Odds Ratio/Examples/Smoking Disease

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Example of Empirical Odds Ratio

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:

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


Proof

By definition of empirical odds ratio:

\(\ds \theta^*\) \(=\) \(\ds \dfrac {12 / \paren {150 - 12} } {2 / \paren {100 - 2} }\)
\(\ds \) \(=\) \(\ds \dfrac {12 / 138} {2 / 98}\)
\(\ds \) \(=\) \(\ds 4 \cdotp 26\)

$\blacksquare$


Sources

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