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 The RELIABILITY Procedure

## Analysis of Binomial Data

This example illustrates the analysis of binomial proportions using capacitor failure data from nine circuit boards given by Nelson (1982, p. 451). The following statements create and list a SAS data set named BINEX containing the data.
```   data binex;
input board sample fail;
datalines;
1 84 2
2 72 3
3 72 5
4 119 19
5 538 21
6 51 2
7 517 9
8 462 18
9 143 2
;

proc print data=binex;
run;
```

Figure 30.23 displays a listing of the data. The variable BOARD identifies the circuit board, the variable SAMPLE provides the number of capacitors on the boards, and the variable FAIL provides the number of capacitors failing on the boards.

 Obs board sample fail 1 1 84 2 2 2 72 3 3 3 72 5 4 4 119 19 5 5 538 21 6 6 51 2 7 7 517 9 8 8 462 18 9 9 143 2
Figure 30.23: Listing of the Capacitor Data

The following statements analyze the proportion of capacitors failing.

```   proc reliability;
distribution binomial;
analyze fail(sample) = board / predict(1000)
tolerance(.05);
run;
```

The DISTRIBUTION statement specifies the binomial distribution. The analysis requested with the ANALYZE statement consists of tabular output only. Graphical output is not available for the binomial distribution. The variable FAIL provides the number of capacitors failing on each board, the variable SAMPLE provides the sample size (number of capacitors) for each board, and the variable BOARD identifies the individual boards. The statement option PREDICT(1000) requests the predicted number of capacotors failing and prediction limits in a future sample of size 1000. The option TOLERANCE(.05) requests the sample size required to estimate the binomial proportion to within 0.05. Figure 30.24 displays the results of the analysis.

The "Pooled Data Analysis" table displays the estimated binomial probability and exact binomial confidence limits when data from all boards are pooled. The chi-squared value and p-value for a test of equality of the binomial probabilities for all of the boards are also shown. In this case, the p-value is less than 0.05, so you reject the test of equality at the 0.05 level.

The "Predicted Values and Limits" table provides the predicted failure count and prediction limits for the number of capacitors that would fail in a future sample of size 1000 for the pooled data, as requested with the PREDICT(1000) option. The "Sample Size for Estimation" table gives the sample size required to estimate the binomial probability to within 0.05 for the pooled data, as requested with the TOLERANCE(.05) option.

The "Estimates by Group" table supplies the estimated binomial probability, confidence limits, and the contribution to the total chi-squared for each board. The pooled values are shown on the last line of the table.

The "Predicted Values by Group" table gives the predicted counts in a future sample of size 1000, prediction limits, and the sample size required to estimate the binomial probability to within the tolerance of 0.05 for each board. Values for the pooled data are shown on the last line of the table.

 The RELIABILITY Procedure

 Model Information - All Groups Input Data Set WORK.BINEX Events Variable fail Trials Variable sample Distribution Binomial Confidence Coefficient 95% Observations Used 9

 Binomial Data Analysis Pooled Events 81.0000 Pooled Trials 2058.0000 Estimate of Proportion 0.0394 Lower Limit For Proportion 0.0314 Upper Limit For Proportion 0.0487 ChiSquare 56.8504 Pr>ChiSquare 0.0000

 Predicted Value and Limits Sample Size For Prediction 1000.0000 Predicted Count 39.3586 Lower Prediction Limit 24.8424 Upper Prediction Limit 56.3237

 Sample Size For Estimation Tolerance 0.0500 Sample Size For Tolerance 58.0975

 Estimates By Group Group Events Trials Prop 95% Confidence Limits X2 Lower Upper 1 2 84 0.0238 0.0029 0.0834 0.5371 2 3 72 0.0417 0.0087 0.1170 0.0101 3 5 72 0.0694 0.0229 0.1547 1.7237 4 19 119 0.1597 0.0990 0.2381 45.5528 5 21 538 0.0390 0.0243 0.0590 0.0015 6 2 51 0.0392 0.0048 0.1346 0.0000 7 9 517 0.0174 0.0080 0.0328 6.5884 8 18 462 0.0390 0.0233 0.0609 0.0019 9 2 143 0.0140 0.0017 0.0496 2.4348 Pooled 81 2058 0.0394 0.0314 0.0487 56.8504

 Predicted/Tolerance Values By Group Group PredictedCount 95% Prediction Limits ToleranceSample Size Lower Upper 1 23.81 1.5476 88.5824 35.71 2 41.67 6.9416 124.6142 61.36 3 69.44 20.4052 165.3499 99.30 4 159.66 91.9722 254.5444 206.17 5 39.03 20.1599 64.7140 57.64 6 39.22 3.3970 144.2494 57.90 7 17.41 5.3506 36.7531 26.28 8 38.96 19.3343 66.3850 57.53 9 13.99 0.3851 53.0715 21.19 Pooled 39.36 24.8424 56.3237 58.10
Figure 30.24: Analysis of the Capacitor Data

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