Recurrence Data with Exact Ages

See the section Analysis of Recurrence Data on Repairs and the section Comparison of Two Samples of Repair Data for examples of the analysis of recurrence data with exact ages.

Formulas for the MCF estimator and the variance of the estimator Var are given in Nelson (1995). Table 12.61 shows a set of artificial repair data from Nelson (1988). For each system, the data consist of the system and cost for each repair. If you want to compute the MCF for the number of repairs, rather than cost of repairs, then you should set the cost equal to 1 for each repair. A plus sign (+) in place of a cost indicates that the age is a censoring time. The repair history of each system ends with a censoring time.

Table 12.62 illustrates the calculation of the MCF estimate from the data in Table 12.61. The RELIABILITY procedure uses the following rules for computing the MCF estimates.

1. Order all events (repairs and censoring) by age from smallest to largest.

   • If the event ages of the same or different systems are equal, the corresponding data are sorted from the largest repair cost to the smallest. Censoring events always sort as smaller than repair events with equal ages.

   • When event ages and values of more than one system coincide, the corresponding data are sorted from the largest system identifier to the smallest. The system IDs can be numeric or character, but they are always sorted in ASCII order.

2. Compute the number of systems in service at the current age as the number in service at the last repair time minus the number of censored units in the intervening times.

3. For each repair, compute the mean cost as the cost of the current repair divided by the number in service .

4. Compute the MCF for each repair as the previous MCF plus the mean cost for the current repair.

The variance of the estimator of the MCF Var is computed as in Nelson (1995). If the VARIANCE=LAWLESS or VARMETHOD2 option is specified, the method of Lawless and Nadeau (1995) is used to compute the variance of the estimator of the MCF. This method is recommended if the number of systems or events is large or if a FREQ statement is used to specify a frequency variable.

Default approximate two-sided pointwise confidence limits for are computed as

where represents the percentile of the standard normal distribution.

If you specify the LOGINTERVALS option in the MCFPLOT statement, alternative confidence intervals based on the asymptotic normality of , rather than of , are computed. Let

Then the limits are computed as

These alternative limits are always positive, and can provide better coverage than the default limits when the MCF is known to be positive, such as for counts or for positive costs. They are not appropriate for MCF differences, and are not computed in this case.

The following SAS statements create the tabular output shown in Figure 12.47 and the plot shown in Figure 12.48:

   data Art;               
      input Sysid$ Time Cost; 
      datalines;                  
      sys1 19 2               
      sys1 39 2               
      sys1 42 -1              
      sys2 8  2               
      sys2 14 1               
      sys2 26 1               
      sys2 33 -1              
      sys3 18 3               
      sys3 29 -1              
      sys4 16 2               
      sys4 2  1               
      sys4 20  -1             
      sys4 8  1               
      sys5 16 -1              
      sys6 5 3                
      sys6 12 1               
      sys6 12 -1              
      ;                       
   run;                       

proc reliability data=Art;
   unitid Sysid;
   mcfplot Time*Cost(-1) ;
run;

The first table in Figure 12.47 displays the input data set, the number of observations used in the analysis, the number of systems (units), and the number of repair events. The second table displays the system age, MCF estimate, standard error, approximate confidence limits, and system ID for each event.