Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Count data ▷ Binary contingency table ▷ Odds ratio

Definition: Consider a $2 \times 2$ contingency table characterized by random events $A$ and $B$. Then, the odds ratio (OR) is defined as the ratio of the odds of event $A$ taking place in the presence of $B$, and the odds of $A$ taking place in the absence of $B$:

\[\label{eq:or} \begin{split} \mathrm{OR} &= \frac{\mathrm{Pr}(A|B)}{\mathrm{Pr}(\overline{A}|B)} \bigg/ \frac{\mathrm{Pr}(A|\overline{B})}{\mathrm{Pr}(\overline{A}|\overline{B})} \\ &= \frac{p_{11}/(p_{11} + p_{01})}{p_{01}/(p_{11} + p_{01})} \bigg/ \frac{p_{10}/(p_{10} + p_{00})}{p_{00}/(p_{10} + p_{00})} \\ &= \frac{p_{11} / p_{01}}{p_{10} / p_{00}} = \frac{p_{11} \, p_{00}}{p_{01} \, p_{10}} \; . \end{split}\]

Given observed data from a $2 \times 2$ contingency table, its sample estimate is given by

\[\label{eq:or-samp} \hat{\mathrm{OR}} = \frac{y_{11} / y_{01}}{y_{10} / y_{00}} = \frac{y_{11} \, y_{00}}{y_{01} \, y_{10}} \; .\]
 
Sources:

Metadata: ID: D240 | shortcut: or | author: JoramSoch | date: 2026-09-11, 10:34.