Analysis of Interval Age Recurrence Data

You can analyze recurrence data when the recurrence ages are grouped into intervals, instead of being exact ages. Figure 14.38 shows a listing of a SAS data set containing field data on replacements of defrost controls in 22,914 refrigerators, whose ages are grouped by months in service. Nelson (2002, problem 5.2, chapter 5) presents these data. Grouping the control data on the 22,914 refrigerators into age intervals enables you to represent the data by 29 data records, instead of requiring a single data record for each refrigerator, as required for exact recurrence data.

The variables Lower and Upper are the lower and upper monthly interval endpoints, Recurrences is the number of defrost control replacements in each month, and Censored is the number of refrigerator histories censored in each month—that is, the number with current age in the monthly interval. Data are entered as shown in Figure 14.38.

Figure 14.38 Listing of the Defrost Controls Data
Obs Lower Upper Recurrences Censored
1 0 1 83 0
2 1 2 35 0
3 2 3 23 0
4 3 4 15 0
5 4 5 22 0
6 5 6 16 3
7 6 7 13 36
8 7 8 12 24
9 8 9 15 29
10 9 10 15 37
11 10 11 24 40
12 11 12 12 20041
13 12 13 7 14
14 13 14 11 17
15 14 15 15 13
16 15 16 6 28
17 16 17 8 22
18 17 18 9 27
19 18 19 9 64
20 19 20 5 94
21 20 21 6 119
22 21 22 6 118
23 22 23 6 138
24 23 24 5 1188
25 24 25 7 17
26 25 26 5 28
27 26 27 5 99
28 27 28 6 128
29 28 29 3 590

The following SAS statements create the plot of the sample MCF of defrost control replacement shown in Figure 14.39 and the tabular listing in Figure 14.40:
proc reliability data=defrost;
   mcfplot ( interval    = Lower Upper
             recurrences = Recurrences
             censor      = Censored ) / plotsymbol = X
                                        vaxis = 0 to .12 by .04
                                        interpolate = join;
run;

Pointwise confidence limits are included on the plot and in the tabular listing. These limits are approximate, and are usually shorter than the correct limits, which have not been developed for interval data.

Here, INTERVAL = LOWER UPPER specifies the input data set variables Lower and Upper as the age interval endpoints. The variable Recurrences identifies the number of recurrences (defrost control replacements) in each time interval, and Censored identifies the number of units censored in each interval (number in an age interval or removed from the sample in an age interval).

Figure 14.39 MCF Plot for the Defrost Controls
MCF Plot for the Defrost Controls

Figure 14.40 Listing of the Output for the Defrost Controls Data
Recurrence Data Summary
Input Data Set WORK.DEFROST
Observations Used 29
Number of Units 22914
Number of Events 404

Recurrence Data Analysis
  Endpoints Sample MCF Standard Error Naive 95% Confidence
Limits
Lower Upper Lower Upper
  0.00 1.00 0.004 0.000 0.003 0.004
  1.00 2.00 0.005 0.000 0.004 0.006
  2.00 3.00 0.006 0.001 0.005 0.007
  3.00 4.00 0.007 0.001 0.006 0.008
  4.00 5.00 0.008 0.001 0.007 0.009
  5.00 6.00 0.008 0.001 0.007 0.010
  6.00 7.00 0.009 0.001 0.008 0.010
  7.00 8.00 0.010 0.001 0.008 0.011
  8.00 9.00 0.010 0.001 0.009 0.012
  9.00 10.00 0.011 0.001 0.010 0.012
  10.00 11.00 0.012 0.001 0.011 0.013
  11.00 12.00 0.013 0.001 0.011 0.014
  12.00 13.00 0.015 0.001 0.013 0.018
  13.00 14.00 0.020 0.002 0.016 0.023
  14.00 15.00 0.025 0.002 0.021 0.030
  15.00 16.00 0.027 0.002 0.023 0.032
  16.00 17.00 0.031 0.003 0.025 0.036
  17.00 18.00 0.034 0.003 0.028 0.040
  18.00 19.00 0.038 0.003 0.031 0.044
  19.00 20.00 0.040 0.003 0.033 0.046
  20.00 21.00 0.042 0.003 0.035 0.049
  21.00 22.00 0.045 0.004 0.038 0.052
  22.00 23.00 0.048 0.004 0.040 0.055
  23.00 24.00 0.051 0.004 0.043 0.059
  24.00 25.00 0.059 0.005 0.049 0.069
  25.00 26.00 0.065 0.006 0.054 0.077
  26.00 27.00 0.072 0.006 0.059 0.084
  27.00 28.00 0.081 0.007 0.066 0.096
* 28.00 29.00 0.091 0.010 0.072 0.110
* The estimate and limits for this interval may not be appropriate.

The last interval is always marked with a footnote indicating that estimates for the last interval may be biased since censoring ages often are not uniformly spread over that interval.