Overview
Reliability Data Analysis Using S-PLUS
William Q. Meeker
Department of Statistics and
Iowa State University
Ames, IA 50011
32nd Symposium on the Interface:
Computing Science and Statistics
New Orleans, LA
April 5-8, 2000
History, other software
Specic user needs and GUI development goals
Data objects and outline of menu structure
Examples
Numerical Methods
Concluding remarks and future work
2
1
History and Development of SLIDA
Motivated by GE's STATPAC (Nelson et al. 1972,...)
CENSOR Fortran program (Meeker and Duke 1978-1980 at
ISU)
STAR written in Old S (Meeker and others at Bell Labs
QAC 1985-1986) then translated into C to be a commercial
product (Others at Bell Labs QAC 1986-1989)
GENMAX Fortran (Meeker 1986-1992)
SLIDA S-PLUS functions for Meeker and Escobar (1998)
examples (Meeker 1992-Present)
SLIDA object-oriented S-PLUS commands (Meeker 1997present)
SLIDA S-PLUS GUI (Meeker 1998-present)
Other Reliability Data Analysis Software
Terry Therneau's SURVIVAL package (in S-PLUS)
SAS, JMP, MINITAB
Special purpose packages:
I WEIBULL++ and ALTA by Relisoft
I WinSmith
Various biomedical packages that handle censored data
3
4
General Needs for Reliability Data Analyses
Complicated censoring and/or truncation; multiple failure
modes.
Wide range of standard and nonstandard models (e.g., nonnormal distributions, nonlinear relationships)
Estimates and statistical intervals for failure probabilities, distribution quantiles, failure rates, predictions for future number of failures, etc.
Integration of analytical methods by graphically displaying
data and tted models together.
Methods for planning reliability studies
Use of simulation in inference and planning
5
SLIDA User Interface
Command User Interface: All functionality in Meeker and Escobar (1998) plus recent developments.
Most outputs given in graphical form.
With numerous options, command argument specication is
complicated.
GUI simplies option choice. The most important functionality in Meeker and Escobar (1998) plus recent developments
(driven by courses).
6
SLIDA Data Objects
Bearing Cage Failure-Time Data
Data objects contain important and useful (optional) information about a data set (denes response, censoring, truncation, weights, explanatory variables, title, units, notes, etc.)
Bearing Cage Failure Data
Count
Multiple methods (dierent types of analyses) can be performed on particular data objects
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0
500
1000
1500
288
148
1
124
1
111
1
106
99
110
114
119
127
1
1
123
93
47
41
27
1
11
6
1
2
2000
Thu Sep 30 18:02:24 CDT 1999
Hours
7
8
Earth-Moving Machine Maintenance Actions
Event Plot
Earth-Moving Machine Maintenance Action
Mean Cumulative Labor Hours.
Mean Cumulative Function for Mean Cumulative Number of Labor Hours
Mean Cumulative Number of Labor Hours
140
Mean Cumulative Failures
System ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
120
100
80
60
40
20
0
0
2000
4000
6000
8000
0
Thu Sep 30 18:02:27 CDT 1999
Age in Hours of Operation
2000
4000
6000
Age in Hours of Operation
8000
Thu Sep 30 18:02:29 CDT 1999
9
Accelerated Degradation Test Results Giving Power
Drop in Device-B Output for a Sample of Units
Tested a Three of Levels Temperature.
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Goals for the SLIDA GUI Design
0.0
-0.2
Power drop in dB
Unit ID
I Life data objects
Single distribution
Multiple failure modes
Single explanatory variable with a few levels
Comparison explanatory variable
General explanatory variables
I Recurrence data (point process) objects
I Repeated measures (degradation) data objects
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
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23
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25
-0.4
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-0.6
-0.8
-1.0
-1.2
-1.4
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237 Degrees C
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2000
150 Degrees C
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195 Degrees C
3000
4000
Hours
11
Easy for occasional new/users.
Organize according to how users want to do their work.
Hide complexity for common users and every-day tasks; allow
access for experts/experienced users.
Develop structure to guide unfamiliar users through their
work (without restricting the experts).
Minimize required inputs; use defaults whenever possible.
Minimize the need for typing/remembering. Present some or
all choices whenever possible; eliminate inappropriate choices.
Recall previous inputs for defaults, when appropriate.
Signal bad (or questionable) input as soon as possible.
Result: Some complicated analyses much easier to do through
the GUI than through commands; engineers like it.
