The RELIABILITY Procedure
 Analysis of Recurrence Data on Repairs

This example illustrates analysis of recurrence data from repairable systems. Repair data analysis differs from life data analysis, where units fail only once. As a repairable system ages, it accumulates repairs and costs of repairs. The RELIABILITY procedure provides a nonparametric estimate and plot of the mean cumulative function (MCF) for the number or cost of repairs for a population of repairable systems.

The nonparametric estimate of the MCF, the variance of the MCF estimate, and confidence limits for the MCF estimate are based on the work of Nelson (1995) . The MCF, also written as , is defined by Nelson (1995) to be the population mean of the distribution of the cumulative number or cost of repairs at age . The method does not assume any underlying structure for the repair process.

The SAS statements that follow create the listing of the SAS data set VALVE shown in Figure 12.31, which contains repair histories of 41 diesel engines in a fleet (Nelson; 1995) . The valve seats in these engines wear out and must be replaced. The variable Id is a unique identifier for individual engines. The variable Days provides the engine age in days. The value of the variable Value is 1 if the age is a valve seat replacement age or -1 if the age is the end of history, or censoring age, for the engine.

``` data valve;
input id days value @@;
datalines;
251 761 -1      252 759 -1      327  98  1      327 667 -1
328 326  1      328 653  1      328 653  1      328 667 -1
329 665 -1      330  84  1      330 667 -1      331  87  1
331 663 -1      389 646  1      389 653 -1      390  92  1
390 653 -1      391 651 -1      392 258  1      392 328  1
392 377  1      392 621  1      392 650 -1      393  61  1
393 539  1      393 648 -1      394 254  1      394 276  1
394 298  1      394 640  1      394 644 -1      395  76  1
395 538  1      395 642 -1      396 635  1      396 641 -1
397 349  1      397 404  1      397 561  1      397 649 -1
398 631 -1      399 596 -1      400 120  1      400 479  1
400 614 -1      401 323  1      401 449  1      401 582 -1
402 139  1      402 139  1      402 589 -1      403 593 -1
404 573  1      404 589 -1      405 165  1      405 408  1
405 604  1      405 606 -1      406 249  1      406 594 -1
407 344  1      407 497  1      407 613 -1      408 265  1
408 586  1      408 595 -1      409 166  1      409 206  1
409 348  1      409 389 -1      410 601 -1      411 410  1
411 581  1      411 601 -1      412 611 -1      413 608 -1
414 587 -1      415 367  1      415 603 -1      416 202  1
416 563  1      416 570  1      416 585 -1      417 587 -1
418 578 -1      419 578 -1      420 586 -1      421 585 -1
422 582 -1
;
run;
```

Figure 12.31 Partial Listing of the Valve Seat Data
Obs id days value
1 251 761 -1
2 252 759 -1
3 327 98 1
4 327 667 -1
5 328 326 1
6 328 653 1
7 328 653 1
8 328 667 -1
9 329 665 -1
10 330 84 1
11 330 667 -1
12 331 87 1
13 331 663 -1
14 389 646 1
15 389 653 -1
16 390 92 1
17 390 653 -1
18 391 651 -1
19 392 258 1
20 392 328 1

The following statements produce the graphical display in Figure 12.32.

```proc reliability;
unitid id;
mcfplot days*value(-1) / nocenprint;
run;
```

The UNITID statement specifies that the variable Id uniquely identifies each system. The MCFPLOT statement requests a plot of the MCF estimates as a function of the age variable Days, and it specifies -1 as the value of the variable Value, which identifies the end of history for each engine (system). The option NOCENPRINT specifies that only failure times, and not censoring times, be printed in the tabular output.

In Figure 12.32, the MCF estimates and confidence limits are plotted versus system age in days. The end-of-history ages are plotted in an area at the top of the plot. Except for the last few points, the plot is essentially a straight line, suggesting a constant replacement rate. Consequently, the prediction of future replacements of valve seats can be based on a fitted line in this case.

