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The RELIABILITY Procedure

Comparison of Two Samples of Repair Data

Nelson (1995) and Doganaksoy and Nelson (1991) show how the difference of MCFs from two samples can be used to compare the populations from which they are drawn. The RELIABILITY procedure provides Doganaksoy and Nelson's confidence intervals for the pointwise difference of the two MCFs, which can be used to assess whether the difference is statistically significant.

Doganaksoy and Nelson (1991) give an example of two samples of locomotives with braking grids from two different production batches. Figure 30.20 is a listing of the data. The variable ID is a unique identifier for individual locomotives. The variable DAYS provides the locomotive age in days. The variable VALUE is 1 if the age corresponds to a valve seat replacement or -1 if the age corresponds to the locomotive's latest age (the current end of its history). The variable SAMPLE is a group variable that identifies the grid production batch.

 
Obs sample ID days value
1 Sample1 S1-01 462 1
2 Sample1 S1-01 730 -1
3 Sample1 S1-02 364 1
4 Sample1 S1-02 391 1
5 Sample1 S1-02 548 1
6 Sample1 S1-02 724 -1
7 Sample1 S1-03 302 1
8 Sample1 S1-03 444 1
9 Sample1 S1-03 500 1
10 Sample1 S1-03 730 -1
11 Sample1 S1-04 250 1
12 Sample1 S1-04 730 -1
13 Sample1 S1-05 500 1
14 Sample1 S1-05 724 -1
15 Sample1 S1-06 88 1
16 Sample1 S1-06 724 -1
17 Sample1 S1-07 272 1
18 Sample1 S1-07 421 1
19 Sample1 S1-07 552 1
20 Sample1 S1-07 625 1
21 Sample1 S1-07 719 -1
22 Sample1 S1-08 481 1
23 Sample1 S1-08 710 -1
24 Sample1 S1-09 431 1
25 Sample1 S1-09 710 -1
26 Sample1 S1-10 367 1
27 Sample1 S1-10 710 -1
28 Sample1 S1-11 635 1
29 Sample1 S1-11 650 1
30 Sample1 S1-11 708 -1
31 Sample1 S1-12 402 1
32 Sample1 S1-12 700 -1
33 Sample1 S1-13 33 1
34 Sample1 S1-13 687 -1
35 Sample1 S1-14 287 1
36 Sample1 S1-14 687 -1
37 Sample1 S1-15 317 1
38 Sample1 S1-15 498 1
39 Sample1 S1-15 657 -1
40 Sample2 S2-01 203 1
41 Sample2 S2-01 211 1
42 Sample2 S2-01 277 1
43 Sample2 S2-01 373 1
44 Sample2 S2-01 511 -1
45 Sample2 S2-02 293 1
46 Sample2 S2-02 503 -1
47 Sample2 S2-03 173 1
48 Sample2 S2-03 470 -1
49 Sample2 S2-04 242 1
50 Sample2 S2-04 464 -1
51 Sample2 S2-05 39 1
52 Sample2 S2-05 464 -1
53 Sample2 S2-06 91 1
54 Sample2 S2-06 462 -1
55 Sample2 S2-07 119 1
56 Sample2 S2-07 148 1
57 Sample2 S2-07 306 1
58 Sample2 S2-07 461 -1
59 Sample2 S2-08 382 1
60 Sample2 S2-08 460 -1
61 Sample2 S2-09 250 1
62 Sample2 S2-09 434 -1
63 Sample2 S2-10 192 1
64 Sample2 S2-10 448 -1
65 Sample2 S2-11 369 1
66 Sample2 S2-11 448 -1
67 Sample2 S2-12 22 1
68 Sample2 S2-12 447 -1
69 Sample2 S2-13 54 1
70 Sample2 S2-13 441 -1
71 Sample2 S2-14 194 1
72 Sample2 S2-14 432 -1
73 Sample2 S2-15 61 1
74 Sample2 S2-15 419 -1
75 Sample2 S2-16 19 1
76 Sample2 S2-16 185 1
77 Sample2 S2-16 419 -1
78 Sample2 S2-17 187 1
79 Sample2 S2-17 416 -1
80 Sample2 S2-18 93 1
81 Sample2 S2-18 205 1
82 Sample2 S2-18 264 1
83 Sample2 S2-18 415 -1
Figure 30.20: Listing of the Braking Grids Data

The following statements request the Nelson (1995) nonparametric estimate and confidence limits for the difference of the MCF functions shown in Figure 30.21 for the braking grids. 

   symbol c=blue v=plus;
   proc reliability data=grids;
      unitid ID;
      mcfplot days*value(-1) = sample / mcfdiff
                                        cframe=ligr ;
      inset / cfill  = ywh;
   run;

The MCFPLOT statement requests a plot of each MCF estimate as a function of age (provided by DAYS), and it specifies that the end of history for each system is identified by VALUE equal to -1. The variable SAMPLE identifies the two samples of braking grids. The option MCFDIFF requests that the difference between the MCFs of the two groups given in the variable SAMPLE be computed and plotted. Confidence limits for the MCF difference are also computed and plotted. The UNITID statement specifies that the variable ID uniquely identifies each system.

