Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 13L (Statistical Modelling)

With $p_i=\mathbb{P}({\mathrm{dis}}_i=1)$, the fitted logistic model isRisk category 1 is the reference level, so its effect is included in the intercept; adding a separate coefficient for both levels would make the design matrix linearly dependent.

Holding risk category fixed, increasing pollution by one unit multiplies the disease odds byThus the fitted odds increase by about twenty percent per unit of pollution.

There are $53$ observations. The null model estimates one intercept, so its residual degrees of freedom are $53-1=52$. The fitted model estimates three coefficients, so its residual degrees of freedom are $53-3=50$.

Fit the intercept-only model withIts deviance is $73.5$. Since it has one fitted coefficient, its AIC on the same convention is

The first likelihood-ratio test compareswith the alternative that at least one is nonzero. The deviance drop $73.5-55.3=18.2$ is compared with ${\chi }_2^2$, giving $p=0.000114$; there is strong evidence that the covariates improve the model.

The second test compares the additive model withagainst a model with a pollution-by-risk interaction. Its ${\chi }_1^2$ likelihood-ratio p-value is $0.871$, so there is no evidence for an interaction.