Chapter 8A
Population and Process Comparison
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
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.
January 13, 2014
3h 41min
8A - 1
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.
• Describe generalization of the procedures to three or more
populations.
8A - 2
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)
8A - 3
Separate Normal Distribution ML Estimates for the
Snubber Designs (Nelson 1982)
.6
OldDesign
NewDesign
.5
.4
Fraction Failing
.3
.2
.1
.05
.02
.01
.005
-400
-200
0
200
400
600
800
1000
1200
1400
Toaster Cycles
8A - 4
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).
b new − µ
b old = 1126 − 908 = 218
µ
sceµbnew −µbold =
r
2
c
sce2
+
s
e
µ
bnew
µ
bold =
q
(76.2)2 + (123)2 = 144.7
• Approximate 95% confidence interval for ∆ = µnew − µold
is
[∆,
e
˜ = µ
b new − µ
b old ± z(1−α/2)sc
∆]
eµbnew −µbold
= 218 ± 1.96 × 144.7 = [−66,
501].
Interval contains 0 thus the difference between the means
could be zero.
8A - 5
Common σ Normal Distribution ML Estimates from
the Old and New Snubber Designs
.8
Old
New
.7
.6
.5
.4
Fraction Failing
.3
.2
.1
.05
.02
.01
.005
.002
.001
.0005
.0001
-500
0
500
1000
1500
Toaster Cycles
8A - 6
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.
• Substituting x = 0, 1 into the model gives
µ(0) = β0, for the old design
µ(1) = β0 + β1, for the new design
• Model assumes that σ is the same for both designs.
• Note that ∆ = tp(1) − tp(0) = µ(1) − µ(0) = β1, so ∆ does
not depend on the choice of which quantile to compare.
• [β1,
e
β˜1] = βb1±z1−α/2sceβb = 86.7±1.96×114 = [−137,
1
311]
8A - 7
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.
8A - 8
R Proc Reliability
SAS
Weibull Probability Plot for 6MP Drug
(Gehan 1965)
99.9
Percent
99
95
90
80
70
60
50
40
30
20
10
5
2
1
1
GROUP
10
Weeks
1
2
50
8A - 9