Chapter 8A
Population and Process Comparison
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
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Copyright 1998-2008 W. Q. Meeker and L. A. Escobar.
Based on the authors’ text Statistical Methods for Reliability
Data, John Wiley & Sons Inc. 1998.
December 14, 2015
8h 9min
Snubber Life Test Data
• A snubber is a component in a toaster.
• Multiple censoring due to another failure mode
• Purpose of the test was to compare the two different designs.
• Data first presented in Nelson (1982)
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Comparison of Snubber Designs—Separate Analyses
(Model 1: different σ’s)
• In general comparison complicated. What should we compare? Typical choice: specified quantile or F (t) at a specified t.
• Compare the t.5 (also µ for the normal distribution).
r
q
2
2
2
c2
sceµ
=
bnew + seµ
bold = (76.2) + (123) = 144.7
b new − µ
b old = 1126 − 908 = 218
µ
sceµbnew −µbold
501].
˜ = µ
b new − µ
b old ± z(1−α/2)sc
eµbnew −µbold
∆]
• Approximate 95% confidence interval for ∆ = µnew − µold
is
e
[∆,
= 218 ± 1.96 × 144.7 = [−66,
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Interval contains 0 thus the difference between the means
could be zero.
Chapter 8A
Comparing Populations or Processes
Objectives
• Describe general issues in comparing two or more processes
or populations.
• Describe the comparison of two population means without
making assumptions on the population variances.
• Describe the comparison of two population means assuming
that the population variances are equal.
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• Describe generalization of the procedures to three or more
populations.
-400
-200
OldDesign
NewDesign
0
200
600
Toaster Cycles
400
800
1000
1200
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1400
Separate Normal Distribution ML Estimates for the
Snubber Designs (Nelson 1982)
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
.001
.0005
.0001
-500
Old
New
0
Toaster Cycles
500
1000
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1500
Common σ Normal Distribution ML Estimates from
the Old and New Snubber Designs
Fraction Failing
Fraction Failing
Comparison of Snubber Designs—Dummy Variable
Regression Analyses (Model 2: common σ)
• Simple regression model using µ = β0 + β1x where x = 0
for old design and x = 1 for the new design.
for the new design
for the old design
• Substituting x = 0, 1 into the model gives
µ(0) = β0,
µ(1) = β0 + β1,
• Model assumes that σ is the same for both designs.
1
10
Weeks
1
2
R Proc Reliability
SAS
Weibull Probability Plot for 6MP Drug
(Gehan 1965)
99
99.9
95
90
80
70
60
50
40
30
20
5
10
2
1
1
GROUP
50
311]
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β˜1] = βb1±z1−α/2sceβb = 86.7±1.96×114 = [−137,
• Note that ∆ = tp(1) − tp(0) = µ(1) − µ(0) = β1, so ∆ does
not depend on the choice of which quantile to compare.
e
• [β1,
Percent
Example: 6MP Drug
• Gehan (1965) gives remission-times for leukemia patients.
• Notice the greater dispersion in the treated group. Also
censoring occurs in the treated group but not in the control
group.
• It is of interest to assess the drug effect.
• Also want to find a parametric model to describe the treated
group.
• A question of interest is the existence of a threshold parameter for the treated group.
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