Odds Ratio

Definition

In a $2\times 2$ contingency table, the odds ratio is defined as

$$\theta =\frac{{\text{odds}}_1}{{\text{odds}}_2}=\frac{{\pi }_1/(1-{\pi }_1)}{{\pi }_2/(1-{\pi }_2)}$$

, where:

• ${\text{odds}}_1$ : The odds of row 1
• ${\text{odds}}_2$ : The odds of row 2
• ${\pi }_1$ : Probability of success of variable

Alternative formulas

$$\begin{array}{rl}\theta & =\frac{{\pi }_{11}{\pi }_{22}}{{\pi }_{12}{\pi }_{21}}\\ & =\frac{{\pi }_{1|1}/{\pi }_{2|1}}{{\pi }_{1|2}/{\pi }_{2|2}}\\ & =\end{array}$$

Remark

The further away $\theta$ is from 1, the more evidence there is against independency between the two variables.

If the two variables are perfectly independent, then ${\pi }_1={\pi }_2$, thus $\theta =1$.