Chapter 10D
Minimum Sample Size Demonstration Tests
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
10D - 3
10D - 1
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.
December 14, 2015
8h 9min
3.5
0.8
1
1.5
2
3
4.0
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.
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• Risk: The probability of passing the test will be small unless your reliability is MUCH larger than the level to be
demonstrated.
3.0
Test Length Factor k
2.5
3.5
Sample Size Needed to Demonstrate R = 0.9
as a Function of the Test Time−Length Factor
2.0
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4.0
3
2
1.5
1
0.8
Time and Sample Size Needed to Demonstrate
R = 0.90 to a Certain Time for Different Values of β.
1.5
Most Common Implementation
1 log(α)
×
.
k log(R)
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• Smaller sample sizes are possible if you can bound β higher.
• Is conservative if β > 1.
• Requires the assumption that there is no infant mortality.
n=
Assume β = 1 (constant hazard, or exponential distribution)
Number of Units Tested with Zero Failures
Sample Size Formula
log(α)
1
n= β×
k
log(R)
3.0
Test Length Factor k
2.5
Sample Size Needed to Demonstrate R = 0.99
as a Function of the Test Time−Length Factor
2.0
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15
10
where
• tc = k × t0 is the test length.
Number of Units Tested with Zero Failures
5
0
• (1 − α) is the confidence level (α = 0.05 for 95%).
• R is the reliability to be demonstrated.
• β is the Weibull shape parameter.
1.5
Time and Sample Size Needed to Demonstrate
R = 0.99 to a Certain Time for Different Values of β.
150
100
50
0
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.
10000
12000
Testing n = 2 Units Assuming beta = 1
8000
Failure−Free Testing Cycles
6000
14000
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16000
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• Detecting infant mortality is nearly impossible with small samples.
4000
Testing n = 2 Units Assuming β = 1
50%
80%
90%
95%
2000
log(α)
× log(R)
Probability of Successful Demonstration
1
Pr(pass test) = (True R at tc) kβ
= (True R at t0)log(α)/ log(R).
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Demonstrated Reliability at 4000 Cycles
"
log(α)
.
n × kβ
#
Solving the Sample Size Formula for R
R = exp
• tc = k × t0 is the test length.
• (1 − α) is the confidence level (0.05 for 95%).
• R is the reliability to be demonstrated.
50%
80%
95%
90%
80%
50%
90%
95%
2000
0.95
8000
10000
12000
Failure−Free Testing Cycles
6000
Testing n = 2 Units Assuming beta = 2
0.96
0.98
Actual Reliability
0.97
14000
0.99
Successfully Demonstrating That R= 0.9
Probability of Successfully
Demonstrating that R = 0.90
4000
Testing n = 2 Units Assuming β = 2
• β is the Weibull shape parameter.
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
Demonstrated Reliability at 4000 Cycles
Probability of Successful Demonstration
1.0
0.8
0.6
0.4
0.2
0.0
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16000
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1.00
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1.0
0.8
0.6
0.4
Probability of Successful Demonstration
0.995
50%
80%
90%
95%
Probability of Successfully
Demonstrating that R = 0.99
0.998
Actual Reliability
0.997
0.999
Successfully Demonstrating That R= 0.99
0.996
References
1.000
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10D - 15
• August 2004 Quality Progress article by Meeker, Hahn,
and Doganaksoy.
• Chapter 10 of Meeker and Escobar (1998).
0.2
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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