Chapter 10A
Planning Tests to Compare Populations or Processes
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
Copyright 1998-2010 W. Q. Meeker and L. A. Escobar.
Complements to the authors’ text Statistical Methods for
Reliability Data, John Wiley & Sons Inc., 1998.
January 13, 2014
3h 42min
10A - 1
Chapter 10A
Planning Comparison of Populations or Processes
Objectives
• Describe general issues in planning two or more processes
or populations.
• Describe the planning to compare two population means or
quantiles assuming that the population variances are equal.
• Describe generalization of the procedures to compare populations with different variances.
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Comparison of 2 Location Parameters
µ2
µ1
∆
d= ∆ /σ
µ2
µ1
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Comparing Populations or Processes
• Product design decisions often require choosing the best
from among k different populations or processes.
• Suppose that response (e.g., failure time or strength)
from population i follows a log-location-scale (e.g., Weibull
or lognormal) distribution
!
log(t) − µi
F (t; µi, σ) = Φ
.
σ
• Parameter µi varies; constant σ.
• Specific interest in choosing the population with the largest
distribution quantile value (tp)i = exp[µi + σΦ−1(p)], i =
1, . . . , k among the k populations.
• With constant σ the ordering of the (tp)i values is the same
as the ordering of the µi values.
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Tests Comparing k Populations or Processes
• Samples of equal size n to be taken from each population.
• If interest centers on the lower tail of the distribution, there
is little reason to wait for all units to fail (in a life test).
• In some situations data may be censored. For example
◮ Testing all units from all populations simultaneously
until a given censoring time tc results in Type I censored
data.
◮ Testing each population (perhaps sequentially because
of a limitation in test positions) until a given number of
failures occurs results in Type II censored data.
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Tests Comparing k Populations or
Processes–Continued
• To select the population with the largest quantile, if σ
is the same in each group, choose the population with
b i. To plan the test, supposing that µ1 >
the largest µ
max {µ2, . . . , µk }, one would want an assessment of
b 1 > max {µ
b 2, . . . , µ
b k }) .
Pr(CS) = Pr(Correct Selection) = Pr (µ
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Comparison of 5 Location Parameters
µ2
µ4
µ1
µ5 µ3
d= ∆ /σ
µ2 = µ3 = µ4 = µ5
∆
µ1
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General Formula to Compute Pr(CS)
when Selecting the Population with the Largest µ
• Notation:
b 1, . . . , µ
bk) :
f (µ
b 1, . . . , µ
bk)
joint distribution of (µ
b 1) : marginal of µ
b1
f (µ
b 2, . . . , µ
bk | µ
b 1) : conditional distribution of (µ
b 2, . . . , µ
b k ) given µ
b1
f (µ
• The general formula to compute Pr(CS) is
b2 ≤ µ
b 1, . . . , µ
bk ≤ µ
b 1)
Pr(CS) = Pr (µ
=
Z ∞
−∞
b 1)dµ
b 1.
b2 ≤ µ
b 1, . . . , µ
bk ≤ µ
b1 | µ
b 1 ) f (µ
Pr (µ
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Probability of Correct Selection when Choosing the
b i Based on Type II Censored Data
Largest µ
from k Populations
• With Type II censoring resulting in r of n failures from each
population and any log-location-scale distribution, there is
a relatively simple expression for Pr(CS).
• For the case µ1 > µ2 = µ3 = · · · = µk , which is the least
favorable situation to choose the correct population, the
probability of correct selection simplifies to
b 1 > max {µ
b 2, . . . , µ
b k })
Pr(CS) = Pr (µ
h
n
o
= Pr Zb1 > max Zb2, . . . , Zbk − d
b i − µi)/σ and d = (µ1 − µ2)/σ.
where Zbi = (µ
i
,
• The joint distribution of (Zb1, . . . , Zbk ) depends only on the
definition of Φ, n, and r and can be evaluated effectively
with simulation without having to specify d.
• Simulate with given n and r. Plot Pr(CS) versus d. Repeat
for different n and r combinations.
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Probability of Correct Selection when Choosing the
b i Based on Type II Censored Data
Smallest µ
from k Populations
• There is a similar expression for the probability of correct
selecting the smallest µi from a log-location-scale distribution.
• For the case µ1 < µ2 = µ3 = · · · = µk , the probability of
correct selection simplifies to
b 1 < min {µ
b 2, . . . , µ
b k })
Pr(CS) = Pr (µ
n
o
i
b
b
b
= Pr Z1 < min Z2, . . . , Zk − d ,
h
b i − µi)/σ and d = (µ1 − µ2)/σ.
where Zbi = (µ
• The joint distribution of (Zb1, . . . , Zbk ) depends only on the
definition of Φ, n, and r and can be evaluated effectively
with simulation without having to specify d.
• Simulate with given n and r. Plot Pr(CS) versus d. Repeat
for different n and r combinations.
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Elicitation of a Planning Value for d
• For a log-location-scale distribution, a planning value d✷ for
d can be obtained as a function of a planning value for σ
and a specified increase on life of interest.
