 Solution

The stated mean and variance identify ${\lambda }_i$ as a gamma distribution with shape ${\nu }_i$ and scale ${\alpha }_i$. A Poisson-gamma mixture is negative binomial, withWhen ${\nu }_i=\nu$, this is exactly the constant-shape variance model fitted by the negative binomial regression, so the most appropriate printed p-value is $0.0798$ from model3.