# Statistical Data Analysis: Primer ```Stat 470-18
•
Today: Start Chapter 10
A wood products company wishes to investigate the affect of several factors on the thickness swell of a manufactured wood
product. The goal of the experiment is to minimize the swelling after the wood has bee saturated with th water.
The manufacturing process has 2 steps. First, pieces of wood and additives are combined to form a batch. Next, the batch is split
into several sub-batches and processed into s sheet or plank of wood. It is possible to make between 2 and 4 sub-batches per
batch (it is the experimenter's choice).
A designed experiment is to be run to investigate the impact of factors at both stages. The factors in the first stage are
A,B,C,D,E. The second stage factors are labeled p, q and r.
Due to operational restrictions, the experiment is run by first fixing 5 factors affecting the batch and then creating a batch of raw
materials. Next the mixture is separated into sub-batches, keeping the first stage factors fixed. Thus a split-plot design results.
a.
b.
c.
d.
e.
f.
g.
h.
i.
Construct a 2(5+3)-(2+1) 32-run fractional factorial split-plot design.
Find the defining contrast sub-group of the design.
Write out the 32 treatments to be run and identify which trials are performed in the same sub-plot.
Which effects are assessed using the WP error, and which effects are assessed using the SP error?
Construct a 2(5+3)-(1+2) 32-run fractional factorial split-plot design.
Find the defining contrast sub-group of the design.
Which effects are assessed using the WP error, and which effects are assessed using the SP error?
Which design has less aberration?
If the mixing stage is the most expensive part of the experiment, which design is cheaper to run? Why?
Chapter 10:
Robust Parameter Design
• Robust parameter design is an experimentation technique which aims
to reduce system variation and also optimize the mean system response
• Idea is to use control factors to make the system robust to the
influences of noise factors
Example: Leaf Spring Experiment (p. 438)
•
Experiment was conducted to investigate the impact of a heat treatment
process on truck leaf springs where the target height of the springs is 8 inches
•
Experiment considered 5 factors, each at 2 levels:
– B: High heat treatment
– C: Heating time
– D: Transfer time
– E: Hold down time
– Q: Quench oil temperature
•
In regular production Q is not controllable, but can be in the experiment
Example: Leaf Spring Experiment (p. 438)
• 25-1 fractional factorial design was performed: I=BCDE
• Experiment has 3 replicates
Control Factors
B
C
D
E
-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
Q Level
7.78
8.15
7.50
7.59
7.94
7.69
7.56
7.56
-1
7.78
8.18
7.56
7.56
8.00
8.09
7.62
7.81
7.81
7.88
7.50
7.75
7.88
8.06
7.44
7.69
7.50
7.88
7.50
7.63
7.32
7.56
7.18
7.81
+1
7.25
7.88
7.56
7.75
7.44
7.69
7.18
7.50
7.12
7.44
7.50
7.56
7.44
7.62
7.25
7.59
Noise Factors
• Noise factors are factors that impact the system response, but in
practice are not controllable
• Examples include environmental factors, differing user conditions,
variation in process parameter settings, …
• Example: refrigerators are manufactured so that the interior
temperature remains close to some target
Variance Reduction Via Parameter Design
• Let x denote the control factor settings and z denote the noise factor
settings
• Relationship between the system response and the factors: y = f(x,z)
• If noise factors impact the response, then variation in the levels of z
will transmit this variance to the response, y
• If some noise and control factors interact, can potentially adjust levels
of control factors to dampen impact of noise factor variation
Variance Reduction Via Parameter Design
• Suppose there is one noise factor and two control factors
y    x1  x2 z  z  
• What is variance of y in practice?
• What does this imply?
