Poisson generalized linear mixed model

A Poisson generalized linear mixed model gives a count response a conditional Poisson distribution whose logarithmic mean contains both fixed effects and latent random effects.

Let the positive multiplier $U_i$ have a gamma distribution with mean $\tau$ and variance $\theta$, and conditionally let the repeated counts be independent with means $U_i{a}_{ij}$. Total expectation and variance giveThe Poisson-gamma mixture gives negative-binomial marginal counts of size ${\tau }^2/\theta$. Despite the shared multiplier, the marginal correlations generally depend on both marginal means and need not be exchangeable.

When $a_{ij}=\exp ({\beta }_0+{\beta }^{\mathsf{T}}{x}_{ij})$, replacing $U_i$ by $c{U}_i$ and the fixed intercept by ${\beta }_0-\log c$ leaves every conditional count mean and the integrated data law unchanged. Thus without fixing the random multiplier's mean, the gamma random-intercept Poisson model identifies the marginal intercept ${\beta }_0+\log \tau$ and relative heterogeneity $\theta /{\tau }^2$, rather than these three parameters separately. Normalizing $\tau =1$ removes this scale ambiguity.

Given Gaussian random intercept $b_0$ and random slope $b_1$, let the conditional incidence rate be $e^{{\beta }_0+{\beta }_{1}s+b_0+b_{1}s}$. The normal moment-generating function gives the displayed marginal rate, with $\mathrm{Var}(b_0+b_{1}s)={\tau }_0^2+2s{\tau }_{01}+s^2{\tau }_1^2$. 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.