Tests for equality of proportions (independent samples)

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.

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.

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.