Example 6.3: Probability Plotting -Arbitrarily Censoring

The LIFEREG Procedure

Example 6.3: Probability Plotting -Arbitrarily Censoring

Table 6.2 contains microprocessor failure data from Nelson (1990). Units were inspected at predetermined time intervals. The data consist of inspection interval endpoints (in hours) and the number of units failing in each interval. A missing (.) lower endpoint indicates left censoring, and a missing upper endpoint indicates right censoring. These can be thought of as semi-infinite intervals with a lower (upper) endpoint of negative (positive) infinity for left (right) censoring.

Table 6.2: Interval-Censored Data

Lower	Upper	Number
Endpoint	Endpoint	Failed
.	6	6
6	12	2
24	48	2
24	.	1
48	168	1
48	.	839
168	500	1
168	.	150
500	1000	2
500	.	149
1000	2000	1
1000	.	147
2000	.	122

The following SAS program will compute the Turnbull estimate and create a lognormal probability plot.

   data micro;                                
      input t1 t2 f ;                          
      datalines;                                   
   . 6 6                                    
   6 12 2                                   
   12 24 0                                  
   24 48 2                                  
   24 .  1                                  
   48 168 1                                 
   48 .   839                               
   168 500 1                                
   168 .   150                              
   500 1000 2                               
   500 .    149                             
   1000 2000 1                              
   1000 . 147                               
   2000 . 122                               
   ;  
   run;



   title;
   symbol v=dot c=white;  
   proc lifereg data=micro;               
      model ( t1 t2 ) = / d=lognormal intercept=25 scale=5;  
      weight f;                                  
      probplot 
         cframe = ligr
         cfit   = blue
         pupper = 10         
         itprintem              
         printprobs 
         maxitem = (1000,25)
         ppout;    
      inset / cfill = white;                 
   run; quit;

The two initial values INTERCEPT= 25 and SCALE= 5 in the MODEL statement are used to aid convergence in the model-fitting algorithm.

The following tables are created by the PROBPLOT statement in addition to the standard tabular output from the MODEL statement. Output 6.3.1 shows the iteration history for the Turnbull estimate of the CDF for the microprocessor data. With both options ITPRINTEM and PRINTPROBS specified in the PROBPLOT statement, this table contains the log likelihoods and interval probabilities for every 25th iteration and the last iteration. It would only contain the log likelihoods if the option PRINTPROBS were not specified.

Output 6.3.1: Iteration History for the Turnbull Estimate

The LIFEREG Procedure

Iteration History for the Turnbull Estimate of the CDF
Iteration	Loglikelihood	(., 6)	(6, 12)	(24, 48)	(48, 168)	(168, 500)	(500, 1000)	(1000, 2000)	(2000, .)
0	-1133.4051	0.125	0.125	0.125	0.125	0.125	0.125	0.125	0.125
25	-104.16622	0.00421644	0.00140548	0.00140648	0.00173338	0.00237846	0.00846094	0.04565407	0.93474475
50	-101.15151	0.00421644	0.00140548	0.00140648	0.00173293	0.00234891	0.00727679	0.01174486	0.96986811
75	-101.06641	0.00421644	0.00140548	0.00140648	0.00173293	0.00234891	0.00727127	0.00835638	0.9732621
100	-101.06534	0.00421644	0.00140548	0.00140648	0.00173293	0.00234891	0.00727125	0.00801814	0.97360037
125	-101.06533	0.00421644	0.00140548	0.00140648	0.00173293	0.00234891	0.00727125	0.00798438	0.97363413
130	-101.06533	0.00421644	0.00140548	0.00140648	0.00173293	0.00234891	0.00727125	0.007983	0.97363551

Output 6.3.2: Summary for the Turnbull Algorithm

The LIFEREG Procedure

Lower Lifetime	Upper Lifetime	Probability	Reduced Gradient	Lagrange Multiplier
.	6	0.0042	0	0
6	12	0.0014	0	0
24	48	0.0014	0	0
48	168	0.0017	0	0
168	500	0.0023	0	0
500	1000	0.0073	-7.219342E-9	0
1000	2000	0.0080	-0.037063236	0
2000	.	0.9736	0.0003038877	0

The table in Output 6.3.2 summarizes the Turnbull estimates of the interval probabilities, the reduced gradients, and Lagrange multipliers as described in the section "Arbitrarily Censored Data".

Output 6.3.3: Final CDF Estimates for Turnbull Algorithm

The LIFEREG Procedure

Cumulative Probability Estimates
Lower Lifetime	Upper Lifetime	Cumulative Probability	Pointwise 95% Confidence Limits		Standard Error
Lower Lifetime	Upper Lifetime	Cumulative Probability	Lower	Upper	Standard Error
6	6	0.0042	0.0019	0.0094	0.0017
12	24	0.0056	0.0028	0.0112	0.0020
48	48	0.0070	0.0038	0.0130	0.0022
168	168	0.0088	0.0047	0.0164	0.0028
500	500	0.0111	0.0058	0.0211	0.0037
1000	1000	0.0184	0.0094	0.0357	0.0063
2000	2000	0.0264	0.0124	0.0553	0.0101

Output 6.3.3 shows the final estimate of the CDF, along with standard errors and nonparametric confidence limits. Two kinds of nonparametric confidence limits, pointwise or simultaneous, are available. The default is the pointwise nonparametric confidence limits. You can specify the simultaneous nonparametric confidence limits by the NPINTERVALS= SIMUL option.

Output 6.3.4 shows the CDF estimates, the maximum likelihood fit, and the pointwise parametric confidence limits plotted on a lognormal probability plot.

Output 6.3.4: Lognormal Probability Plot for the Microprocessor Data

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