Fisher's Exact Test
Fisher's Exact Test is a statistical significance test primarily employed in the analysis of contingency tables. This test is widely used in scenarios where sample sizes are small, although it remains valid for any sample size. The test is named after the eminent statistician Ronald Fisher, who developed it following a remark made by Muriel Bristol. She claimed to discern whether tea or milk was added first to her cup, thus spurring Fisher's interest in creating a test to examine such claims.
Fisher's Exact Test is part of a class of tests known as exact tests, which allow the calculation of the significance of deviations from a null hypothesis exactly. This contrasts with many other tests that rely on approximations becoming exact only as sample sizes increase.
Application
The test is particularly relevant when the chi-squared test is not suitable, such as when the expected values in any cell of a contingency table fall below 5, or below 10 when there is one degree of freedom. These criteria are considered overly conservative today. For small, sparse, or unbalanced datasets, the exact and asymptotic p-values can differ significantly, potentially leading to differing conclusions about the hypothesis being tested.
Methodology
Fisher's Exact Test assumes that the row and column sums of the contingency table are fixed by design. This feature makes the test exact, allowing its use regardless of the sample characteristics. It calculates the probability of obtaining a table at least as extreme as the observed data under the null hypothesis, given the fixed margins.
Relationship to Other Tests
- Boschloo's Test: An alternative to Fisher's Exact Test, Boschloo's Test is noted for being uniformly more powerful. Proposed by R. D. Boschloo in 1970, it uses a 2 × 2 contingency table.
- Barnard's Test: Another related test, Barnard's Test, is an exact test used for analyzing 2 × 2 contingency tables with one margin fixed.
- Permutation Test: A type of exact statistical hypothesis test that involves two or more groups, often used when the assumptions for Fisher's Exact Test are not met.
- Hypergeometric Distribution: The one-tailed version of the hypergeometric test is identical to the corresponding one-tailed version of Fisher's Exact Test.
Use Cases
Fisher's Exact Test is widely applied beyond simple contingency table analysis. It is used in diverse fields such as genetics to test for Hardy–Weinberg equilibrium and in A/B testing to compare binomial distributions like click-through rates.