10.5
Fitting Distributions to Reliability Data
SAS Code for Appliance Cycle Data Example
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\CRSNOTES\REL1.PDF’;
ODS LISTING;
OPTIONS NODATE NONUMBER LS=78;
********************************************************************;
*** Multiply censored appliance cycle data example (from Nelson) ***;
********************************************************************;
DATA example1;
INPUT cycles censor n @@;
LABEL CYCLES = ’NUMBER OF CYCLES TO FAILURE’;
LINES;
45 1 1
47 0 1
73 0 1
136 1 5
145 0 1
190 1 2
281 1 1
311 0 1
417 1 1
485 1 2
490 0 1
569 1 1
571 1 1
571 0 1
575 0 1
608 1 12
608 0 2
630 0 1
670 0 2
731 1 1
838 0 1
964 0 2
1164 1 7
1198 1 1
1198 0 1
1300 1 3
;
PROC LIFETEST DATA=example1 PLOTS=(LS,LLS,S) OUTSURV=survive ;
TITLE F=SWISSB H=.4 CM ’RELIABILITY ANALYSIS: APPLIANCE CYCLE DATA’;
TIME cycles*censor(1);
FREQ n;
SYMBOL1 H=1 V=DOT W=2;
PROC PRINT DATA=survive;
RUN;
PROC RELIABILITY DATA=example1;
DISTRIBUTION EXPONENTIAL;
PROBPLOT cycles*censor(1) / WAXIS=2 WFIT=2 FONT=SWISSB;
FREQ n;
SYMBOL1 H=1.5 V=CIRCLE W=2;
TITLE F=SWISSB H=.4 CM ’APPLIANCE CYCLE DATA: FITTING AN EXPONENTIAL
DISTRIBUTION’;
RUN;
/*
PROC RELIABILITY DATA=example1;
DISTRIBUTION EXTREME;
PROBPLOT cycles*censor(1) / WAXIS=2 WFIT=2 FONT=SWISSB;
FREQ n;
SYMBOL1 H=1.5 V=CIRCLE W=2;
TITLE F=SWISSB H=.4 CM ’APPLIANCE CYCLE DATA: FITTING AN EXTREME VALUE
DISTRIBUTION’;
RUN;
*/
263
/*
PROC RELIABILITY DATA=example1;
DISTRIBUTION LOGISTIC;
PROBPLOT cycles*censor(1) / WAXIS=2 WFIT=2
FREQ n;
SYMBOL1 H=1.5 V=CIRCLE W=2;
TITLE F=SWISSB H=.4 CM ’APPLIANCE CYCLE DATA:
DISTRIBUTION’;
RUN;
*/
/*
PROC RELIABILITY DATA=example1;
DISTRIBUTION LOGLOGISTIC;
PROBPLOT cycles*censor(1) / WAXIS=2 WFIT=2
FREQ n;
SYMBOL1 H=1.5 V=CIRCLE W=2;
TITLE F=SWISSB H=.4 CM ’APPLIANCE CYCLE DATA:
DISTRIBUTION’;
RUN;
*/
/*
PROC RELIABILITY DATA=example1;
DISTRIBUTION LOGNORMAL;
PROBPLOT cycles*censor(1) / WAXIS=2 WFIT=2
FREQ n;
SYMBOL1 H=1.5 V=CIRCLE W=2;
TITLE F=SWISSB H=.4 CM ’APPLIANCE CYCLE DATA:
DISTRIBUTION’;
RUN;
*/
/*
PROC RELIABILITY DATA=example1;
DISTRIBUTION NORMAL;
PROBPLOT cycles*censor(1) / WAXIS=2 WFIT=2
FREQ n;
SYMBOL1 H=1.5 V=CIRCLE W=2;
TITLE F=SWISSB H=.4 CM ’APPLIANCE CYCLE DATA:
RUN;
*/
/*
PROC RELIABILITY DATA=example1;
DISTRIBUTION WEIBULL;
PROBPLOT cycles*censor(1) / WAXIS=2 WFIT=2
FREQ n;
SYMBOL1 H=1.5 V=CIRCLE W=2;
TITLE F=SWISSB H=.4 CM ’APPLIANCE CYCLE DATA:
DISTRIBUTION’;
RUN;
*/
264
FONT=SWISSB;
FITTING A LOGISTIC
FONT=SWISSB;
FITTING A LOGLOGISTIC
FONT=SWISSB;
FITTING A LOGNORMAL
FONT=SWISSB;
FITTING A NORMAL DISTRIBUTION’;
FONT=SWISSB;
FITTING A WEIBULL
10.5.1
Fitting Distributions to the Appliance Cycle Data
265
266
267
10.5.2
Exponential Data
• Suppose the sample times of the n units are y1 , y2 , . . . , yn of which r are failure
times and n − r are censored survival times.
n
• The maximum likelihood estimate (MLE) for the mean θ is
1X
b
θ =
yi .
r i=1
Thus, the MLE is the total time for all n units divided by the number of
failures r.
