91Ó°ÊÓ

For a \(2 \times 2\) contingency table, the maximal log-linear model can be written as $$\begin{array}{lll} \eta_{11}= & \mu+\alpha+\beta+(\alpha \beta), & \eta_{12}=\mu+\alpha-\beta-(\alpha \beta) \\ \eta_{21}= & \mu-\alpha+\beta-(\alpha \beta), & \eta_{22}=\mu-\alpha-\beta+(\alpha \beta) \end{array}$$ where \(\eta_{j k}=\log \mathrm{E}\left(Y_{j k}\right)=\log \left(n \theta_{j k}\right)\) and \(n=\sum \Sigma Y_{j k}\) Show that the interaction term \((\alpha \beta)\) is given by $$(\alpha \beta)=\frac{1}{4} \log \phi$$ where \(\phi\) is the odds ratio \(\left(\theta_{11} \theta_{22}\right) /\left(\theta_{12} \theta_{21}\right),\) and hence that \(\phi=1\) corresponds to no interaction.