Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 102 / 4 / d / Solution

Use the population as exposure regardless of whether a population-size category is also a covariate. Define a demographic factor with levels $\mathrm{Low}$ for at most 20%, $\mathrm{Medium}$ for $(20,30]$%, and $\mathrm{High}$ above 30%; define $H_{it}=1_{\{P_{it}>250\}}$ if following the suggested population dichotomy. Treat village-size codes as a categorical variable, not equally spaced areas.

A candidate multivariate Poisson generalized linear mixed model retains the random intercept and slope from part (c) and usesThis specifies the suggested village-size-by-population interaction while keeping its main effects. One may add selected time-by-covariate interactions to explain slope heterogeneity, provided information from 32 clusters supports them. Compare continuous demographic and population formulations as a sensitivity analysis: categorization loses information, and the coefficient of a size category measures association beyond the already included exposure effect.

Fit models with the same likelihood convention and conditional distribution when comparing Akaike information criteria; use joint fixed-effect likelihood-ratio tests, boundary-aware parametric bootstrap tests for random effects, residual diagnostics and village-level cross-validation to justify a parsimonious final model. Preserve interactions' main effects and report interval estimates from the final multivariate model, rather than interpreting separate univariate fits. For example, the adjusted inland-versus-coastal rate ratio is $e^{{\theta }_I}$, while a medium-versus-small village ratio is $e^{{\gamma }_2+{\kappa }_{2}H}$; a Wald interval for that contrast uses the full coefficient covariance matrix. Conditional time effects and standardized marginal predictions should be distinguished.

The displayed rows cannot identify this final model: population exceeds 250 in every displayed village-year, no large village is shown, and inland status is confounded with the identity of the two displayed villages. The complete 160-row file is needed for fitted effects, model selection, all-village plots and uncertainty. No numerical best model or evidence of the specified covariate associations can honestly be reported from this excerpt.

Finally, even a significant adjusted association with young-male percentage would not show that young males were involved in most incidents. This is the ecological fallacy, specifically that aggregate composition does not identify offender composition: the same village counts can be assigned to different individual age and sex groups without changing any observed aggregate predictor. Individual incident-level age, sex and involvement records are required to answer that last question. The aggregate dataset cannot establish the claimed majority involvement.