Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 5J (Statistical Modelling)

Let $Y_i$ be the indicator that man $i$ develops heart disease and let ${\pi }_i=\mathbb{P}(Y_i=1)$. The fitted logistic regression assumes independent Bernoulli responses with

Two packs contain $40$ cigarettes. Holding height fixed, increasing average consumption from zero to two packs per day changes the log odds byThe corresponding odds ratio is thereforeThus the model estimates that a two-pack-per-day smoker has about $2.52$ times the odds of heart disease of an otherwise comparable nonsmoker. This is an association in the fitted observational model, rather than by itself a causal conclusion.

Let ${\epsilon }_i$ be independent standard logistic random variables and define the latent variableSet $Y_i=1$ exactly when $Z_i>0$. If $F$ is the logistic cumulative distribution function, its symmetry and the identity $F^{-1}(p)=\mathrm{logit}(p)$ givewhich is precisely the fitted logistic regression.