Use of Sensitivity Analysis
Overview
to Assess the Effect of Model Uncertainty
in Analyzing Accelerated Life Test Data
• Accelerated life tests
• Spring accelerated life test data
William Q. Meeker
• Graphical analysis
Department of Statistics
• Acceleration model and analysis
Iowa State University
• Inference at use conditions and assessment of sampling
error
Ames, IA 50011
www.public.iastate.edu/˜wqmeeker
• Using sensitivity analysis to assess possible model error
• Software
August 3, 2001
• Concluding remarks
Work done in collaboration with Luis Escobar, Louisiana State University
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2
Spring Accelerated Life Test Data
Pairs Plot
Spring Fatigue Data
55
60
65
70
New
Old
5000
50
3000
Weibull Distribution
Log Location Scale Family Member
70
0 1000
Kilocycles
60
65

F (t) = Pr(T ≤ t) = 1 − exp −
50
55
Stroke
900
= Φsev
η
log(t) − µ
,
σ
t>0
where µ = log(η), σ = 1/β, and Φsev (z) = 1 − exp[− exp(z)].
Old
500
700
Temp
β 
t 
New
Method
0
1000
2000
3000
4000
5000
500
600
700
800
900
1000
3
4
Spring Accelerated Life Test Data
Weibull Multiple Probability Plot
Individual ML Estimates of F (t)
Spring Accelerated Life Test Data
Weibull Multiple Probability Plot
Individual ML Estimates of F (t)
Spring Fatigue Data
With Individual Weibull Distribution ML Estimates
Weibull Probability Plot
50Stroke;500Temp;OldMethod
50Stroke;1000Temp;NewMethod
50Stroke;1000Temp;OldMethod
60Stroke;500Temp;NewMethod
60Stroke;500Temp;OldMethod
60Stroke;1000Temp;NewMethod
60Stroke;1000Temp;OldMethod
70Stroke;500Temp;NewMethod
70Stroke;500Temp;OldMethod
70Stroke;1000Temp;NewMethod
70Stroke;1000Temp;OldMethod
.9
.7
.5
.3
.2
.98
.9
.7
.5
Fraction Failing
.98
Fraction Failing
Spring Fatigue Data
With Individual Weibull Distribution ML Estimates
Weibull Probability Plot
.1
.3
.2
.1
.05
.05
.03
.03
.02
.02
.01
.01
1
10
100
1000
10000
Kilocycles
50
100
200
500
1000
2000
5000
Kilocycles
5
6
Weibull Multiple Probability Plot
Individual ML Estimates of F (t),
Common Weibull Shape Parameter (Floating Scale)
Spring Accelerated Life Test Data
Weibull Residual Probability Plot
Floating Scale Model
Spring Fatigue Data subset Estimable Subsets Model MLE
Stroke;Temp;Methodclass, Dist:Weibull
Weibull Probability Plot
Spring Fatigue Data subset Estimable Subsets
Residual Probability Plot
Stroke;Temp;Methodclass, Dist:Weibull
Weibull Probability Plot
.98
.98
.9
.9
.7
.7
.5
.5
Fraction Failing
.3
Probability
.3
.2
.2
.1
.1
.05
.05
.03
.02
.03
.02
.01
.01
.005
50
100
200
500
1000
2000
5000
0.02
0.05
0.10
Kilocycles
0.20
0.50
1.00
2.00
5.00
Standardized Residuals
7
8
Spring Accelerated Life Test Data
Weibull Multiple Probability Plot
ML Response Surface Estimates of F (t)
Spring Accelerated Life Test
Weibull Distribution Response Surface Model
F (t; µ, σ) = Φsev
Spring Fatigue Data Model MLE
StrokeLog, TempLinear, MethodClass, Dist:Weibull
Weibull Probability Plot
log(t) − µ
,
σ
.98
t>0
.9
.7
Fraction Failing
.5
µ = β0 + β1 log(Stroke) + β2Temp + β3Method
σ = constant
.3
.2
.1
.05
.03
where Method = 0 for New and Method = 1 for Old.
