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

Use incident counts $Y_{it}$ divided by that village-year's population exposure $P_{it}$, and graph $r_{it}=1000Y_{it}/P_{it}$ against calendar year. Show separate small panels or labelled lines for all villages; a population-weighted overall line answers a different question from an equally weighted average village rate. A plot of counts alone can mistake population change for a rate change.

For the two printed villages, the observed rates per thousand residents are approximatelyThe first trajectory increases throughout; the second falls initially and then rises sharply. The last-to-first rate ratios are $1.472$ and $1.711$, respectively. The figure displays these two observed trajectories only. The other thirty village trajectories cannot be drawn from the PDF, because their observations were supplied in the separate emailed dataset.

For a population-averaged rate trend, use the marginal incidence trend with clustered observations modelFit a generalized estimating equation with village as cluster, a log link and an appropriate working correlation; estimate uncertainty with a village-level sandwich covariance matrix. Then $e^{\beta }$ is the fitted annual marginal rate ratio, $100(e^{\beta }-1)$ is the annual percentage rate change, and $e^{4\beta }$ is the four-year endpoint ratio. Test $\beta =0$ using cluster-robust uncertainty, with a small-sample correction or village-resampling bootstrap given only 32 independent clusters. Also compare year as a factor with a linear trend, since a single slope can conceal nonlinear behaviour.

Report which population is averaged over. The directly pooled rate ${\sum }_i{Y}_{it}/{\sum }_i{P}_{it}$ is exposure weighted; a mean of village rates gives each village equal weight. For standardized comparisons one can use the same village weights at every year, preventing changing population composition from masquerading as a rate trend.

A conditional random-slope coefficient cannot simply be substituted for the marginal slope: the marginal mean of a Poisson random-slope model contains time-dependent variance terms. A direction, magnitude and confidence interval for the trend across all villages require the missing full dataset. The increase in the two displayed villages does not determine the trend in the other thirty.

Center time at 2008 by setting $s=t-4$, and initially ignore the explanatory covariates. A Poisson generalized linear mixed model addressing both heterogeneity questions isHere ${\tau }_0$ describes between-village variation of log rates in 2008, while ${\tau }_1$ describes heterogeneity of annual log-rate changes. The latent 2008 rate has median $e^{{\beta }_0}$, mean $e^{{\beta }_0+{\tau }_0^2/2}$ and coefficient of variation $\sqrt{e^{{\tau }_0^2}-1}$. A central 95% interval for latent village rates is $e^{{\beta }_0\pm 1.96{\tau }_0}$; the corresponding interval for village-specific annual rate ratios is $e^{{\beta }_1\pm 1.96{\tau }_1}$. These describe heterogeneity, not confidence intervals for estimated fixed effects.

Fit an intercept-only random-effects model and then add a correlated random slope. Compare them with a parametric bootstrap of the likelihood improvement: the null slope variance is on the boundary, so an ordinary fixed-effect chi-squared reference is not automatically valid. Inspect uncertainty in both variance components, shrinkage of village-specific estimates, residual overdispersion and remaining serial correlation. A negative-binomial conditional model can distinguish additional within-village count variation from persistent village differences when the data support it.

By the marginal mean of a Poisson random-slope model, the marginal log rate is ${\beta }_0+{\beta }_{1}s+({\tau }_0^2+2s{\tau }_{01}+s^2{\tau }_1^2)/2$, so marginal curvature can arise even when each village has a straight conditional log trend. Numerical variance estimates and evidence of heterogeneous slopes cannot be obtained for 32 villages from two displayed trajectories.

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.