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 chisquared 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
chisquared 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 
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 

Figure 30.24: Analysis of the Capacitor Data
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.