The RELIABILITY Procedure
 Weibull Analysis of Interval Data with Common Inspection Schedule

Table 12.1 shows data for 167 identical turbine parts provided by Nelson (1982, p. 415). The parts were inspected at certain times to determine which parts had cracked since the last inspection. The times at which parts develop cracks are to be fitted with a Weibull distribution.

Table 12.1 Turbine Part Cracking Data

Inspection (Months)

Number

Start

End

Cracked

Cumulative

0

6.12

5

5

6.12

19.92

16

21

19.92

29.64

12

33

29.64

35.40

18

51

35.40

39.72

18

69

39.72

45.24

2

71

45.24

52.32

6

77

52.32

63.48

17

94

63.48

Survived

73

167

Table 12.1 shows the time in months of each inspection period and the number of cracked parts found in each period. These data are said to be interval censored since only the time interval in which failures occurred is known, not the exact failure times. Seventy-three parts had not cracked at the last inspection, which took place at 63.48 months. These 73 lifetimes are right censored, since the lifetimes are known only to be greater than 63.48 months.

The interval data in this example are read from a SAS data set with a special structure. All units must have a common inspection schedule. This type of interval data is called readout data. The following SAS program creates the SAS data set named CRACKS, shown in Figure 12.11, and provides the data in Table 12.1 with this structure:

```data cracks;
input time units fail;
datalines;
6.12  167 5
19.92 162 16
29.64 146 12
35.4  134 18
39.72 116 18
45.24 98  2
52.32 96  6
63.48 90 17
;
```

The variable Time is the inspection time—that is, the upper endpoint of each interval. The variable Units is the number of unfailed units at the beginning of each interval, and the variable Fail is the number of units with cracks at the inspection time.

Figure 12.11 Listing of the Turbine Part Cracking Data
Obs time units fail
1 6.12 167 5
2 19.92 162 16
3 29.64 146 12
4 35.40 134 18
5 39.72 116 18
6 45.24 98 2
7 52.32 96 6
8 63.48 90 17

The following statements use the RELIABILITY procedure to produce the probability plot in Figure 12.12 for the data in the data set CRACKS:

```proc reliability data=cracks;
freq fail;
nenter units;
distribution weibull;
ppout
pconfplt
noconf;
run;
```

The FREQ statement specifies that the variable Fail provides the number of failures in each interval. The NENTER statement specifies that the variable Units provides the number of unfailed units at the beginning of each interval. The DISTRIBUTION statement specifies that the Weibull distribution be used for parameter estimation and probability plotting. The PROBPLOT statement requests a probability plot of the data.

The PROBPLOT statement option READOUT indicates that the data in the CRACKS data set are readout (or interval) data. The option PCONFPLT specifies that confidence intervals for the cumulative probability of failure be plotted. The confidence intervals for the cumulative probability are based on the binomial distribution for time intervals until right censoring occurs. For time intervals after right censoring occurs, the binomial distribution is not valid, and a normal approximation is used to compute confidence intervals.

The option NOCONF suppresses the display of confidence intervals for distribution percentiles in the probability plot.

Figure 12.12 Weibull Probability Plot for the Part Cracking Data

A listing of the tabular output produced by the preceding SAS statements is shown in Figure 12.13 and Figure 12.14. By default, the specified Weibull distribution is fitted by maximum likelihood. The line plotted on the probability plot and the tabular output summarize this fit. For interval data, the estimated cumulative probabilities and associated confidence intervals are tabulated. In addition, general fit information, parameter estimates, percentile estimates, standard errors, and confidence intervals are tabulated.

Figure 12.13 Partial Listing of the Tabular Output for the Part Cracking Data
The RELIABILITY Procedure

Model Information
Input Data Set WORK.CRACKS
Analysis Variable time
Frequency Variable fail
NENTER Variable units
Distribution Weibull
Estimation Method Maximum Likelihood
Confidence Coefficient 95%
Observations Used 8

Cumulative Probability Estimates
Probability
Pointwise 95% Confidence
Limits
Standard Error
Lower Upper
. 6.12 0.0299 0.0125 0.0699 0.0132
6.12 19.92 0.1257 0.0834 0.1852 0.0257
19.92 29.64 0.1976 0.1440 0.2649 0.0308
29.64 35.4 0.3054 0.2403 0.3793 0.0356
35.4 39.72 0.4132 0.3410 0.4893 0.0381
39.72 45.24 0.4251 0.3524 0.5013 0.0383
45.24 52.32 0.4611 0.3869 0.5370 0.0386
52.32 63.48 0.5629 0.4868 0.6361 0.0384

 Algorithm converged.

Summary of Fit
Observations Used 8
Right Censored Values 73
Left Censored Values 5
Interval Censored Values 89
Maximum Loglikelihood -309.6684

Figure 12.14 Partial Listing of the Tabular Output for the Part Cracking Data
Weibull Parameter Estimates
Parameter Estimate Standard Error Asymptotic Normal
95% Confidence Limits
Lower Upper
EV Location 4.2724 0.0744 4.1265 4.4182
EV Scale 0.6732 0.0664 0.5549 0.8168
Weibull Scale 71.6904 5.3335 61.9634 82.9444
Weibull Shape 1.4854 0.1465 1.2242 1.8022

Other Weibull Distribution Parameters
Parameter Value
Mean 64.7966
Mode 33.7622
Median 56.0144
Standard Deviation 44.3943

Weibull Percentile Estimates
Percent Estimate Standard Error Asymptotic Normal
95% Confidence Limits
Lower Upper
0.1 0.68534385 0.29999861 0.29060848 1.61625083
0.2 1.09324674 0.42889777 0.50673224 2.3586193
0.5 2.02798319 0.67429625 1.05692279 3.8912169
1 3.23938972 0.93123832 1.84401909 5.69063837
2 5.18330703 1.2581604 3.22101028 8.34106988
5 9.70579945 1.78869256 6.76335893 13.9283666
10 15.7577991 2.22445157 11.9491109 20.7804776
20 26.1159906 2.6327383 21.4337103 31.821134
30 35.8126238 2.90557264 30.547517 41.9852137
40 45.6100472 3.27409792 39.6239146 52.5005271
50 56.0143651 3.89410377 48.8792027 64.1910859
60 67.5928125 4.90210777 58.6364803 77.917165
70 81.2334227 6.46932648 69.4938134 94.9562075
80 98.7644937 8.95137184 82.6900902 117.963654
90 125.694556 13.5078386 101.821995 155.164133
95 150.057755 18.2060035 118.300075 190.340791
99 200.437864 29.1957544 150.658574 266.66479
99.9 263.348102 44.7205513 188.791789 367.347666

In this example, the number of unfailed units at the beginning of an interval minus the number failing in the interval is equal to the number of unfailed units entering the next interval. This is not always the case since some unfailed units might be removed from the test at the end of an interval; that is, they might be right censored. The special structure of the input SAS data set required for interval data enables the RELIABILITY procedure to analyze this more general case.

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