# Fisher Exact Test > Use when you must test a 2x2 contingency table with small expected counts: run the exact Fisher test under the fixed-margin null, compute the hypergeometric probability of the observed table, enumerate every table with the same margins, sum the one-tailed and two-tailed exact p-values, compute the odds ratio with the Haldane-Anscombe zero-cell correction, and return the small-expected-count verdict that recommends the exact test over the large-sample approximation. Produces p_obs, p_one_tail, p_two_tail, direction, odds ratio and the verdict. Trigger: fisher exact test, 2x2 contingency table, hypergeometric tail probability, small expected count, two by two table, odds ratio, exact independence test. - Skill: `ashfordeou/fisher-exact-test` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/fisher-exact-test` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/fisher-exact-test/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/fisher-exact-test --- # Fisher Exact Test (cross-cutting/numerics/fisher-exact-test) Use when you must test a 2x2 contingency table whose expected cell counts are small (below 5), where the large-sample approximation is unreliable: the exact Fisher test conditions on the fixed row and column margins and works directly from the hypergeometric distribution of the top-left cell, so the p-values are exact at any sample size. This leaf computes the observed table probability, enumerates every table with the same margins, sums the one-tailed and two-tailed exact p-values, forms the odds ratio with the Haldane-Anscombe zero-cell correction, and returns the small-expected-count verdict that gates the independence conclusion. Pure Python stdlib (math.comb), offline and deterministic. It pairs with cross-cutting/numerics/hypothesis-testing for the large-sample verdict on the same tables and with cross-cutting/numerics/probability-distributions for the hypergeometric family context. ## Domain quick reference - Convention: table [[a, b], [c, d]] with row margins a+b and c+d, column margins a+c and b+d and total n = a+b+c+d. Under the fixed- margin null the top-left count follows the hypergeometric distribution: P(a) = C(a+b, a) * C(c+d, c) / C(n, a+c), with C = math.comb. - Enumeration: every table with the same margins has a top-left count a' in [max(0, (a+c) - (c+d)), min(a+b, a+c)]; the worked example has seven feasible tables, a' = 1..7, and their probabilities sum to 1. - Direction and one-tailed p: direction is "low" when the odds ratio is below 1 (small top-left counts are the more extreme direction), "high" when above 1, "symmetric" when equal to 1. p_one_tail sums table probabilities with a' <= a_obs (low), a' >= a_obs (high), and the common sum in the symmetric case. - Two-tailed p: p_two_tail sums every table probability <= p_obs (the documented two-sided definition), so it is always >= p_one_tail. - Odds ratio: (a*d) / (b*c); a zero cell makes the raw ratio 0 or infinite, so any zero cell triggers the Haldane-Anscombe correction, +0.5 added to every cell before forming the ratio. - Small-count verdict: the minimum expected cell count under independence is min over the four cells of row_total * col_total / n; below 5 the verdict is "exact-test-recommended", otherwise "chi-square-adequate". - NACA-TR-824 is the numerics sibling standards reference; the relations above are standard engineering methodology, summary-only. ## Workflow 1. Arrange the two groups and two outcomes as a 2x2 table [[a, b], [c, d]] with non-negative integer counts and confirm the margins (a+b, c+d) and (a+c, b+d). 2. Get the probability of the observed table under the null with hypergeometric_p(a, b, c, d); this is p_obs. 3. See how many tables the margins admit with enumerate_tables(a, b, c, d): the a' values and the margin-preserving tuples. 4. Run the exact test with fisher_exact_p_value(a, b, c, d): read the dict {p_obs, p_one_tail, p_two_tail, direction} and use p_two_tail for the two-sided significance conclusion, or p_one_tail for a directional claim on the side the odds ratio points to. 5. Quantify the association with odds_ratio(a, b, c, d); the Haldane-Anscombe correction keeps zero-cell tables finite. 6. Gate the method choice with small_count_verdict(a, b, c, d): when the minimum expected cell count is below 5, report the exact p-values rather than the large-sample approximation. 