Tests for equality of proportions (independent samples)Tests for the equality of proportions of independent samples and their properties and assumptions.

Test	Purpose
Fisher exact	Test if the two proportions are equal. Assumes fixed marginal distributions. Although not naturally fixed in most studies, this test still applies by conditioning on the marginal totals. Uses hypergeometric distribution and computes an exact p-value. The exact p-value is conservative, that is, the actual rejection rate is below the nominal significance level. Recommended for small sample sizes or sparse data.
Score Z	Test if the two proportions are equal. Uses the score statistic and computes an asymptotic p-value. Note that the two-sided Score Z test is equivalent to Pearson's X² test.
Pearson X²	Test if the proportions are equal in each category. Uses the score statistic and computes an asymptotic p-value. Note that the asymptotic distribution of the X² test is the same regardless of whether only the total is fixed, both marginal distributions are fixed, or only one marginal distribution is fixed.

Test

Purpose

Fisher exact

Test if the two proportions are equal.

Assumes fixed marginal distributions. Although not naturally fixed in most studies, this test still applies by conditioning on the marginal totals.

Uses hypergeometric distribution and computes an exact p-value. The exact p-value is conservative, that is, the actual rejection rate is below the nominal significance level. Recommended for small sample sizes or sparse data.

Score Z

Test if the two proportions are equal.

Uses the score statistic and computes an asymptotic p-value.

Note that the two-sided Score Z test is equivalent to Pearson's X² test.

Pearson X²

Test if the proportions are equal in each category.

Uses the score statistic and computes an asymptotic p-value.

Note that the asymptotic distribution of the X² test is the same regardless of whether only the total is fixed, both marginal distributions are fixed, or only one marginal distribution is fixed.

Related concepts

Equality of proportions hypothesis test

Exact and asymptotic p-values

Continuity correction

Chi-square, Fisher exact or McNemar?

Cohen's kappa and weighted kappa

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