Scale identifiability in a gamma random-intercept Poisson model ($({\beta }_0,\tau ,\theta )\mapsto ({\beta }_0-\log c,c\tau ,c^2\theta )$)

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.