### Analysis of Binomial Data

This example illustrates the analysis of binomial proportions of capacitor failures from nine circuit boards. The data are 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
;
```

Figure 16.43 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.

Figure 16.43: Listing of the Capacitor Data

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

The following statements analyze the proportion of capacitors failing:

```proc reliability data=Binex;
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 capacitors 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 16.44 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-square 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-square for each board. The pooled values are shown in 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 in the last line of the table.

Figure 16.44: Analysis of the Capacitor Data

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 Predicted
Count
95% Prediction Limits Tolerance
Sample 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