Odds Ratio/Examples

Examples of Odds Ratios

Smoking Disease

Let $P$ be a population of $250$ people.

Let $P_1$ be the set consisting of the $150$ members of $P$ who are smokers.

Let $P_2$ be the set consisting of the $100$ members of $P$ who do not smoke.

Let $Q$ be the property of the members of $P$ defined as:

$\forall x \in P: \map Q x$ means that $x$ has a certain disease supposed to be caused by smoking.

Let:

the number of members of $P_1$ with $Q$ be $12$

the number of members of $P_2$ with $Q$ be $2$.

The empirical odds ratio of having $Q$ relative to smoking or non-smoking is then:

$\theta^* = 4 \cdotp 26$

It is common to use the empirical odds ratio as the basis of a hypothesis test.

The null hypothesis in such a case is whether $P_1 = P_2$.

This is equivalent to testing whether $\theta^*$ is significantly different from $\theta = 1$.

The above can be placed into a contingency table:

$\begin{array}{r|cc|c} & \text {Diseased} & \text {Not diseased} & \text {totals} \\ \hline \text {Smokers} & 12 & 138 & 150 \\ \text {Nonsmokers} & 2 & 98 & 100 \\ \hline \text {Totals} & 14 & 236 & 250 \end{array}$

$2 \times 2$ Contingency Table

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:

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