• Suppose that there is interest in detecting a percent increase of p, 0 < p < 100, or more on life and that a planning
value σ ✷ for σ is available. Then
◮ Equating 1 + p/100 to the increase on life, one gets
exp µ1 + σz(1−α)
p
1+
=
100
exp µ2 + σz(1−α)
Taking logarithms
= exp (µ1 − µ2) .
p
∆ = µ1 − µ2 = log 1 +
.
100
◮ Then
1
p
∆
.
=
log
1
+
d✷ =
✷
✷
σ
σ
100
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Probability of Correct Selection when Choosing the
Largest Weibull Characteristic Life Based on Type II
Censored Data from k = 2 Populations
Probability of Correctly Selecting the Largest
Weibull Population out of 2 Populations
Pr(Correct Selection)
1.0
0.9
0.8
n=40, r=24
n=20, r=12
n=10, r=6
n=5, r=3
0.7
0.6
0.5
0.0
0.5
1.0
1.5
2.0
Standardized Difference (d)
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Probability of Correct Selection when Choosing the
Largest Weibull Characteristic Life Based on Type II
Censored Data from k = 5 Populations
Probability of Correctly Selecting the Largest
Weibull Population out of 5 Populations
Pr(Correct Selection)
1.0
0.8
0.6
n=40, r=24
n=20, r=12
n=10, r=6
n=5, r=3
0.4
0.2
0.0
0.5
1.0
1.5
2.0
Standardized Difference (d)
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Probability of Correct Selection when Choosing the
Largest Log-Mean based on Type I Censored Data
from k Populations
• With Type I censoring, test units from all populations are
tested until a prespecified censoring time tc.
• The expression for Pr(CS) is the same as with Type II censoring but the joint distribution of (Zb1, . . . , Zbk ) depends on
the definition of Φ, n, (log(tc) − µ1)/σ, and (log(tc) − µ2)/σ.
• Pr(CS) can be evaluated with simulation, but a new simulation is needed for every different specified combination of
n, µ1, µ2, σ, and tc (or n, µ1/σ, µ2/σ, and log(tc)/σ).
• There may be a possibility of 0 failures (if so, this will often
show up in the simulation).
• If the sample size n in each group is large enough, the
Pr(CS) versus d curves for Type II censoring provide an approximation to the sample size needed for Type I censoring.
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Extensions
• Results for the selection of the distribution with the smallest quantile are similar.
• Evaluations when σ values differ among populations. Specification of population characteristics is much more complicated.
• Other censoring schemes, and stopping rules. For example,
stop testing when at least r units fail in each group or at
time tc, which ever comes first.
• Dynamic rule to allow termination of simultaneous testing
when there is confidence that Pr(CS) is at least a specified
value.
• Non-log-location-scale distributions (e.g., gamma).
• Bayesian approaches to allow incorporating prior information.
• Estimation of Pr(CS) after data have been collected.
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Special Case: Probability of Correct Selection for
Complete Samples from a Lognormal Distribution
b i is
• For complete samples from a lognormal distribution µ
the mean of the logtimes to failure.
• Then
b i ∼ NOR(µi, σ 2/n)
µ
• In this case there are some simpler formulas to compute
Pr(CS) and the required sample size to obtain a prespecified
Pr(CS).
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b 2, . . . , µ
bk)
Special Case: Conditional Distribution of (µ
for Underlying Lognormal Distributions and
Complete Data
b i denote the sample mean of the log failure times from
Let µ
the sample from population i.
b2 ≤ µ
b1 . . . , µ
bk ≤ µ
b1 | µ
b 1) =
Pr (µ
=
=
where z1 =
√
k
Y
i=2
k
Y
i=2
k
Y
i=2
bi ≤ µ
b 1)
Pr (µ
√
Φnor
b 1 − µi )
n(µ
σ
!
√ Φnor z1 + di n ,
b 1 − µ1)/σ and di = (µ1 − µi)/σ.
n(µ
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Special Case: Probability of Correct Selection when
Choosing the Largest Log-Mean with Complete Data
from k Lognormal Populations
• Although we would expect all of the µi values to be different,
a conservative (i.e., lower) value of Pr(CS) is obtained by
supposing that there there is a best population and that all
of the k − 1 other populations have the same value of µ
(i.e., µ1 > µ2 = µ3 = · · · = µk ).
• In this case,
Pr(CS) =
Z ∞ h
−∞
√ ik−1
Φnor z1 + d n
φnor (z1) dz1,
where d = (µ1 − µ2)/σ.
• In this case, Pr(CS) is easy to compute numerically.
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Special Case: Sample Size to Give a Specified
Probability of Correct Selection when Choosing the
Largest Log-Mean with Complete Data from k
Lognormal Populations
• The sample size required to obtain a probability of correct
selection equal to 1 − α, when detection of a difference of
µ1 − µ2 is important, is given by
n = c2/d2
where d = (µ1 − µ2)/σ and c is the solution to the equation
1−α =
Z ∞
−∞
[Φnor (z1 + c)]k−1 φnor (z1) dz1.
• To identify small differences one needs large samples and
to identify large differences smaller samples are sufficient.
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