Figure 30.21 shows the plot of the MCF difference function and pointwise 95% confidence intervals. Since the pointwise confidence limits do not include zero for some system ages, the difference between the two populations is statistically significant.

A partial listing of the tabular output is shown in Figure 30.22. It contains a summary of the repair data for the two samples, estimates, standard errors, and confidence intervals for the MCF difference.

grids.gif (5926 bytes)

Figure 30.21: Mean Cumulative Function Difference

 
The RELIABILITY Procedure

MCF Difference Data Summary
Input Data Set WORK.GRIDS
Group 1 Sample1
Observations Used 39
Number of Units 15
Number of Events 24
Group 2 Sample2
Observations Used 44
Number of Units 18
Number of Events 26
 
Sample MCF Differences
Age MCF Difference Standard Error 95% Confidence Limits Unit ID
Lower Upper
19.00 -0.056 0.056 -0.164 0.053 S2-16
22.00 -0.111 0.076 -0.261 0.038 S2-12
33.00 -0.044 0.101 -0.243 0.154 S1-13
39.00 -0.100 0.112 -0.320 0.120 S2-05
54.00 -0.156 0.121 -0.392 0.081 S2-13
61.00 -0.211 0.127 -0.461 0.039 S2-15
88.00 -0.144 0.142 -0.422 0.133 S1-06
91.00 -0.200 0.146 -0.486 0.086 S2-06
93.00 -0.256 0.149 -0.548 0.037 S2-18
119.00 -0.311 0.151 -0.607 -0.015 S2-07
148.00 -0.367 0.172 -0.703 -0.030 S2-07
173.00 -0.422 0.171 -0.758 -0.087 S2-03
185.00 -0.478 0.188 -0.846 -0.110 S2-16
187.00 -0.533 0.185 -0.897 -0.170 S2-17
192.00 -0.589 0.182 -0.946 -0.232 S2-10
194.00 -0.644 0.177 -0.992 -0.297 S2-14
203.00 -0.700 0.172 -1.037 -0.363 S2-01
205.00 -0.756 0.183 -1.115 -0.396 S2-18
211.00 -0.811 0.194 -1.191 -0.432 S2-01
242.00 -0.867 0.185 -1.230 -0.503 S2-04
250.00 -0.856 0.185 -1.218 -0.494 S1-04
264.00 -0.911 0.208 -1.319 -0.503 S2-18
272.00 -0.844 0.214 -1.264 -0.424 S1-07
277.00 -0.900 0.234 -1.359 -0.441 S2-01
287.00 -0.833 0.238 -1.300 -0.367 S1-14
293.00 -0.889 0.228 -1.337 -0.441 S2-02
302.00 -0.822 0.231 -1.275 -0.369 S1-03
306.00 -0.878 0.248 -1.364 -0.391 S2-07
317.00 -0.811 0.250 -1.300 -0.322 S1-15
364.00 -0.744 0.250 -1.233 -0.255 S1-02
367.00 -0.678 0.248 -1.164 -0.191 S1-10
369.00 -0.733 0.237 -1.199 -0.268 S2-11
373.00 -0.789 0.265 -1.309 -0.269 S2-01
382.00 -0.844 0.254 -1.342 -0.347 S2-08
391.00 -0.778 0.269 -1.306 -0.250 S1-02
402.00 -0.711 0.266 -1.232 -0.190 S1-12
421.00 -0.644 0.279 -1.191 -0.098 S1-07
431.00 -0.578 0.273 -1.113 -0.043 S1-09
444.00 -0.511 0.283 -1.066 0.044 S1-03
462.00 -0.444 0.275 -0.984 0.095 S1-01
481.00 -0.378 0.266 -0.899 0.143 S1-08
498.00 -0.311 0.273 -0.846 0.224 S1-15
500.00 -0.244 0.261 -0.756 0.267 S1-05
500.00 -0.178 0.283 -0.733 0.377 S1-03
548.00 -0.111 0.303 -0.704 0.482 S1-02
552.00 -0.044 0.320 -0.672 0.583 S1-07
625.00 0.022 0.349 -0.663 0.707 S1-07
635.00 0.089 0.336 -0.569 0.747 S1-11
650.00 0.156 0.335 -0.500 0.811 S1-11
Figure 30.22: Partial Listing of the Output for the Braking Grids Data

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