---
title: "Phi coefficient (2x2 association)"
method_id: SM.REL.COR.2X2.PHI
family: Relationship
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/phi_coefficient.md
source: Clarus method library
---

# Phi coefficient (2x2 association)

> A classical statistical method in the Clarus library (Relationship 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:** phi; phi coefficient; Pearson phi; Matthews correlation coefficient (binary); mean square contingency coefficient.

## Hypotheses

- **Null (H0):** phi = 0 (the two binary variables are independent; equivalently the 2x2 odds ratio = 1).
- **Alternative (H1):** phi != 0 (the two binary variables are associated).

## When to use it

Use this method when your goal is to:
- quantify association two binary
- measure relationship two dichotomous
- effect size for 2x2 table

## Data it expects

- **dependent:** not_applicable (symmetric association)
- **independent:** 2 nominal variables, each with exactly 2 levels (a 2x2 contingency table)
- **pairing:** independent (each observation contributes to exactly one cell)

## Assumptions Clarus checks

- **expected frequency** (gate: expected_frequency; severity: warning; on violation: warn_only)
- **independence of observations** (severity: critical; on violation: reroute)

## Effect size reported

- **phi** (Pearson (1900); Yule (1912); phi = sqrt(chi2 / N), equivalently (a*d - b*c)/sqrt((a+b)(c+d)(a+c)(b+d)) for the signed version)

## Honest limitations

- phi is symmetric: report it as the strength of association between the two variables, not as one variable 'affecting' or 'differing on' the other.
- The sign of phi depends entirely on how the two levels of each variable were ordered/coded — always state the coding when reporting direction.
- phi is attenuated by unequal marginal splits: with a lopsided row or column distribution |phi| cannot reach 1 even under perfect dependence, so a 'small' phi can coexist with a large odds ratio. Read the odds ratio alongside.
- phi's significance rests on the chi-square approximation, which requires adequately large expected counts; for sparse 2x2 tables use Fisher's exact test for the p-value.
- With a large sample even a negligible association becomes statistically significant; always read |phi| (or the odds ratio) for practical magnitude.
- If p >= alpha, this is absence of evidence of association, not proof that the variables are independent; the minimum detectable phi describes what the study could realistically rule out.
- A significant association is not evidence of causation unless subjects were randomly assigned.
- Results generalize only to the sampled population.

## Primary sources

- Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, 50(302), 157-175. https://doi.org/10.1080/14786440009463897
- Yule, G. U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75(6), 579-652. https://doi.org/10.2307/2340126
- Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press.
- Cochran, W. G. (1954). Some methods for strengthening the common chi-square tests. Biometrics, 10(4), 417-451. https://doi.org/10.2307/3001616
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley, Ch. 3.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.), Ch. 18.

## 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, "Phi coefficient (2x2 association)", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/phi_coefficient.md

**APA**

Clarus. (2026). Phi coefficient (2x2 association) (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/phi_coefficient.md

**BibTeX**

```bibtex
@misc{clarus-phi-coefficient,
  author       = {Clarus},
  title        = {Phi coefficient (2x2 association)},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/phi_coefficient.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.REL.COR.2X2.PHI · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library