Gamma random-intercept Poisson model ($Y_{ij}\mid U_i\sim \mathrm{Poisson}(U_i{a}_{ij})$)

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.