Analysis of Binomial Data

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
Group	Events	Trials	Prop	Lower	Upper	X2
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
Group	Predicted Count	Lower	Upper	Tolerance Sample Size
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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