• The following table contains data on 70 diesel engine fans with 344,400 hours
of service. Only 12 of the 70 failed, with 58 censored times. Management
wanted an estimate and a confidence interval for the fraction failing on an
8000 hour warranty.
Example: Time to Fan Failure in Hours (+ denotes censored times)
268
Weibull Data
261
261
269
Lognormal Data
• Formulas for MLEs and confidence intervals for µ and σ are complex, and can be found in
Applied Life Data Analysis (1982) by Wayne Nelson.
• Statistical packages, like SAS, provide MLEs and approximate confidence intervals for µ, σ,
percentiles, and reliabilities.
• The following table contains the singly-censored life data on 96 locomotive controls. There is
also a lognormal probability plot of the data with 95% confidence bands about the percentiles
from the fitted MLE-based lognormal distribution.
• Management wanted an estimate and confidence intervals for the fraction of controls failing
on an 80 thousand mile warranty.
Example: Locomotive Control Time to Failure Data
Lognormal Probability Plot of Locomotive Control Failures
270
10.5.3
Fitting Distributions to the Grinder Lifetime Data
271
272
273
10.5.4
Fitting Distributions to the Fan Failure Lifetime Data
274
275
276
10.5.5
Fitting Distributions to the Locomotive Control Lifetime Data
277
278
279
Competing Failure Modes
10.6
Failure
Modes
10.6Competing
Competing
Failure
Modes
We now consider a case with two modes of failure with the mode of each failure known. We will
Competing Failure Modes
(i) consider fitting a life distribution ignoring the mode of failure and
10.6
(ii) fitting separate life distributions for each failure mode.
280
273
281
282
283
284
285
286
10.7
Confidence Intervals for Distribution Parameters
• Let θ = (θ1 , θ2 , . . . , θk ) be the vector of distribution parameters, and θbM LE = (θb1 , θb2 , . . . , θbk )
be the vector of maximum likelihood estimates (MLEs).
• The estimated covariance matrix of θbM LE is:
2
−1
∂ l(θ)
−1
b
Σ = [c
σij ] = −H
= −
.
∂θi ∂θj θ=θbM LE
This is the negative of the matrix of second derivatives of the log likelihood l(θ) evaluated at
the MLE parameter estimate vector θbM LE .
• The negative of the matrix of second derivatives of the log likelihood l(θ) is called the Fisher
information matrix.
• The σ
bii diagonal entry of θbM LE is an estimate of the variance of the ith MLE θbi .
p
bii .
• The standard error of the ith MLE θbi is SE(θbi = σ
• The following table summarizes the computation used by SAS to generate confidence intervals
for the MLEs of the distribution parameters.
Distribution
Normal
Lognormal
Extreme Value
µL
µU
µL
µU
µL
µU
Exponential
Location
=µ
b − z ∗ SE(b
µ)
∗
=µ
b + z SE(b
µ)
∗
=µ
b − z SE(b
µ)
=µ
b + z ∗ SE(b
µ)
∗
=µ
b − z SE(b
µ)
=µ
b + z ∗ SE(b
µ)
Weibull
Logistic
Log-logistic
µL
µU
µL
µU
=µ
b − z ∗ SE(b
µ)
=µ
b + z ∗ SE(b
µ)
∗
=µ
b − z SE(b
µ)
∗
=µ
b + z SE(b
µ)
Parameter Type
Scale
σL = σ
b / exp[z ∗ SE(b
σ )/b
σ]
∗
σU = σ
b × exp[z SE(b
σ )/b
σ]
∗
σL = σ
b / exp[z SE(b
σ )/b
σ]
σU = σ
b × exp[z ∗ SE(b
σ )/b
σ]
∗
σL = σ
b / exp[z SE(b
σ )/b
σ]
σU = σ
b × exp[z ∗ SE(b
σ )/b
σ]
αL = exp[b
µ − z ∗ SE(b
µ)]
αU = exp[b
µ + z ∗ SE(b
µ)]
αL = exp[b
µ − z ∗ SE(b
µ)]
αU = exp[b
µ + z ∗ SE(b
µ)]
∗
σL = σ
b / exp[z SE(b
σ )/b
σ]
σU = σ
b × exp[z ∗ SE(b
σ )/b
σ]
∗
σL = σ
b / exp[z SE(b
σ )/b
σ]
∗
σU = σ
b × exp[z SE(b
σ )/b
σ]
Shape
βL = exp[−z ∗ SE(b
σ )/b
σ] / σ
b
βU = exp[ z ∗ SE(b
σ )/b
σ] / σ
b
• For the exponential distribution, µ
b is the location parameter of the extreme value distribution
for the logarithm of lifetime. This is denoted as “EV location” in the SAS output.
• For the Weibull distribution, µ
b and σ
b are the location and scale parameters of the extreme
value distribution for the logarithm of lifetime. These are denoted as “EV location” and “EV
scale” in the SAS output.
287