Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 13J (Statistical Modelling)

Kilometres is an ordinal categorical variable: its five levels have a meaningful order. Treating it as an ordinary factor would use four indicator variables and hence four unrelated regression coefficients. Converting the levels to numerical scores instead imposes a single linear trend across successive mileage bands. This is more parsimonious and gives greater precision when that trend is a reasonable approximation, although it also assumes that adjacent levels are equally spaced in their effect.

The graph is a profile likelihood for the parameter $\lambda$ of a Box–Cox transformation. Its maximum is close to $\lambda =0$, and the displayed likelihood interval contains zero but excludes $\lambda =1$. Since $\lambda =0$ gives a logarithmic transformation whereas $\lambda =1$ leaves the response unchanged up to an affine transformation, the plot supports fittingas the response in model2.

Because model2 uses a logarithmic transformation, the logarithm of the payment ratio is the difference of two predicted log payments. The cars have equal Brand and Bonus values and differ by one Kilometres level, so its fitted mean is the displayed mileage regression coefficient, ${\hat{\beta }}_K=0.057132$.

For two independent future payments, the prediction interval must include both the standard error $0.032654$ of ${\hat{\beta }}_K$ and two independent residual errors. The estimated variance of the predicted log ratio is thereforeUsing the residual degrees of freedom shown in the output, a 95% prediction interval for a ratio of log-normal responses isWith $t_{284,0.975}\simeq 1.97$, this is approximately $[0.120,9.33]$. The interval is wide because it predicts the ratio of two individual future payments; an interval for the ratio of their fitted mean responses would omit the $2{\hat{\sigma }}^2$ term.

A suitable generalized linear model is a Gamma regression with logarithmic link:withThe gamma distribution is appropriate for a positive, continuous, right-skewed payment, and its variance function allows the conditional variance to grow like the square of the conditional mean instead of assuming constant variance. The logarithmic link function also guarantees positive fitted means and represents covariate effects multiplicatively, which accords with the Box–Cox transformation evidence for a logarithmic scale.