### 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 16.40 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 16.40.

Figure 16.40: 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 16.41 and the tabular listing in Figure 16.42:

```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 16.42: 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 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.