Odds Ratio/Examples/2x2 Contingency Table

Example of Empirical Odds Ratio

In general, as empirical odds ratio can be evaluated from a contingency table:

$\begin{array}{r|cc|c} & & & \\ \hline & a & b & a + b \\ & c & d & c + d \\ \hline \text {Totals} & a + c & b + d & \end{array}$

The empirical odds ratio is then the ratio $a d : b c$.

Let the expected numbers in the $4$ cells be $m_{11}$, $m_{12}$, $m_{21}$ and $m_{22}$.

Then:

$\theta = \dfrac {m_{11} m_{22} } {m_{12} m_{21} }$

When $\theta = 1$ this implies no association.

In such a $2 \times 2$ contingency table:

$0 \le \theta^* \le \infty$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): odds ratio
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): odds ratio