Figure 12.32 Mean Cumulative Function for the Number of Repairs

A partial listing of the tabular output is shown in Figure 12.33 and Figure 12.34. It contains a summary of the repair data, estimates of the MCF, the Nelson (1995) standard errors, and confidence intervals for the MCF.

Figure 12.33 Partial Listing of the Output for the Valve Seat Data
Recurrence Data Summary
Input Data Set WORK.VALVE
Observations Used 89
Number of Units 41
Number of Events 48

Figure 12.34 Partial Listing of the Output for the Valve Seat Data
Recurrence Data Analysis
Age Sample MCF Standard Error 95% Confidence Limits Unit ID
Lower Upper
61.00 0.024 0.024 -0.023 0.072 393
76.00 0.049 0.034 -0.018 0.116 395
84.00 0.073 0.041 -0.008 0.154 330
87.00 0.098 0.047 0.006 0.190 331
92.00 0.122 0.052 0.021 0.223 390
98.00 0.146 0.056 0.037 0.256 327
120.00 0.171 0.059 0.054 0.287 400
139.00 0.195 0.063 0.072 0.318 402
139.00 0.220 0.074 0.074 0.365 402
165.00 0.244 0.076 0.094 0.394 405
166.00 0.268 0.078 0.115 0.422 409
202.00 0.293 0.080 0.136 0.449 416
206.00 0.317 0.089 0.143 0.491 409
249.00 0.341 0.090 0.165 0.517 406
254.00 0.366 0.091 0.188 0.544 394
258.00 0.390 0.092 0.211 0.570 392
265.00 0.415 0.092 0.234 0.595 408
276.00 0.439 0.099 0.245 0.633 394
298.00 0.463 0.111 0.246 0.681 394
323.00 0.488 0.111 0.270 0.706 401
326.00 0.512 0.111 0.294 0.730 328
328.00 0.537 0.116 0.309 0.765 392
344.00 0.561 0.116 0.333 0.788 407
348.00 0.585 0.126 0.339 0.832 409
349.00 0.610 0.125 0.364 0.855 397
367.00 0.634 0.125 0.390 0.879 415
377.00 0.659 0.133 0.397 0.920 392
404.00 0.684 0.138 0.414 0.953 397
408.00 0.709 0.142 0.431 0.986 405
410.00 0.734 0.141 0.457 1.010 411
449.00 0.759 0.144 0.475 1.042 401
479.00 0.784 0.148 0.494 1.073 400
497.00 0.809 0.151 0.512 1.105 407
538.00 0.834 0.154 0.531 1.136 395
539.00 0.859 0.157 0.551 1.166 393
561.00 0.884 0.164 0.563 1.205 397
563.00 0.909 0.166 0.583 1.234 416
570.00 0.934 0.172 0.596 1.272 416
573.00 0.959 0.171 0.623 1.294 404
581.00 0.985 0.173 0.645 1.325 411
586.00 1.014 0.176 0.669 1.359 408
604.00 1.060 0.188 0.692 1.427 405
621.00 1.119 0.211 0.705 1.532 392
635.00 1.181 0.210 0.768 1.594 396
640.00 1.244 0.231 0.791 1.696 394
646.00 1.320 0.233 0.864 1.777 389
653.00 1.432 0.259 0.923 1.940 328
653.00 1.543 0.324 0.908 2.177 328

Parametric modeling of the repair process requires more assumptions than nonparametric modeling, and considerable work has been done in this area. Ascher and Feingold (1984), Tobias and Trindade (1995), and Rigdon and Basu (2000) describe parametric models for repair processes. For example, repairs are sometimes modeled as a nonhomogeneous Poisson process. The current release of the RELIABILITY procedure does not include this type of parametric modeling, although it is planned for future releases. The MCF plot might be a first step in modeling a repair process, but, in many cases, it provides the required answers without further analysis. An estimate of the MCF for a sample of systems aids engineers in determining the repair rate at any age and the increase or decrease of repair rate with population age. The estimate is also useful for predicting the number of future repairs.

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