Chapter Contents Previous Next
 The RELIABILITY Procedure

## Analysis of Right-Censored Data from a Single Population

The Weibull distribution is used in a wide variety of reliability analysis applications. This example illustrates the use of the Weibull distribution to model product life data from a single population. The following statements create a SAS data set containing observed and right-censored lifetimes of 70 diesel engine fans (Nelson 1982, p. 318).
```   data fan;
input lifetime censor@@;
lifetime = lifetime / 1000;
datalines;
450 0    460 1   1150 0   1150 0   1560 1
1600 0   1660 1   1850 1   1850 1   1850 1
1850 1   1850 1   2030 1   2030 1   2030 1
2070 0   2070 0   2080 0   2200 1   3000 1
3000 1   3000 1   3000 1   3100 0   3200 1
3450 0   3750 1   3750 1   4150 1   4150 1
4150 1   4150 1   4300 1   4300 1   4300 1
4300 1   4600 0   4850 1   4850 1   4850 1
4850 1   5000 1   5000 1   5000 1   6100 1
6100 0   6100 1   6100 1   6300 1   6450 1
6450 1   6700 1   7450 1   7800 1   7800 1
8100 1   8100 1   8200 1   8500 1   8500 1
8500 1   8750 1   8750 0   8750 1   9400 1
9900 1  10100 1  10100 1  10100 1  11500 1
;
run;
```

Some of the fans had not failed at the time the data were collected, and the unfailed units have right-censored lifetimes. The variable LIFETIME represents either a failure time or a censoring time in thousands of hours. The variable CENSOR is equal to 0 if the value of LIFETIME is a failure time, and it is equal to 1 if the value is a censoring time. The following statements use the RELIABILITY procedure to produce the graphical output shown in Figure 30.1:

```   symbol c=blue v=plus;
proc reliability data=fan;
distribution weibull;
pplot lifetime*censor( 1 ) /  covb
cfit    = yellow
cframe  = ligr
ccensor = megr;
inset / cfill = ywh;
run;
```

The DISTRIBUTION statement specifies the Weibull distribution for probability plotting and maximum likelihood (ML) parameter estimation. The PROBPLOT statement produces a probability plot for the variable LIFETIME and specifies that the value of 1 for the variable CENSOR denotes censored observations. You can specify any value, or group of values, for the censor-variable (in this case, CENSOR) to indicate censoring times. The option COVB requests the ML parameter estimate covariance matrix. The graphical output, displayed in Figure 30.1, consists of a probability plot of the data, an ML fitted distribution line, and confidence intervals for the percentile (lifetime) values. An inset box containing summary statistics, Weibull scale and shape estimates, and other information is displayed on the plot by default. The locations of the right-censored data values are plotted in an area at the top of the plot.

Figure 30.1: Weibull Probability Plot for the Engine Fan Data

The tabular output produced by the preceding SAS statements is shown in Figure 30.2. This consists of summary data, fit information, parameter estimates, distribution percentile estimates, standard errors, and confidence intervals for all estimated quantities.

 Probability Plot for Fan Data

 The RELIABILITY Procedure

 Model Information Input Data Set WORK.FAN Analysis Variable lifetime Fan Life (1000s of Hours) Censor Variable censor Distribution Weibull Estimation Method Maximum Likelihood Confidence Coefficient 95% Observations Used 70

 Algorithm converged.

 Summary of Fit Observations Used 70 Uncensored Values 12 Right Censored Values 58 Maximum Loglikelihood -42.248

 Weibull Parameter Estimates Parameter Estimate Standard Error Asymptotic Normal 95% Confidence Limits Lower Upper EV Location 3.2694 0.4659 2.3563 4.1826 EV Scale 0.9448 0.2394 0.5749 1.5526 Weibull Scale 26.2968 12.2514 10.5521 65.5344 Weibull Shape 1.0584 0.2683 0.6441 1.7394

 Other Weibull DistributionParameters Parameter Value Mean 25.7156 Mode 1.7039 Median 18.6002

 Estimated Covariance MatrixWeibull Parameters EV Location EV Scale EV Location 0.21705 0.09044 EV Scale 0.09044 0.05733

 Estimated Covariance MatrixWeibull Parameters Weibull Scale Weibull Shape Weibull Scale 150.09724 -2.66446 Weibull Shape -2.66446 0.07196

 Weibull Percentile Estimates Percent Estimate Standard Error Asymptotic Normal 95% Confidence Limits Lower Upper 0.1 0.03852697 0.05027782 0.002985 0.49726229 0.2 0.07419554 0.08481353 0.00789519 0.69725757 0.5 0.17658807 0.16443381 0.02846732 1.09540855 1 0.34072273 0.2635302 0.07482449 1.55152389 2 0.65900116 0.40845639 0.19556981 2.22060107 5 1.58925244 0.68465855 0.68311002 3.69738878 10 3.13724079 0.99379006 1.68620756 5.83693255 20 6.37467675 1.74261908 3.73051433 10.8930029 30 9.92885165 3.00353842 5.48788931 17.9635721 40 13.9407124 4.85766683 7.04177638 27.5986417 50 18.6002319 7.40416922 8.52475116 40.5840149 60 24.2121441 10.8733301 10.0408557 58.3842593 70 31.3378076 15.750336 11.7018888 83.9230489 80 41.2254517 23.1787018 13.6956839 124.092954 90 57.8253251 36.9266698 16.5405275 202.156081 95 74.1471722 51.6127806 18.9489625 290.137423 99 111.307797 88.1380261 23.5781482 525.462197 99.9 163.265082 144.264145 28.8905203 922.637827
Figure 30.2: Tabular Output for the Fan Data Analysis

 Chapter Contents Previous Next Top