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 |
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.