Shared zero-inflated Gamma-Poisson count model ($Y_j\mid B\sim \mathrm{Pois}(B{\mu }_j)$)

Let $B=0$ with probability $\pi$, and otherwise let $B$ have a Gamma distribution with mean one and variance $\tau$. Given $B$, counts $Y_j$ are conditionally independent Poisson random variables with means $B{\mu }_j$. For $q=1-\pi$, the law of total variance and law of total covariance giveEach component has a zero-inflated negative binomial distribution, but the components are not independent after marginalizing $B$. The zero component is shared by the whole profile. At $\tau =0$, interpret the positive Gamma component as a point mass at one.

If $\mathbb{E}(Y_j\mid B,x)=B\exp ({\beta }_0+{\beta }^{T}x+{\alpha }_j)$ and the distribution of $B$ is the same at each observation time with finite positive mean, integrating out $B$ adds $\log \mathbb{E}B$ to the marginal log-mean intercept. The conditional positive-$B$ mean ratio and the marginal mean ratio both equal $e^{{\alpha }_j-{\alpha }_k}$. This is a property of the multiplicative logarithmic link function; it need not hold for nonlinear links such as the logit link.