 Solution

The third fit is a negative binomial regression. The Poisson model is obtained at the boundary where the negative-binomial overdispersion tends to zero, so the likelihood-ratio test has the asymptotic null distribution $\frac{1}{2}{\chi }_0^2+\frac{1}{2}{\chi }_1^2$. Since model3 has one additional parameter,The supplied output gives $\mathbb{P}({\chi }_1^2\ge D)=0.04608555$, and hence the boundary-corrected p-value is