Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 218 / 1

Let $H_i$ be the number healthy among $m_i=25$ patients and $Y_i=H_i/m_i$. The fitted grouped-binomial logistic regression isFor $y=h/m$ its mass has exponential dispersion family formwhereThus $\mathbb{E}Y=p=b^{\prime }(\theta )$ and

The positive intercept means the fitted baseline probability for untreated women exceeds $1/2$. The positive coefficient ${\hat{\beta }}_S=0.1854$ makes each fitted male log odds exceed the corresponding female log odds, while ${\hat{\beta }}_T=-0.3698$ makes treatment lower the fitted log odds for either sex. Numerically the fitted probabilities are approximatelyTheir order is $F1<M1<F0<M0$, which differs from the observed order $F1<M0<M1<F0$. The additive model therefore cannot reproduce the observed ordering.

Add a sex-by-treatment interaction term:In R the formula becomes prop ~ sex * treatment. There are four parameters for four cell probabilities, so this model is saturated and, provided the cell proportions lie strictly between zero and one, fits each observed proportion exactly. It therefore recovers all the stated order relations.

For observed proportions $y_i$ and fitted proportions ${\hat{p}}_i$, the binomial deviance isA second-order Taylor expansion around $y_i={\hat{p}}_i$ givesthe generalized Pearson chi-squared statistic. Under the fitted binomial model this is approximately ${\chi }_{4-3}^2={\chi }_1^2$. The observed deviance is $1.6102$, whose upper-tail probability is about $0.20$. Since this exceeds $0.05$, there is no significant evidence of overdispersion.