7. Confirm the deterministic checks with the contract test scripts/test_fisher_exact_test.py. ## Worked example Table [[2, 6], [5, 1]] (row margins 8 and 6, column margins 7 and 7, n = 14), the wave-38 anchor: - hypergeometric_p(2, 6, 5, 1) = 0.048951, the observed table probability (168 / 3432 = 28 * 6 / C(14, 7)). - enumerate_tables(2, 6, 5, 1) returns 7 tables with a' = 1..7 and preserved margins; the full probability mass sums to 1.0. - fisher_exact_p_value(2, 6, 5, 1) returns p_obs 0.048951, direction "low" (odds ratio 0.0667 < 1), p_one_tail = P(a' <= 2) = 0.051282 (0.002331 + 0.048951) and p_two_tail = 0.102564, the sum of the four tables with probability <= p_obs (a' in {1, 2, 6, 7}). - odds_ratio(2, 6, 5, 1) = 0.066667 = 2/30, no zero cells so the raw ratio stands. - small_count_verdict(2, 6, 5, 1) reports min_expected 3.0 (the bottom-row cells carry 6*7/14 = 3.0, the smallest of the four; the top row carries 8*7/14 = 4.0) and verdict "exact-test-recommended". - The identity table [[2, 2], [2, 2]] gives odds ratio 1.0 and direction "symmetric", with p_one_tail 0.757143 (53/70) and both tail directions equal. ## Verification - Confirm hypergeometric_p(2, 6, 5, 1) returns 0.048951 and that the seven-table probability mass sums to 1.0. - Confirm fisher_exact_p_value(2, 6, 5, 1) returns p_one_tail 0.051282 and p_two_tail 0.102564 within 1e-4, direction "low", and that the mirrored table [[6, 2], [1, 5]] gives direction "high" with the same p-values. - Confirm odds_ratio(2, 6, 5, 1) returns 0.066667, the identity table [[2, 2], [2, 2]] returns 1.0, and zero-cell tables return the Haldane-Anscombe corrected value. - Confirm small_count_verdict(2, 6, 5, 1) returns min_expected 3.0 with verdict "exact-test-recommended", while a large-count table such as [[40, 60], [60, 40]] returns "chi-square-adequate". - Confirm p_two_tail >= p_one_tail across a spread of tables, and that a table at the extreme end of its direction has p_one_tail == p_obs. - Confirm every negative cell raises ValueError from each function and that a non-positive table total raises ValueError from small_count_verdict. - Run the contract test offline: python3 scripts/test_fisher_exact_test.py (35 tests, deterministic). ## Related leaves - cross-cutting/numerics/hypothesis-testing: the large-sample independence test for tables whose expected counts are not small, the complement verdict layer for the same 2x2 data. - cross-cutting/numerics/probability-distributions: the hypergeometric family behind the exact p-values and its moments. - cross-cutting/numerics/rank-based-hypothesis-testing: distribution- free alternatives for ordinal and non-normal comparisons. - cross-cutting/numerics/grubbs-outlier-test: exact small-sample reasoning applied to outlier detection rather than tables. - cross-cutting/numerics/proportion-confidence-interval: attribute- data rate bounds at a stated confidence level, the estimation side of the same pass-fail data. ## Pitfalls - Doubling the one-tailed p for a two-sided test: the documented two-sided definition sums every table probability <= p_obs (0.102564 in the worked example); twice the one-tailed p is a different, conservative convention and can disagree with this value. - Trusting the large-sample verdict near 5: min_expected < 5 means the exact test is recommended, and the threshold sits exactly at the boundary of common textbook guidance; report the exact p-values when the verdict says so. - Letting a zero cell collapse the odds ratio: the raw (a*d)/(b*c) becomes 0 or division by zero; apply the +0.5-per-cell Haldane- Anscombe correction and report that the correction was used. - Summing the one-tailed p on the wrong side: direction is read from the odds ratio (low means small a' is the extreme side), so a directional claim must sum a' <= a_obs or a' >= a_obs accordingly; the data decide the side, not the label of the claim. - Quoting p_obs as the significance: p_obs (0.048951) is the mass of one table only and understates the evidence; the significance conclusion uses p_two_tail (0.102564) against the level. - Reading the minimum expected count from the top row only: with unequal row margins the smallest expected cell can sit in the other row (6*7/14 = 3.0, not the top-row 8*7/14 = 4.0); take the minimum over all four cells. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_fisher_exact_test.py The test covers the worked-example anchors (p_obs 0.048951, one-tailed p 0.051282, two-tailed p 0.102564, odds ratio 0.066667, seven tables with a' = 1..7, minimum expected count 3.0 with the exact-test- recommended verdict), the large-count chi-square-adequate verdict, the symmetric identity table with odds ratio 1.0, high- and low-direction tails on mirrored tables, the p_two_tail >= p_one_tail ordering across tables, the extreme-table identity p_one_tail == p_obs in both directions, the documented two-tailed selection rule, exact dict keys, the Haldane-Anscombe zero-cell corrections, alternative value validation, and ValueError rejection of negative cells and non-positive totals. ## Compliance - Standards referenced, not reproduced: NACA-TR-824 is the numerics pack reference standard; the hypergeometric and odds-ratio relations above are standard engineering methodology, summary-only per standards-map.yaml. - compliance: STANDARDS-REF, gated: false.