12
SLIDA Top-Level Menu
Plot nonparametric estimate of cdf and condence bands
Probability plot with nonparametric condence bands
Probability plot with parametric ML t
Likelihood contour plot
Compare distribution ML ts on probability plot
Threshold parameter probability plot with parametric ML t
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14
Bearing Cage Failure-Time Data Lognormal
Probability Plot and MLE
Bearing Cage Failure-Time Data Lognormal
Probability Plot and MLE
Log Hours
Bearing Cage Failure Data
with Lognormal MLE and Pointwise 95% Confidence Intervals
Lognormal Probability Plot
2.4
2.6
2.8
3.0
3.2
.05
.05
-2.0
.02
.02
.01
.01
.005
.005
.002
.001
-2.5
.002
-3.0
.001
.0005
.0005
Standard quantile
Probability
SLIDA ! Single distribution life data analyses !
-3.5
.0001
.0001
.00005
.00005
Muhat = 10.75
.00001
200
400
600
800
1000
1200
Muhat = 10.75
.00001
1600
200
400
Thu Sep 30 18:02:33 CDT 1999
Hours
600
800
1000
1200
1600
Thu Sep 30 18:02:37 CDT 1999
Hours
15
16
Bearing Cage Failure-Time Data
Weibull-Lognormal Comparison
Bearing Cage Failure-Time Data
Weibull-Lognormal Comparison
Bearing Cage Failure Data
Lognormal Probability Plot
.05
-4.0
Sigmahat = 1.554
Sigmahat = 1.554
Bearing Cage Failure Data
Lognormal Probability Plot
.6
Lognormal Distribution ML Fit
Weibull Distribution ML Fit
95% Pointwise Confidence Intervals
Lognormal Distribution ML Fit
Weibull Distribution ML Fit
95% Pointwise Confidence Intervals
.5
.4
.3
.02
.2
.01
Probability
Make/summary/view/modify data object !
Plan single a distribution study !
Single distribution life data analyses !
Multiple failure mode life data analysis !
Comparison of distributions life data analysis !
Plan an accelerated life test (ALT) !
Simple regression (ALT) data analysis !
Multiple regression (ALT) life data analysis !
Regression residual analysis !
Recurrence (point process) data analysis !
Degradation (repeated measures) data analysis !
Preferences (change SLIDA default options) !
Probability
Probability
.005
.002
.1
.05
.02
.01
.001
.005
.0005
.002
.001
.0005
.0001
.0001
200
400
600
Hours
800
1000
1200
1600
Thu Sep 30 18:02:40 CDT 1999
17
200
500
1000
2000
Hours
5000
10000
Thu Sep 30 18:02:44 CDT 1999
18
Bearing Cage Failure-Time Data
Lognormal Joint Condence Region
Bearing Cage Failure-Time Data
Lognormal Relative Likelihood
Bearing Cage Failure Data
Lognormal Distribution Joint Confidence Region
Bearing Cage Failure Data
Lognormal Distribution Relative Likelihood
4
4.5
3
4.0
95
2
sigma
3.0
a
sigm
3.5
2.5
50
d
kelihoo
ive Li
0.8
Relat
0.6
2 0.4
0 0.
1
1
2.0
1.5
1.0
0.5
6
8
10
12
14
12
10
8
14
mu
Thu Sep 30 18:02:51 CDT 1999
mu
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Numerical Methods
SLIDA ! Plan single distribution study !
Algorithms
I Stable optimization
I Accurate mathematical/statistical functions
I Analytical and numerical derivatives
Maximum likelihood estimation
I Stable parameterization
I Good starting values
I Dealing with unbounded or at likelihood functions.
Specify life test planning information (planning values)
Plot life test planning information (planning values)
Plot of approximate required sample size
Simulate a life test
Probability of successful demonstration
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Life Test Planning Values
Needed Sample Size for a Life Test
Weibull Distribution with eta= 6464 and beta= 0.8037
Weibull Probability Plot
Needed sample size giving approximatley a 50% chance of having
a confidence interval factor for the 0.1 quantile that is less than R
weibull Distribution with eta= 6464 and beta= 0.804
Test censored at 1000 Hours with 20 percent failing
.98
10000
.9
5000
.7
2000
Sample.size
.5
Probability
.3
.2
.1
.05
1000
500
200
100
99%
50
.03
.02
20
.01
10
20
50
200
500
2000
Hours
5000
20000
50000
Thu Sep 30 18:03:00 CDT 1999
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95%
90%
80%
1.0
1.5
2.0
2.5
3.0
3.5
Thu Sep 30 18:03:02 CDT 1999
Confidence Interval Precision Factor R
