Fisher's Exact Test

For a 2x2 contingency table with fixed row and column margins, Fisher's exact test computes the exact probability of the observed table under the hypergeometric distribution, then sums this probability over the observed table and every other table with the same margins that is as extreme or more extreme in the direction of the alternative hypothesis, yielding an exact p-value with no large-sample approximation involved. The test generalizes beyond 2x2 tables to $r\times c$ tables, though the computation becomes substantially more demanding as the table grows, requiring network algorithms rather than direct enumeration of every possible table.

In a 2x2 table with cells 8, 2 / 1, 9 ($n=20$), several expected counts fall below 5, making the chi-square approximation unreliable. Fisher's exact test instead sums the hypergeometric probabilities of this table and every more extreme table with the same margins, producing an exact p-value directly, with no distributional approximation involved.

• Fisher's exact test conditions on both margins of the table being fixed, treating the table as a single draw from the hypergeometric distribution given those fixed margins, a framing that is exactly correct when both margins genuinely are fixed by the study design, and only an approximation otherwise.
• Because it is exact rather than asymptotic, Fisher's exact test remains valid at any sample size, including the very small samples and sparse tables where the Pearson chi-square test's large-sample approximation breaks down.
• The test can be run one-sided or two-sided, and two-sided p-values are conventionally computed by summing the probabilities of all tables at least as extreme as the observed one, a definition that is unambiguous for 2x2 tables but admits more than one reasonable convention for larger tables.