---
title: "Fisher's exact test (2x2)"
method_id: SM.DIF.CAT.2X2.FISHER
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/fisher_exact_test.md
source: Clarus method library
---

# Fisher's exact test (2x2)

> A classical statistical method in the Clarus library (Difference family).
> Clarus selects, assumption-checks and runs this method on your data with a
> deterministic rule engine and real numerical libraries (scipy / statsmodels);
> it never generates numbers, citations or results.

**Also known as:** Fisher-Irwin test; exact test of independence (2x2).

## Hypotheses

- **Null (H0):** OR = 1 (row and column classifications are independent)
- **Alternative (H1):** OR != 1 (row and column classifications are associated)

## When to use it

Use this method when your goal is to:
- test association two binary variables
- compare proportions two independent groups

## Data it expects

- **dependent:** 1 nominal/2 levels
- **independent:** 1 nominal/2 levels
- **pairing:** independent
- **table_shape:** 2x2
- **cell_size:** small_expected_counts
- **sample_size:** [object Object]

## Assumptions Clarus checks

- **independence of observations** (severity: error; on violation: reroute)
- **fixed total or random sampling** (severity: warning; on violation: warn_only)

## Effect size reported

- **odds ratio** (Agresti (2013), Sec. 2.3)

## Honest limitations

- Fisher's exact test conditions on the observed row and column totals; the p-value is exact under that conditioning and is known to be slightly conservative.
- A significant association is not causal unless the grouping variable was set by random assignment.
- The sample odds ratio is undefined (0 or infinite) when any cell is zero; the conditional maximum-likelihood estimate and its exact CI are reported instead.
- Results generalize only to the sampled population; the test says nothing about the magnitude beyond the odds ratio and its CI.

## Primary sources

- Fisher, R. A. (1922). On the interpretation of chi-square from contingency tables, and the calculation of P. Journal of the Royal Statistical Society, 85(1), 87-94. https://doi.org/10.2307/2340521
- Fisher, R. A. (1935). The Design of Experiments. Oliver & Boyd.
- Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley.
- Cramer, H. (1946). Mathematical Methods of Statistics. Princeton University Press.

## How to cite this

To cite this Clarus method page (the page itself — for the method's own primary sources, see above):

**Plain text**

Clarus, "Fisher's exact test (2x2)", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/fisher_exact_test.md

**APA**

Clarus. (2026). Fisher's exact test (2x2) (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/fisher_exact_test.md

**BibTeX**

```bibtex
@misc{clarus-fisher-exact-test,
  author       = {Clarus},
  title        = {Fisher's exact test (2x2)},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/fisher_exact_test.md}
}
```

_No DOI is minted for Clarus method pages yet; this citation uses the canonical URL, version and date. Cite the primary sources above for the method's scientific provenance._

---

METHOD SM.DIF.CAT.2X2.FISHER · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library