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Simulated Life Test = 10
Simulated Life Test = 40
n
n
30 simulated life tests of size n = 10 : Weibull Distribution with eta= 6464 and beta= 0.804
Weibull Probability Plot
30 simulated life tests of size n = 40 : Weibull Distribution with eta= 6464 and beta= 0.804
Weibull Probability Plot
.999
.999
Censor Time ->
Censor Time ->
.9
.7
.7
.5
.5
Probability
.98
.9
Probability
.98
.3
.2
.1
.05
.3
.2
.1
.05
.03
.02
.03
.02
1 samples out of 30 with 0 failures
.01
.01
Conditional average 95% confidence interval
precision factor R for t_0.1 = 173
.005
5x10
0
10
1
1
2x10
1
5x10
10
2
2x10
2
5x10
2
3
10
2x10
3
5x10
3
10
4
4
2x10
4
5x10
10
average 95% confidence interval
precision factor R for t_0.1 = 6.18
.005
5
5x10
0
10
1
1
2x10
1
5x10
10
2
2x10
2
5x10
Thu Sep 30 18:03:07 CDT 1999
Hours
2
3
10
2x10
3
5x10
3
10
4
4
2x10
4
5x10
10
5
Thu Sep 30 18:03:30 CDT 1999
Hours
25
26
Simulated Life Test = 160
n
SLIDA ! Multiple regression (ALT) life data analysis
30 simulated life tests of size n = 160 : Weibull Distribution with eta= 6464 and beta= 0.804
Weibull Probability Plot
.999
Censor Time ->
.98
.9
.7
Probability
.5
.3
.2
.1
.05
.03
.02
.01
average 95% confidence interval
precision factor R for t_0.1 = 1.69
.005
5x10
0
10
1
1
2x10
1
5x10
10
2
2x10
2
5x10
2
3
10
2x10
3
5x10
3
10
4
4
2x10
4
5x10
10
Censored data pairs plot
Censored data scatter plot
Probability plot and ML t for individual conditions
Prob plot and ML t for indiv cond: common shapes (slopes)
Probability plot and ML t of a regression (acceleration)
model
Conditional stress plot
Sensitivity analysis plot
5
Thu Sep 30 18:03:56 CDT 1999
Hours
28
27
New Spring Multiple
Probability Plot Lognormal Regression Model
1 = (Stroke Temp Method)
New Spring Experiment Pairs Plot
Spring Life, Stroke, Temp, Method
55
60
65
70
New
t:
g
;
;
Old
Spring Fatigue Data
with Lognormal Log,Linear,Class Model MLE
Lognormal Probability Plot
4000
50
0
2000
Kilocycles
60
65
70
.99
.98
.95
Stroke
500
700
Temp
Probability
900
50
55
.9
.8
.7
.6
.4
.3
.2
Old
.1
New
Method
.05
.02
Lot4
.005
Lot1
Lot
0
1000
3000
5000
500
600
700
800
900
1000
Lot1
Lot3
Lot5
.001
20
50
200
500
Kilocycles
29
2000
5000
20000
50000
Mon Sep 27 18:30:23 CDT 1999
wq meeker
30
New Spring Experiment
Lognormal Regression Model Life vs Stress Plot
New Spring Experiment
Sensitivity Analysis Plot
Fixed values of Temp=600, Method=New
for the Spring Fatigue Data
Spring Fatigue Data
with Lognormal Stroke:log, Temp:linear, Method:class at 30,600,New
Power Transformation Sensitivity Analysis on Stroke
6
0.1 Quantile of Life in Kilocycles
10
5
5x10
5
Kilocycles
2x10
5
10
4
5x10
4
2x10
4
10
3
5x10
90%
3
50%
2x10
3
10
2
5x10
2x10
10%
2
5x10
6
2x10
6
10
5
2x10
5
40
50
60
70
80
5
5x10
4
2x10
4
10
30
6
5x10
10
ML estimate of the 0.1 quantile
95% Pointwise confidence intervals
4
-1.0
Thu Sep 30 18:04:31 CDT 1999
-0.5
0.0
0.5
1.0
1.5
Box-Cox Transformation Power
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32
Reliability Data Analysis
Concluding Remarks and Future Work
New Spring Experiment
.1 Quantile Sensitivity Analysis Likelihood Prole
Profile Likelihood and 95% Confidence Interval
for Box-Cox Transformation Power from the Lognormal Distribution
Profile Likelihood
0.8
0.50
0.60
0.6
0.70
0.80
0.4
0.90
Confidence Level
1.0
0.2
0.95
0.99
0.0
-1.0
-0.5
0.0
0.5
1.0
2.0
Thu Sep 30 18:05:28 CDT 1999
Stroke on Log scale
1.5
2.0
Box-Cox Transformation Power
33
Considerable progress has been made in this area in the past
10 years; much remains to be done.
Improvements in the user interface. Make it easy to learn
how to use.
Better approximate methods for censored-data condence intervals (e.g., likelihood ratio and/or bootstrap).
Better and more methods for recurrence and degradation
data.
Methods for incorporating prior information (exible, easy to
use, Bayes methods).
Better (i.e., easier to implement and more robust) methods
for tting user-specied models the system.
34