Cross Array Strategy
• We will consider two types of design/analysis techniques for robust
parameter design
• The first one uses location-dispersion modeling (e.g., have a model for
the mean response and another for the variance) similar to the epitaxial
layer growth experiment in Chapter 3
• The design strategy for this technique is based on a cross array
Cross Array Strategy
• Consider the leaf spring example
• We can view this experiment as the combination of two separate
experimental designs
– Control array: design for the control factors
– Noise array: design for the noise factors
• Cross array: design consisting of all level combinations between the
control array and the noise array
• If there are N1 runs in the control array and N2 trials in the noise array,
then the cross array has N1 N2 trials
Cross Array Strategy
• Design for control factors:
• Design for noise factors:
Cross Array Strategy
• The responses are modeled using the location-dispersion approach
• The models include ONLY the control factors
2
• At each control factor setting, yi and ln( si ) are used as measures of
location and dispersion
• Factors that impact the mean are called location factors and those that
impact the variance are dispersion factors
• Location factors that are not dispersion factors are called adjustment
factors
Example: Leaf Spring
Control Factors
B
C
D
E
-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
yi
lnsi2 
7.540
7.902
7.520
7.640
7.670
7.785
7.372
7.660
-2.4075
-2.6488
-6.9486
-4.8384
-2.3987
-2.9392
-3.2697
-4.0582
Example: Leaf Spring
Effect
B
C
D
E
BC=DE
BD=CE
BE=CD
Estimate (Location)
0.22125
0.17625
0.02875
0.10375
0.01725
0.01975
-0.03525
Estimate (Dispersion)
0.1350
2.1802
-1.0444
-0.6499
-0.5259
0.7995
1.1852
-1.0 -0.5 0. DisperonEfcts 0.5 1.0 1.5 2.0
0. 0.5 LocatinEfes 0.1 0.15 0.2
Example: Leaf Spring
-1 .0
-0 .5
0 .0
0 .5
1 .0
Qu a n ti l e s
of
-1 .0
-0 .5
0 .0
0 .5
1 .0
S ta n d a rd
No rm
Qu a n ti l e s
of
Example: Leaf Spring
• Location Model:
yˆ  7.6360  0.1106 xB  0.0881xC  0.0519 xE
• Dispersion Model:
zˆ  3.6886  1.0901xC
• Level settings:
Two-Step Optimization Procedures
• Nominal the best problem:
– Select the levels of the dispersion factors to minimize the
dispersion
– The select the levels of the adjustment factors to move the process
on target
• Larger (Smaller) the better problem:
– Select levels of location factors to optimize process mean
– Select levels of dispersion factors that are not location factors to
minimize dispersion
• Leaf Spring Example was a nominal the best problem
Response Modeling
• There may be several noise factors and control factors in the
experiment
• The cross array approach identifies control factors to help adjust the
dispersion and location models, but does not identify which noise
factors interact with which control factors
• Cannot deduce the relationships between control and noise factors
• The response model approach explicitly model both control and noise
factors in a single model (called the response model)
Response Modeling
• Steps:
– Model response, y, as a function of both noise and control factors
(I.e., compute regression model with main effects and interactions
of both types of factors)
• make control by noise interaction plots for the significant
control by noise interactions. The control factor setting that
results in the flattest relationship gives the most robust setting.
• construct the variance model, and choose control factor
settings that minimize the variance
Example: Leaf Spring Experiment (p. 438)
• 25-1 fractional factorial design was performed: I=BCDE
• Experiment has 3 replicates
Control Factors
B
C
D
E
-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
Q Level
7.78
8.15
7.50
7.59
7.94
7.69
7.56
7.56
-1
7.78
8.18
7.56
7.56
8.00
8.09
7.62
7.81
7.81
7.88
7.50
7.75
7.88
8.06
7.44
7.69
7.50
7.88
7.50
7.63
7.32
7.56
7.18
7.81
+1
7.25
7.88
7.56
7.75
7.44
7.69
7.18
7.50
7.12
7.44
7.50
7.56
7.44
7.62
7.25
7.59
Example: Leaf Spring Experiment (p. 438)
•
25-1 fractional factorial design was performed: I=BCDE
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
E
-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
7.79
8.07
7.52
7.63
7.94
7.95
7.54
7.69
7.29
7.73
7.52
7.65
7.40
7.62
7.20
7.63
Example: Leaf Spring Experiment (p. 438)
Effect
B
C
D
E
Q
BC=DE
BD=CE
BE=CD
BQ
CQ
DQ
EQ
BCQ=DEQ
BDQ=CEQ
BEQ=CDQ
Estimate
0.221
-0.176
-0.029
0.104
-0.260
-0.017
-0.020
-0.035
0.085
0.165
-0.054
0.027
-0.010
0.040
-0.047
E
f
e
c
t
s
i
m
a
-0.2 -0.1 0. 0.1 0.2
Example: Leaf Spring Experiment (p. 438)
-1
0
1
Qu a n ti l e s
of
Example: Leaf Spring Experiment (p. 438)
• Response Model:
Example: Leaf Spring Experiment (p. 438)
m
e
a
n
o
f
Y
7.5 7.6 7. 7.8 7.9
C
-1
1
-1
1
Q
Example: Leaf Spring Experiment (p. 438)
• Variance Model:
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