fitting distributions to reliability data sas code for appliance … · 2016-02-16 · lognormal...

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 2281 1 1 311 0 1 417 1 1 485 1 2 490 0 1 569 1 1571 1 1 571 0 1 575 0 1 608 1 12 608 0 2 630 0 1670 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 EXPONENTIALDISTRIBUTION’;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 VALUEDISTRIBUTION’;RUN;*/

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/*PROC RELIABILITY DATA=example1;

DISTRIBUTION LOGISTIC;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 A LOGISTICDISTRIBUTION’;RUN;*//*PROC RELIABILITY DATA=example1;

DISTRIBUTION LOGLOGISTIC;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 A LOGLOGISTICDISTRIBUTION’;RUN;*//*PROC RELIABILITY DATA=example1;

DISTRIBUTION LOGNORMAL;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 A LOGNORMALDISTRIBUTION’;RUN;*//*PROC RELIABILITY DATA=example1;

DISTRIBUTION NORMAL;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 A NORMAL DISTRIBUTION’;RUN;*//*PROC RELIABILITY DATA=example1;

DISTRIBUTION WEIBULL;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 A WEIBULLDISTRIBUTION’;RUN;*/

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10.5.1 Fitting Distributions to the Appliance Cycle Data

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10.5.2 Exponential Data

• Suppose the sample times of the n units are y1, y2, . . . , yn of which r are failuretimes and n− r are censored survival times.

• The maximum likelihood estimate (MLE) for the mean θ is θ̂ =1

r

n∑i=1

yi.

Thus, the MLE is the total time for all n units divided by the number offailures r.

• The following table contains data on 70 diesel engine fans with 344,400 hoursof service. Only 12 of the 70 failed, with 58 censored times. Managementwanted an estimate and a confidence interval for the fraction failing on an8000 hour warranty.

Example: Time to Fan Failure in Hours (+ denotes censored times)

10.4 Other Distributions

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Weibull Data

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Lognormal Data

• Formulas for MLEs and confidence intervals for µ and σ are complex, and can be found inApplied 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 isalso a lognormal probability plot of the data with 95% confidence bands about the percentilesfrom the fitted MLE-based lognormal distribution.

• Management wanted an estimate and confidence intervals for the fraction of controls failingon an 80 thousand mile warranty.

Example: Locomotive Control Time to Failure Data

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Lognormal Probability Plot of Locomotive Control Failures

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10.5.3 Fitting Distributions to the Grinder Lifetime Data

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10.5.4 Fitting Distributions to the Fan Failure Lifetime Data

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10.5.5 Fitting Distributions to the Locomotive Control Lifetime Data

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10.6 Competing Failure Modes

We now consider a case with two modes of failure with the mode of each failure known. We will

(i) consider fitting a life distribution ignoring the mode of failure and

(ii) fitting separate life distributions for each failure mode.


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10.7 Confidence Intervals for Distribution Parameters

• Let θ = (θ1, θ2, . . . , θk) be the vector of distribution parameters, and θ̂MLE = (θ̂1, θ̂2, . . . , θ̂k)be the vector of maximum likelihood estimates (MLEs).

• The estimated covariance matrix of θ̂MLE is:

Σ̂ = [σ̂ij] = −H−1 = −[∂2l(θ)

∂θi∂θj

]−1

θ=θ̂MLE

.

This is the negative of the matrix of second derivatives of the log likelihood l(θ) evaluated at

the MLE parameter estimate vector θ̂MLE.

• The negative of the matrix of second derivatives of the log likelihood l(θ) is called the Fisherinformation matrix.

• The σ̂ii diagonal entry of θ̂MLE is an estimate of the variance of the ith MLE θ̂i.

• The standard error of the ith MLE θ̂i is SE(θ̂i =√σ̂ii.

• The following table summarizes the computation used by SAS to generate confidence intervalsfor the MLEs of the distribution parameters.

Parameter TypeDistribution Location Scale ShapeNormal µL = µ̂− z∗SE(µ̂) σL = σ̂ / exp[z∗SE(σ̂)/σ̂]

µU = µ̂+ z∗SE(µ̂) σU = σ̂ × exp[z∗SE(σ̂)/σ̂]Lognormal µL = µ̂− z∗SE(µ̂) σL = σ̂ / exp[z∗SE(σ̂)/σ̂]

µU = µ̂+ z∗SE(µ̂) σU = σ̂ × exp[z∗SE(σ̂)/σ̂]Extreme Value µL = µ̂− z∗SE(µ̂) σL = σ̂ / exp[z∗SE(σ̂)/σ̂]

µU = µ̂+ z∗SE(µ̂) σU = σ̂ × exp[z∗SE(σ̂)/σ̂]Exponential αL = exp[µ̂− z ∗ SE(µ̂)]

αU = exp[µ̂+ z ∗ SE(µ̂)]Weibull αL = exp[µ̂− z ∗ SE(µ̂)] βL = exp[−z∗SE(σ̂)/σ̂] / σ̂

αU = exp[µ̂+ z ∗ SE(µ̂)] βU = exp[ z∗SE(σ̂)/σ̂] / σ̂Logistic µL = µ̂− z∗SE(µ̂) σL = σ̂ / exp[z∗SE(σ̂)/σ̂]

µU = µ̂+ z∗SE(µ̂) σU = σ̂ × exp[z∗SE(σ̂)/σ̂]Log-logistic µL = µ̂− z∗SE(µ̂) σL = σ̂ / exp[z∗SE(σ̂)/σ̂]

µU = µ̂+ z∗SE(µ̂) σU = σ̂ × exp[z∗SE(σ̂)/σ̂]

• For the exponential distribution, µ̂ is the location parameter of the extreme value distributionfor the logarithm of lifetime. This is denoted as “EV location” in the SAS output.

• For the Weibull distribution, µ̂ and σ̂ are the location and scale parameters of the extremevalue distribution for the logarithm of lifetime. These are denoted as “EV location” and “EVscale” in the SAS output.

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fitting distributions to reliability data sas code for appliance … · 2016-02-16 · lognormal...

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