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;
label lifetime='Fan Life (1000s of Hours)';
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 12.1:

```proc reliability data=fan;
distribution weibull;
pplot lifetime*censor( 1 ) /  covb ;
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 12.1, 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 12.1 Weibull Probability Plot for Engine Fan Data (Traditional Graphics)

You can optionally create the probability plot by using ODS Graphics. The following SAS statements use ODS Graphics to create the probability plot shown in Figure 12.1:

```ods graphics ON;
proc reliability data=fan;
distribution weibull;
pplot lifetime*censor( 1 ) /  covb;
run;
ods graphics OFF;
```

The plot is shown in Figure 12.2.

Figure 12.2 Weibull Probability Plot for Engine Fan Data (ODS Graphics)

The tabular output produced by the preceding SAS statements is shown in Figure 12.3 and Figure 12.4. This consists of summary data, fit information, parameter estimates, distribution percentile estimates, standard errors, and confidence intervals for all estimated quantities.

Figure 12.3 Tabular Output for the Fan Data Analysis
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 Distribution Parameters
Parameter Value
Mean 25.7156
Mode 1.7039
Median 18.6002
Standard Deviation 24.3066

Estimated Covariance Matrix
Weibull Parameters
EV Location EV Scale
EV Location 0.21705 0.09044
EV Scale 0.09044 0.05733

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

Figure 12.4 Percentile Estimates for the Fan Data
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

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