When you specify the BINOMIAL option in the TABLES
statement, PROC FREQ computes a binomial proportion
for one-way tables. This is the proportion of
observations in the first variable level, or class,
that appears in the output.

where *n*_{1} is the frequency for the first level
and *n* is the total frequency for the one-way table.
The standard error for the binomial proportion is
computed as

Using the normal approximation to the binomial
distribution, PROC FREQ constructs asymptotic
confidence limits for *p* according to

where is the percentile of the
standard normal distribution. The confidence level is
determined by the ALPHA= option, which, by default, equals 0.05 and
produces 95% confidence limits. Additionally, PROC FREQ computes
exact confidence limits for the binomial proportion using
the *F* distribution method given in Collett (1991) and
also described by Leemis and Trivedi (1996).
PROC FREQ computes an asymptotic test of the hypothesis that
the binomial proportion equals *p*_{0}, where the value of
*p*_{0} is specified by the P= option in the TABLES statement.
If you do not specify a value for the P= option, PROC FREQ uses *p*_{0} = 0.5
by default. The asymptotic test statistic is

PROC FREQ computes one-sided and two-sided *p*-values
for this test. When the test statistic *z* is
greater than zero, its expected value under the null hypothesis,
PROC FREQ computes the right-sided *p*-value, which is
the probability of a larger value of the statistic occurring
under the null hypothesis. A small right-sided *p*-value
supports the alternative hypothesis that the true value of
the proportion is greater than *p*_{0}. When the test statistic
is less than or equal to zero, PROC FREQ computes the
left-sided *p*-value, which is the probability of a smaller
value of the statistic occurring under the null hypothesis.
A small left-sided *p*-value supports the alternative
hypothesis that the true value of the proportion is less
than *p*_{0}. The one-sided *p*-value *P*_{1} can be expressed
as

where *Z* has a standard normal distribution.
The two-sided *p*-value *P*_{2} is computed as

When you specify the BINOMIAL option in the EXACT
statement, PROC FREQ also computes an exact test of the
null hypothesis *H*_{0}: *p* = *p*_{0}. To compute this exact
test, PROC FREQ uses the binomial probability function

where the variable X has a binomial distribution with parameters
*n* and *p*_{0}.
To compute , PROC FREQ sums these binomial
probabilities over *x* from zero to *n*_{1}. To compute ,PROC FREQ sums these binomial probabilities over *x* from *n*_{1} to *n*.
Then the exact one-sided *p*-value is

and the exact two-sided *p*-value is

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.