.02
.01
50
100
200
500
1000
2000
5000
Kilocycles
9
10
Spring Accelerated Life Test Data
Weibull Distribution Response Surface Model
Residuals Versus Fitted Values
Spring Accelerated Life Test Data
Conditional Model Plot
Spring Life versus Method
Spring Fatigue Data Conditional Model Plot
StrokeLog, TempLinear, MethodClass, Dist:Weibull
Fixed values of Stroke=20, Temp=600
Spring Fatigue Data
Residuals versus Fitted Values
StrokeLog, TempLinear, MethodClass, Dist:Weibull
5.00
Old
1.00
0.50
Method
0.20
0.10
New
Standardized Residuals
2.00
0.05
10%
50%
90%
0.02
500
1000
2000
5000
10000
20000
50000
Fitted Values
5
10
5
2x10
5
5x10
6
10
2x10
6
5x10
6
10
7
Kilocycles
11
12
Weibull Multiple Probability Plot
Response Surface ML Estimates
Extrapolation in Stroke Displacement
Spring Accelerated Life Test Data
Conditional Model Plot
Spring Life versus Stroke Displacement
Spring Fatigue Data Model MLE
StrokeLog, TempLinear, MethodClass, Dist:Weibull
Weibull Probability Plot
Spring Fatigue Data Conditional Model Plot
StrokeLog, TempLinear, MethodClass, Dist:Weibull
Fixed values of Temp=600, Method=New
.999
75
65
.9
.5
55
Stroke on Log Scale
Fraction Failing
.2
.1
.05
.02
.01
.003
.001
.0003
.0001
45
40
35
30
25
20
.00003
.00001
10%
15
.000003
10
2
10
3
10
4
10
5
10
6
10
2
7
10
10
3
4
10
Kilocycles
10
5
50%
6
10
10
13
14
0.10 Quantile of Spring Life versus
Stroke Displacement Box-Cox Parameter
with 95% Confidence Limits
Profile Likelihood
Stroke Box-Cox Transformation Parameter
Spring Fatigue Life Model
Spring Fatigue Data
Profile Likelihood and 95% Confidence Interval
for Stroke Box-Cox Transformation Power from the Weibull Distribution
9
10
8
10
7
10
6
10
5
10
4
ML estimate of the 0.1 quantile
Approximate 95% Pointwise confidence intervals
1.0
0.8
Profile Likelihood
0.1 Quantile of Kilocycles Distribution
Spring Fatigue Data
with Weibull Stroke:log, Temp:linear, Method:class at 20,600,New
Power Transformation Sensitivity Analysis on Stroke
10
0.50
0.60
0.6
0.70
0.80
0.4
0.90
0.2
0.95
0.99
0.0
-1.0
-0.5
0.0
0.5
1.0
1.5
-2
2.0
-1
0
1
2
Stroke Box-Cox Transformation Power
Stroke Box-Cox Transformation Power
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16
0.10 Quantile of Spring Life versus
Temperature Box-Cox Parameter
with 95% Confidence Limits
Conditional Model Plot
Spring Life versus Processing Temperature
Spring Fatigue Data Conditional Model Plot
StrokeLog, TempLinear, MethodClass, Dist:Weibull
Fixed values of Stroke=20, Method=New
Spring Fatigue Data
with Weibull Stroke:log, Temp:linear, Method:class at 20,600,New
Power Transformation Sensitivity Analysis on Temp
0.1 Quantile of Kilocycles Distribution
1100
1000
Temp on Linear Scale
7
Kilocycles
Confidence Level
10
1
900
800
700
600
500
10%
400
5
10
5
2x10
5
5x10
6
10
50%
2x10
6
90%
5x10
6
10
5x10
6
2x10
6
10
ML estimate of the 0.1 quantile
Approximate 95% Pointwise confidence intervals
6
5x10
5
2x10
5
7
-1.0
Kilocycles
-0.5
0.0
0.5
1.0
1.5
2.0
Temp Box-Cox Transformation Power
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18
0.10 Quantile of Spring Life versus
Stroke Displacement Box-Cox Parameter
Comparing Weibull and Lognormal Distributions
• Collections of S-PLUS functions for reliability data analysis
0.1 Quantile of Kilocycles Distribution
Spring Fatigue Data
with Stroke:log, Temp:linear, Method:class at 20,600,New
Power Transformation Sensitivity Analysis on Stroke
10
9
10
8
10
7
10
6
10
5
10
4
SPLIDA (S-PLUS Life Data Analysis)
Lognormal:0.1 quantile
Weibull:0.1 quantile
• Runs on S-PLUS Windows versions 4.5 and 2000
• Provides ability to do about 95% of the examples in
Meeker and Escobar (1998) plus new capabilities
• Graphical user interface
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
• Available for download from www.public.iastate.edu/˜wqmeeker
Stroke Box-Cox Transformation Power
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Concluding Remarks
• Accelerated life testing is a critically important tool for
the design of high-reliability products
• Extrapolation is required in accelerated testing
• Fundamental knowledge of the mechanisms underlying
failure modes is important
• When fundamental knowledge of the mechanisms underlying is not available, sensitivity analysis and conservative
design decisions are required
• Software is needed to make sensitivity analyses easy to
perform
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