Expected Ranks, Kaplan-Meier, and Modified Kaplan-Meier Methods

Let be ordered observations of a random sample including failure times and censor times. Order the data in increasing order. Label all the data with reverse ranks , with . For the failure corresponding to reverse rank , compute the reliability, or survivor function estimate

with . The expected rank plotting position is computed as . The option PPOS=EXPRANK specifies the expected rank plotting position.

For the Kaplan-Meier method,

The Kaplan-Meier plotting position is then computed as . The option PPOS=KM specifies the Kaplan-Meier plotting position.

For the modified Kaplan-Meier method, use

where is computed from the Kaplan-Meier formula with . The plotting position is then computed as . The option PPOS=MKM specifies the modified Kaplan-Meier plotting position. If the PPOS option is not specified, the modified Kaplan-Meier plotting position is used as the default method.

For complete samples, for the expected rank method, for the Kaplan-Meier method, and for the modified Kaplan-Meier method. If the largest observation is a failure for the Kaplan-Meier estimator, then and the point is not plotted. These three methods are shown for the field winding data in Table 14.54 and Table 14.55.

Nelson-Aalen

Estimate the cumulative hazard function by

with . The reliability is , and the plotting position, or CDF, is . You can show that for all ages. The Nelson-Aalen method is shown for the field winding data in Table 14.56.

Median Ranks

See Abernethy (2006) for a discussion of the methods described in this section. Let be ordered observations of a random sample including failure times and censor times. A failure order number is assigned to the ith failure: , where . The increment is initially 1 and is modified when a censoring time is encountered in the ordered sample. The new increment is computed as

The plotting position is computed for the th failure time as

For complete samples, the failure order number is equal to , the order of the failure in the sample. In this case, the preceding equation for is an approximation to the median plotting position computed as the median of the th-order statistic from the uniform distribution on (0, 1). In the censored case, is not necessarily an integer, but the preceding equation still provides an approximation to the median plotting position. The PPOS=MEDRANK option specifies the median rank plotting position.

For complete data, an alternative method of computing the median rank plotting position for failure is to compute the exact median of the distribution of the th order statistic of a sample of size from the uniform distribution on (0,1). If the data are right censored, the adjusted rank , as defined in the preceding paragraph, is used in place of in the computation of the median rank. The PPOS=MEDRANK1 option specifies this type of plotting position.

Nelson (1982, p. 148) provides the following example of multiply right-censored failure data for field windings in electrical generators. Table 14.57 shows the data, the intermediate calculations, and the plotting positions calculated by exact () and approximate () median ranks.

Readout Data

The RELIABILITY procedure can create probability plots for interval-censored data when all units share common interval endpoints. This type of data is called readout data in the RELIABILITY procedure. Estimates of the cumulative distribution function are computed at times corresponding to the interval endpoints. Right censoring can also be accommodated if the censor times correspond to interval endpoints. See the section Weibull Analysis of Interval Data with Common Inspection Schedule for an example of a Weibull plot and analysis for interval data.

Table 14.58 illustrates the computational scheme used to compute the CDF estimates. The data are failure data for microprocessors (Nelson; 1990, p. 147). In Table 14.58, are the interval upper endpoints, in hours, is the number of units failing in interval , and is the number of unfailed units at the beginning of interval .

Note that there is right censoring as well as interval censoring in these data. For example, two units fail in the interval (24, 48) hours, and there are 1414 unfailed units at the beginning of the interval, 24 hours. At the beginning of the next interval, (48, 168) hours, there are 573 unfailed units. The number of unfailed units that are removed from the test at 48 hours is units. These are right-censored units.

The reliability at the end of interval is computed recursively as

with . The plotting position is .

Arbitrarily Censored Data

The RELIABILITY procedure can create probability plots for data that consists of combinations of exact, left-censored, right-censored, and interval-censored lifetimes. Unlike the method in the previous section, failure intervals need not share common endpoints, although if the intervals share common endpoints, the two methods give the same results. The RELIABILITY procedure uses an iterative algorithm developed by Turnbull (1976) to compute a nonparametric maximum likelihood estimate of the cumulative distribution function for the data. Since the technique is maximum likelihood, standard errors of the cumulative probability estimates are computed from the inverse of the associated Fisher information matrix. A technique developed by Gentleman and Geyer (1994) is used to check for convergence to the maximum likelihood estimate. Also see Meeker and Escobar (1998, chap. 3) for more information.

Although this method applies to more general situations, where the intervals may be overlapping, the example of the previous section will be used to illustrate the method. Table 14.59 contains the microprocessor data of the previous section, arranged in intervals. 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 lower (upper) endpoint of negative (positive) infinity for left (right) censoring.

The following SAS statements 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       
;                

proc reliability data=micro;
   distribution lognormal;
   freq f;
   pplot ( t1 t2 ) / itprintem 
                     printprobs
                     maxitem = ( 1000, 25 )
                     nofit
                     npintervals = simul
                     ppout;
run;

The nonparametric maximum likelihood estimate of the CDF can only increase on certain intervals, and must remain constant between the intervals. The Turnbull algorithm first computes the intervals on which the nonparametric maximum likelihood estimate of the CDF can increase. The algorithm then iteratively estimates the probability associated with each interval. The ITPRINTEM option along with the PRINTPROBS option instructs the procedure to print the intervals on which probability increases can occur and the iterative history of the estimates of the interval probabilities. The PPOUT option requests tabular output of the estimated CDF, standard errors, and confidence limits for each cumulative probability.

Figure 14.52 shows every 25th iteration and the last iteration for the Turnbull estimate of the CDF for the microprocessor data. The initial estimate assigns equal probabilities to each interval. You can specify different initial values with the PROBLIST= option. The algorithm converges in 130 iterations for this data. Convergence is determined if the change in the loglikelihood between two successive iterations less than , where the default value is . You can specify a different value for delta with the TOLLIKE= option. This algorithm is an example of an expectation-maximization (EM) algorithm. EM algorithms are known to converge slowly, but the computations within each iteration for the Turnbull algorithm are moderate. Iterations will be terminated if the algorithm does not converge after a fixed number of iterations. The default maximum number of iterations is 1000. Some data may require more iterations for convergence. You can specify the maximum allowed number of iterations with the MAXITEM= option in the PROBPLOT, ANALYZE, or RPLOT statement.

If an interval probability is smaller than a tolerance ( by default) after convergence, the probability is set to zero, the interval probabilities are renormalized so that they add to one, and iterations are restarted. Usually the algorithm converges in just a few more iterations. You can change the default value of the tolerance with the TOLPROB= option. You can specify the NOPOLISH option to avoid setting small probabilities to zero and restarting the algorithm.

If you specify the ITPRINTEM option, the table in Figure 14.53 summarizing the Turnbull estimate of the interval probabilities is printed. The columns labeled ’Reduced Gradient’ and ’Lagrange Multiplier’ are used in checking final convergence to the maximum likelihood estimate. The Lagrange multipliers must all be greater than or equal to zero, or the solution is not maximum likelihood. See Gentleman and Geyer (1994) for more details of the convergence checking.

Figure 14.54 shows the final estimate of the CDF, along with standard errors and confidence limits. Figure 14.55 shows the CDF and simultaneous confidence limits plotted on a lognormal probability plot.

Figure 14.55 Lognormal Probability Plot for the Microprocessor Data