Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 102 / 3

Only four of the twenty provider records are printed, and the full emailed file has not been recovered. The four observed passenger averages have mean $85.75$, median $82.5$, sample standard deviation $17.67$ and range $68$--$110$. Their bus-mile mean is $112.25$, with range $22$--$160$; the highest passenger average belongs to the provider with the lowest bus miles and zero fare. That is a useful diagnostic of differing service patterns, not evidence of a population-wide negative mileage effect.

For the complete dataset, use scatter plots of passenger average against bus miles, fare and population, and inspect population-scaled demographic counts as well as their raw counts. Compare marginal distributions and the sample correlation matrix, especially among population, older residents, low-income households and poverty households: these measure overlapping aspects of area size and composition and can create multicollinearity. Inspect the zero fare directly; a logarithm of fare is undefined there, so do not silently drop it. All four displayed rows belong to the fixed-route group, so they cannot support a between-service comparison.

The figure labels its bus panel as a four-row excerpt. Twenty-provider summaries and exploratory conclusions cannot be determined from these four rows. In particular a full eight-parameter regression has more parameters than observed rows; reporting a fitted full-data model from this excerpt would be unjustified.

Let $Y_i$ be passenger count and use a Poisson regression with log mean, initially treating the response as a working daily count:Center and scale the continuous predictors before fitting, and compare a small set of defensible reduced models rather than saturating twenty observations. Household counts should not be called household proportions without a household denominator; population-scaled counts are alternative intensity covariates. Bus miles can be an estimated elasticity term rather than an offset, since proportionality of passenger numbers to bus miles has not been established.

For actual counts, the log-likelihood score is $X^T(Y-\mu )=0$, with ${\mu }_i=e^{x_i^T\beta }$. Fitting by iteratively reweighted least squares uses weights ${\mu }_i$ and working response ${\eta }_i+(Y_i-{\mu }_i)/{\mu }_i$. Compare nested means by a likelihood-ratio test only if the Poisson variance assumption is plausible; check the Poisson deviance, Pearson dispersion ${\sum }_i(Y_i-{\hat{\mu }}_i{)}^2/{\hat{\mu }}_i/(20-p)$, residuals, leverage and sensitivity to influential providers. Consider Quasi-Poisson regression or negative binomial regression if variance exceeds the mean.

The recorded response is actually a daily average, so Poisson working models for averaged counts need special care. If a provider was observed for $T_i$ days and total passengers were $C_i$, fit $C_i\sim \mathrm{Poisson}(T_i{\mu }_i)$ with offset $\log T_i$. If only the average and no observation duration are available, a Poisson mean model can still be explored, but its likelihood and standard errors are not automatically justified by integer-looking averages.

A one-unit increase in an untransformed predictor has fitted multiplicative mean effect $e^{{\beta }_j}$; a bus-mile elasticity ${\beta }_B$ gives a doubling effect $2^{{\beta }_B}$. No numerical coefficient estimates or selected twenty-provider model are identifiable from the printed excerpt. The full provider file and, ideally, the averaging periods are needed to execute these model comparisons.

Preserve the original row order and set $G_i=0$ for the first eight fixed-route providers and $G_i=1$ for the twelve on-demand providers. A suitable group comparison builds on a selected small predictor set $z_i$:The model with $\gamma =0,\delta =0$ pools service types; adding $\gamma$ changes the baseline; adding selected components of $\delta$ allows different predictor dependence. Center $z$ at meaningful values so that $e^{\gamma }$ compares groups at those values. The predictor-specific rate ratio is $e^{{\beta }_j}$ in the fixed-route group and $e^{{\beta }_j+{\delta }_j}$ in the on-demand group; the group ratio at predictor value $z$ is $e^{\gamma +z^T\delta }$.

Fit common-slope and selected-interaction models, respecting model hierarchy by retaining corresponding main effects. Test the added interaction block jointly with a likelihood-ratio test, or an appropriate quasi-likelihood comparison under estimated dispersion, and check group-specific residual patterns and leave-one-provider sensitivity. Seven interactions plus all main effects would use 16 parameters for just twenty providers and leave only four residual degrees of freedom; an indiscriminate full interaction model is consequently a poor basis for stable inference.

The excerpt contains only $G=0$, so neither the service-type effect nor any interaction is estimable from it. Full-data estimates could distinguish a baseline difference from genuinely different slopes, but the printed records cannot establish either conclusion.