Example 28.3: Computing Binomial Proportions for OneWay Frequency Tables
The binomial proportion is computed as the proportion
of observations for the first level of the variable that you are
studying. The following statements compute the
proportion of children with brown eyes (from the data set
in Example 28.1) and test this value
against the hypothesis that the proportion is 50%.
Also, these statements test whether the proportion of
children with fair hair is 28%.
proc freq data=Color order=freq;
weight Count;
tables Eyes / binomial alpha=.1;
tables Hair / binomial(p=.28);
title 'Hair and Eye Color of European Children';
run;
The first TABLES statement produces a frequency table for
eye color. The BINOMIAL option computes
the binomial proportion and confidence limits, and it tests the
hypothesis that the proportion for the first eye color level
(brown) is 0.5. The option ALPHA=.1 specifies that
90% confidence limits should be computed. The second
TABLES statement creates a frequency table for hair color
and computes the binomial proportion and confidence limits,
but it
tests that the proportion for the first hair color (fair) is 0.28.
These statements produce Output 28.3.1 and Output 28.3.2.
Output 28.3.1: Binomial Proportion for Eye Color
Hair and Eye Color of European Children 
Eye Color 
Eyes 
Frequency 
Percent 
Cumulative Frequency 
Cumulative Percent 
brown 
341 
44.75 
341 
44.75 
blue 
222 
29.13 
563 
73.88 
green 
199 
26.12 
762 
100.00 
Binomial Proportion for Eyes = brown 
Proportion 
0.4475 
ASE 
0.0180 
90% Lower Conf Limit 
0.4179 
90% Upper Conf Limit 
0.4771 


Exact Conf Limits 

90% Lower Conf Limit 
0.4174 
90% Upper Conf Limit 
0.4779 
Test of H0: Proportion = 0.5 
ASE under H0 
0.0181 
Z 
2.8981 
Onesided Pr < Z 
0.0019 
Twosided Pr > Z 
0.0038 

The frequency table in Output 28.3.1 displays the variable
values in order of descending frequency count.
Since the first variable level is 'brown', PROC FREQ computes
the binomial proportion of children with brown eyes.
PROC FREQ also computes its asymptotic standard error (ASE),
and asymptotic and exact 90% confidence limits. If you do
not specify the ALPHA= option, then PROC FREQ computes the
default 95% confidence limits.
Because the value of Z is less than zero, PROC FREQ
computes a leftsided pvalue (0.0019). This small
pvalue supports the alternative hypothesis that the true
value of the proportion of children with brown eyes is less
than 50%.
Output 28.3.2: Binomial Proportion for Hair Color
Hair and Eye Color of European Children 
Hair Color 
Hair 
Frequency 
Percent 
Cumulative Frequency 
Cumulative Percent 
fair 
228 
29.92 
228 
29.92 
medium 
217 
28.48 
445 
58.40 
dark 
182 
23.88 
627 
82.28 
red 
113 
14.83 
740 
97.11 
black 
22 
2.89 
762 
100.00 
Binomial Proportion for Hair = fair 
Proportion 
0.2992 
ASE 
0.0166 
95% Lower Conf Limit 
0.2667 
95% Upper Conf Limit 
0.3317 


Exact Conf Limits 

95% Lower Conf Limit 
0.2669 
95% Upper Conf Limit 
0.3331 
Test of H0: Proportion = 0.28 
ASE under H0 
0.0163 
Z 
1.1812 
Onesided Pr > Z 
0.1188 
Twosided Pr > Z 
0.2375 

Output 28.3.2 displays the results from the second TABLES
statement. PROC FREQ computes the default 95% confidence limits
since the ALPHA= option is not specified. The value of Z
is greater than zero, so PROC FREQ computes a rightsided pvalue
(0.1188). This large pvalue provides insufficient
evidence to reject the null hypothesis that the proportion
of children with fair hair is 28%.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.