Example 28.3: Computing Binomial Proportions for One-Way 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 |
| One-sided Pr < Z |
0.0019 |
| Two-sided 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 left-sided p-value (0.0019). This small
p-value 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 |
| One-sided Pr > Z |
0.1188 |
| Two-sided 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 right-sided p-value
(0.1188). This large p-value 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.
