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

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.