= Marginal mean of a Poisson random-slope model
{title2=$\mathbb E[Y_{it}]/P_{it}=e^{\beta_0+\beta_1s+\frac12(\tau_0^2+2s\tau_{01}+s^2\tau_1^2)}$}
Given Gaussian <random intercept> $b_0$ and <random slope> $b_1$, let the conditional incidence rate be $e^{\beta_0+\beta_1s+b_0+b_1s}$. The normal moment-generating function gives the displayed marginal rate, with $\operatorname{Var}(b_0+b_1s)=\tau_0^2+2s\tau_{01}+s^2\tau_1^2$. Thus the conditional annual <rate ratio> for a cluster is $e^{\beta_1+b_1}$, whereas the marginal rate ratio also contains covariance and variance terms. Centering $s$ at a chosen year makes intercept heterogeneity refer directly to that year.
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