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 n1 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
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 p0, where the value of
p0 is specified by the P= option in the TABLES statement.
If you do not specify a value for the P= option, PROC FREQ uses p0 = 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 p0. 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 p0. The one-sided p-value P1 can be expressed
where Z has a standard normal distribution.
The two-sided p-value P2 is computed as
When you specify the BINOMIAL option in the EXACT
statement, PROC FREQ also computes an exact test of the
null hypothesis H0: p = p0. To compute this exact
test, PROC FREQ uses the binomial probability function
where the variable X has a binomial distribution with parameters
n and p0.
To compute , PROC FREQ sums these binomial
probabilities over x from zero to n1. To compute ,PROC FREQ sums these binomial probabilities over x from n1 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.