Chapter 10D
Minimum Sample Size Demonstration Tests
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
Copyright 1998-2014 W. Q. Meeker and L. A. Escobar.
Complements to the authors’ text Statistical Methods for
Reliability Data, John Wiley & Sons Inc., 1998.
February 25, 2014
9h 23min
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Basic Ideas
• Want to demonstrate reliability S(t0) = Pr(T > t0) is at
least R (e.g. R = 0.99 or R = 0.999).
• Test a small number of units for a long time (e.g., large
number of operations). Denote this censoring time by tc.
• Pass the test if there are 0 failures.
• Required: Specification of the Weibull shape parameter.
• Risk: The probability of passing the test will be small unless your reliability is MUCH larger than the level to be
demonstrated.
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Sample Size Formula
log(α)
1
n= β×
k
log(R)
where
• tc = k × t0 is the test length.
• (1 − α) is the confidence level (α = 0.05 for 95%).
• R is the reliability to be demonstrated.
• β is the Weibull shape parameter.
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Time and Sample Size Needed to Demonstrate
R = 0.90 to a Certain Time for Different Values of β.
15
10
0.8
5
1
1.5
2
3
0
Number of Units Tested with Zero Failures
Sample Size Needed to Demonstrate R = 0.9
as a Function of the Test Time−Length Factor
1.5
2.0
2.5
3.0
3.5
4.0
Test Length Factor k
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Time and Sample Size Needed to Demonstrate
R = 0.99 to a Certain Time for Different Values of β.
150
100
0.8
50
1
1.5
2
3
0
Number of Units Tested with Zero Failures
Sample Size Needed to Demonstrate R = 0.99
as a Function of the Test Time−Length Factor
1.5
2.0
2.5
3.0
3.5
4.0
Test Length Factor k
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Most Common Implementation
Assume β = 1 (constant hazard, or exponential distribution)
1 log(α)
n= ×
.
k log(R)
• Requires the assumption that there is no infant mortality.
• Is conservative if β > 1.
• Smaller sample sizes are possible if you can bound β higher.
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Common Implementation of the
Minimum Sample Size Test
• Use β = 1 (assume constant hazard), as this is conservative if we are sure that the failure mode is wearout (β > 1).
• Risk: If there is infant mortality, you are anticonservative
by assuming β = 1.
• Detecting infant mortality is nearly impossible with small samples.
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Solving the Sample Size Formula for R
R = exp
"
#
log(α)
.
β
n×k
• tc = k × t0 is the test length.
• (1 − α) is the confidence level (0.05 for 95%).
• R is the reliability to be demonstrated.
• β is the Weibull shape parameter.
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Testing n = 2 Units Assuming β = 1
1.0
0.8
0.6
0.4
50%
0.2
80%
90%
95%
0.0
Demonstrated Reliability at 4000 Cycles
Testing n = 2 Units Assuming beta = 1
2000
4000
6000
8000
10000
12000
14000
16000
Failure−Free Testing Cycles
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Testing n = 2 Units Assuming β = 2
1.0
0.8
0.6
0.4
0.2
0.0
Demonstrated Reliability at 4000 Cycles
Testing n = 2 Units Assuming beta = 2
50%
80%
90%
95%
2000
4000
6000
8000
10000
12000
14000
16000
Failure−Free Testing Cycles
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Probability of Successful Demonstration
Pr(pass test) = (True R at
1 × log(α)
tc) kβ log(R)
= (True R at t0)log(α)/ log(R).
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Probability of Successfully
Demonstrating that R = 0.90
1.0
0.8
0.6
50%
80%
0.4
Probability of Successful Demonstration
Successfully Demonstrating That R= 0.9
90%
95%
0.95
0.96
0.97
0.98
0.99
1.00
Actual Reliability
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Probability of Successfully
Demonstrating that R = 0.99
1.0
0.8
0.6
50%
0.4
80%
90%
95%
0.2
Probability of Successful Demonstration
Successfully Demonstrating That R= 0.99
0.995
0.996
0.997
0.998
0.999
1.000
Actual Reliability
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Alternatives and Extensions
• Designing a test that allows more failures will provide a
higher probability of successful demonstration (but this requires a larger sample size).
• Can develop similar tests that require estimation of both
Weibull parameters (first part of Chapter 10), but sample
size requirements go up dramatically.
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References
• Chapter 10 of Meeker and Escobar (1998).
• August 2004 Quality Progress article by Meeker, Hahn,
and Doganaksoy.
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