Analysis of Right-Censored Data from a Single Population

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 Distribution Parameters
Parameter	Value
Mean	25.7156
Mode	1.7039
Median	18.6002

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

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

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