5.8 Fisher’s Exact Test

Fisher’s exact test is an “exact test” in that the p-value is calculated exactly from the hypergeometric distribution rather than relying on the approximation that the test statistic distribution approaches \(\chi^2\) as \(n \rightarrow \infty\).

The test is applicable in situations where

• the row totals \(n_{i+}\) and the column totals \(n_+j\) are fixed by study design (rarely applies), and
• the expected values of >20% of cells (at least 1 cell in a 2x2 table) have expected cell counts >5, and no expected cell count is <1.

The p-value from the test is computed as if the margins of the table are fixed. This leads under a null hypothesis of independence to a hypergeometric distribution of the numbers in the cells of the table (Wikipedia). Fisher’s exact test is useful for small n-size samples where the chi-squared distribution assumption of the chi-squared and G-test tests fails. Fisher’s exact test is overly conservative (p values too high) for large n-sizes.

The Hypergeometric density function is \[f_X(k|N, K, n) = \frac{{{K}\choose{k}}{{N-K}\choose{n-k}}}{{N}\choose{n}}.\]

The density is the exact hypergeometric probability of observing this particular arrangement of the data, assuming the given marginal totals, on the null hypothesis that the conditional probabilities are equal.