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 chisquare value and pvalue for a test of equality of the binomial probabilities for all of the boards are also shown. In this case, the pvalue 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 chisquare 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
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 