280
-.001
-.002
.212
.424
-.294
-.588
.335
.670
.299
.598
.555
1.110
.108
.216
-.545
-1.090
-.216
-.432
.427
.854
.065
.129
0.0
-0.5
-1.0
x
0.5
1.0
Example: Dispersion Model
-1
0
Quantiles of Standard Normal
1
Example
• Regression equation for the dispersion responses: