Odds Ratio

Definition

Consider a $2\times 2$ Contingency Table

	Event	Non-event
Group 1	$n_{11}$	$n_{12}$	$n_{1.}$
Group 2	$n_{21}$	$n_{22}$	$n_{2.}$
	$n_{.1}$	$n_{.2}$	$n_{..}$

Point Estimation

The odds ratio (OR) is estimated by $OR=\frac{\text{odds in case group}}{\text{odds in control group}}=\frac{(n_{11}/n_{.1})/(n_{21}/n_{.1})}{(n_{12}/n_{.2})/(n_{22}/n_{.2})}=\frac{n_{11}{n}_{22}}{n_{21}{n}_{12}}$

Confidence Interval

The confidence interval for odds ratio is estimated by Delta Method. $\widehat{\mathrm{Var}}(\ln OR)=\frac{1}{n_{11}}+\frac{1}{n_{12}}+\frac{1}{n_{21}}+\frac{1}{n_{22}}$ The $100(1-\alpha )\%$ confidence interval for $OR$ is defined as $\exp [\ln OR\pm z_{\alpha /2}\sqrt{\widehat{\mathrm{Var}}(\ln OR)}]$

Facts

Odds ratio can be used for Case-Control Study.