= Solution
The Pearson <goodness-of-fit test> has null hypothesis that the independent counts follow the fitted <Poisson regression>, in particular $\operatorname{Var}(Y_i\mid d_i)=\mu_i$, against the alternative that the model does not fit; in this setting the scientifically relevant direction is <overdispersion>, $\operatorname{Var}(Y_i\mid d_i)>\mu_i$. Under the null, $X_P^2$ is approximately <chi-squared distribution> with $16$ degrees of freedom. Its tiny $p$-value decisively rejects the Poisson variance assumption.
The <negative binomial regression> keeps the logarithmic mean model but allows $\operatorname{Var}(Y_i\mid d_i)=\mu_i+\mu_i^2/\theta$. It improves the residual deviance from $75.806$ to $18.011$, close to its $16$ residual degrees of freedom, and lowers the <Akaike information criterion> from $172.34$ to $141.66$. Both comparisons strongly favour the negative-binomial fit, although its dose coefficient remains statistically insignificant.
Back to article page
