Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Count data ▷ Binary contingency table ▷ Relative risk

Definition: Consider a $2 \times 2$ contingency table characterized by random events $A$ and $B$. Then, the relative risk (RR) is defined as the ratio of the probability of $A$ happening in the presence of $B$, and the probability of $A$ happening in the absence of $B$:

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

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

\[\label{eq:or-samp} \hat{\mathrm{RR}} = \frac{y_{11}/(y_{11} + y_{01})}{y_{10}/(y_{10} + y_{00})} = \frac{y_{11}\, (y_{10} + y_{00})}{y_{10} \, (y_{11} + y_{01})} \; .\]
 
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Metadata: ID: D241 | shortcut: rr | author: JoramSoch | date: 2026-09-11, 10:47.