---
title: "Odds ratio, 2x2 (case-control)"
method_id: SM.REL.ODD.2X2.OR
family: Relationship
version: 1.0.0
date_modified: 2026-06-21
canonical: https://clarus.ofrencber.com/library/odds_ratio.md
source: Clarus method library
---

# Odds ratio, 2x2 (case-control)

> 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:** OR; cross-product ratio; relative odds; case-control odds ratio.

## Hypotheses

- **Null (H0):** OR = 1 (the odds of {outcome} are the same in the exposed and unexposed groups; exposure and outcome are not associated).
- **Alternative (H1):** OR != 1 (the odds of {outcome} differ between the exposed and unexposed groups).

## When to use it

Use this method when your goal is to:
- quantify association between a binary exposure and a binary outcome as an odds ratio
- estimate an odds ratio from a case control 2x2 table
- report the effect of a binary factor on a binary outcome when risk cannot be estimated

## Data it expects

- **dependent:** 1 binary outcome / disease status (case / control) — counted by the study design
- **independent:** 1 binary exposure / characteristic (present / absent) — the two rows or columns of a 2x2 table
- **pairing:** independent (each subject contributes to exactly one cell); case-control OR cross-sectional OR cohort design — the cross-product odds ratio is valid for all three, and is the ONLY valid association measure under outcome-based (case-control) sampling

## Assumptions Clarus checks

- **independence of observations** (severity: critical; on violation: warn_only)
- **adequate cell counts** (gate: expected_frequency; severity: warning; on violation: correct)

## Effect size reported

- **odds ratio** (Cornfield (1951), JNCI 11(6):1269-1275; OR = (a*d)/(b*c))

## Honest limitations

- The odds ratio is a ratio of ODDS, not of risks; it equals the relative risk only when the outcome is rare. When the outcome is common, the OR is further from 1 than the risk ratio and must not be read as one.
- In case-control studies the odds ratio is the only valid association measure, because risk cannot be estimated from outcome-based sampling (Cornfield 1951).
- Sparse or zero cells make the estimate unstable and the confidence interval wide; a zero cell forces a continuity correction and an approximate result.
- An odds ratio away from 1 is not evidence of causation unless exposure was randomly assigned; case-control and observational data can be confounded.
- If the 95% CI includes 1, this is absence of evidence for an association, not proof that exposure carries no effect; the interval shows which odds ratios remain plausible.
- The odds ratio depends on which outcome level is called the 'event' and which group is 'exposed'; the point estimate is symmetric in rows vs columns, but the verbal interpretation must state the labels.

## Primary sources

- Cornfield, J. (1951). A method of estimating comparative rates from clinical data. Applications to cancer of the lung, breast, and cervix. Journal of the National Cancer Institute, 11(6), 1269-1275.
- Woolf, B. (1955). On estimating the relation between blood group and disease. Annals of Human Genetics, 19(4), 251-253. https://doi.org/10.1111/j.1469-1809.1955.tb01348.x
- Haldane, J. B. S. (1956). The estimation and significance of the logarithm of a ratio of frequencies. Annals of Human Genetics, 20(4), 309-311. https://doi.org/10.1111/j.1469-1809.1955.tb01285.x
- Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley, Ch. 2-3.
- Breslow, N. E., & Day, N. E. (1980). Statistical Methods in Cancer Research, Vol. 1: The Analysis of Case-Control Studies. IARC.

## 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, "Odds ratio, 2x2 (case-control)", version 1.0.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/odds_ratio.md

**APA**

Clarus. (2026). Odds ratio, 2x2 (case-control) (Version 1.0.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/odds_ratio.md

**BibTeX**

```bibtex
@misc{clarus-odds-ratio,
  author       = {Clarus},
  title        = {Odds ratio, 2x2 (case-control)},
  howpublished = {Clarus method library},
  version      = {1.0.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/odds_ratio.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.ODD.2X2.OR · VERSION 1.0.0 · UPDATED 2026-06-21 · SOURCE Clarus method library