Fisher's exact test

Fisher's exact test
Name	Fisher's exact test
Field	Statistics
Introduced	1922
Founder	Ronald Fisher
Table	2×2 contingency table

Fisher's exact test

Fisher's exact test is a statistical significance test for categorical data that assesses the association between two binary variables in a contingency table. It was introduced by Ronald Fisher and widely used in analysis originating from experimental design in contexts involving small sample counts, clinical trials, and genetics. The test is exact in the sense that it conditions on fixed marginal totals and computes probabilities from the hypergeometric distribution.

Introduction

Fisher devised the test while interacting with contemporaries such as Karl Pearson, William Sealy Gosset, and institutions like the University of Cambridge and the British Association for the Advancement of Science. It became influential through discussions with scientists at the Royal Society and practitioners in agricultural research at the Rothamsted Experimental Station. The method influenced later statisticians including Jerzy Neyman, Egon Pearson, and institutions like the Institute of Mathematical Statistics and the Royal Statistical Society.

Definition and formulation

Consider a 2×2 contingency table with fixed row and column sums as in classic designs used by researchers at Harvard University and University of Oxford. Under the null hypothesis of no association, the probability of observing a specific table is given by the hypergeometric distribution, a framework linked historically to work at the University of Cambridge and applications in demographics by scholars at the U.S. Census Bureau. Fisher's approach conditions on margins—an idea related to exact conditional inference developed in mathematical circles including the London School of Economics and the University of Chicago. The test statistic typically used is the probability of the observed table and all tables at least as extreme under the null, a criterion formalized through correspondence with figures like Jerzy Neyman and debated in symposia at institutions such as Princeton University.

Computational methods and algorithms

Early manual calculations relied on factorials and tables circulated among researchers at the Royal Statistical Society and laboratories like Rothamsted Experimental Station. With the advent of electronic computation at centers such as Bell Labs and Los Alamos National Laboratory, algorithms for exact enumeration and cumulative hypergeometric sums became practical. Modern implementations appear in software packages developed by teams at AT&T Bell Laboratories, Microsoft Research, and projects at University of California, Berkeley; they use recursion, network algorithms, and early-stopping criteria inspired by numerical methods from Cambridge University Press texts. Efficient algorithms exploit symmetry and cumulative tail summation; more advanced routines leverage Monte Carlo sampling introduced in work at Los Alamos National Laboratory and variance-reduction techniques influenced by research at the Massachusetts Institute of Technology.

Extensions and generalizations

Generalizations to r×c contingency tables trace to contributions by scholars at Columbia University and Stanford University who extended conditional exact tests using multivariate hypergeometric models. Mid-20th-century developments by statisticians at Cornell University and Yale University led to exact conditional inference methods for stratified tables and matched-pair designs common in clinical research at Johns Hopkins University. Bayesian exact analogues, informed by work at Carnegie Mellon University and University College London, combine prior distributions with hypergeometric likelihoods. Multi-parameter exact tests and permutation-based approaches were advanced in collaborations involving researchers at the Wellcome Trust and the National Institutes of Health.

Applications and examples

The test is standard in biomedical studies at Mayo Clinic and Cleveland Clinic where small-sample comparisons of treatment and control groups arise. In genetics, it is applied to allele association studies in laboratories at the Broad Institute and the Wellcome Sanger Institute. Epidemiologists at the Centers for Disease Control and Prevention and public health units use the test for outbreak investigations and contingency analyses. Conservation biologists affiliated with the Smithsonian Institution and researchers at the World Health Organization employ it for presence–absence data. Classic textbook examples circulated in courses at Princeton University and Imperial College London illustrate comparisons of two proportions in small samples and rare-event analyses used by investigators at the National Cancer Institute.

Limitations and related tests

Limitations were highlighted in critiques from the analytic traditions at Columbia University and University of Michigan emphasizing conservatism and computational burden for larger tables. For moderate-to-large samples, asymptotic tests like Pearson's chi-squared test—developed by Karl Pearson and extended in work at the University of London—and likelihood-ratio tests from research at Bell Labs provide alternatives. Exact unconditional tests and mid-P corrections proposed in papers from Yale University and University of California, San Francisco address some conservativeness. Software implementations from institutions like R Project and packages maintained by groups at The Comprehensive R Archive Network include optimized options and warnings about interpretation in complex survey designs used by analysts at the U.S. Census Bureau.

Category:Statistical tests