= Solution
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 is
$$
Y_{it}\mid b_i\sim\operatorname{Poisson}\{P_{it}\exp(\beta_0+\beta_1s+b_{0i}+b_{1i}s)\},\qquad
\binom{b_{0i}}{b_{1i}}\sim N\left(0,
\begin{pmatrix}\tau_0^2&\tau_{01}\\\tau_{01}&\tau_1^2\end{pmatrix}\right).
$$
Here $\tau_0$ 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 $e^{\beta_0\pm1.96\tau_0}$; the corresponding interval for village-specific annual <rate ratios> is $e^{\beta_1\pm1.96\tau_1}$. 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 $\beta_0+\beta_1s+(\tau_0^2+2s\tau_{01}+s^2\tau_1^2)/2$, so marginal curvature can arise even when each village has a straight conditional log trend. \b[Numerical variance estimates and evidence of heterogeneous slopes cannot be obtained for 32 villages from two displayed trajectories.]
Back to article page
