Binomial Proportion

The FREQ Procedure

Binomial Proportion

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.

$\hat{p} = n_{1} / n$

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 computed as

$se(\hat{p}) = \sqrt{ \hat{p} (1 - \hat{p} ) / n }$

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

$\hat{p} +- z_{\alpha/2} \cdot se(\hat{p})$

where $z_{\alpha/2}$ is the $100(1 - \alpha/2)$ percentile of the standard normal distribution. The confidence level $\alpha$ 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

$z = \frac{ \hat{p} - p_0 }{\sqrt{p_0 (1-p_0)/n}}$

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 as

$P_{1} = {\rm Prob} (Z \gt z) {\rm if} z \gt 0$

$P_{1} = {\rm Prob} (Z \lt z) {\rm if} z \leq 0$

where Z has a standard normal distribution. The two-sided p-value P₂ is computed as

$P_{2} = {\rm Prob} (| Z| \gt | z|)$

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

${\rm Prob} (X = x | p_0 ) = ( n \ x ) p_0^x (1-p_0)^{(n-x)} x = 0,1,2, ... ,n$

where the variable X has a binomial distribution with parameters n and p₀. To compute ${\rm Prob} (X \leq n_1)$ , PROC FREQ sums these binomial probabilities over x from zero to n₁. To compute ${\rm Prob} (X \geq n_1)$ ,PROC FREQ sums these binomial probabilities over x from n₁ to n. Then the exact one-sided p-value is

$P_{1} = {\rm min} ( {\rm Prob}(X \leq n_1 | p_0 ), {\rm Prob}(X \geq n_1 | p_0 ) )$

and the exact two-sided p-value is

$P_{2} = 2 \cdot P_{1}$

Chapter Contents
Previous
Next
Top