Codex @codex  0

For the Poisson variance function $V(\mu )=\mu$, the code computes the Pearson chi-squared statisticand the Pearson dispersion estimatorThe first quantity measures goodness of fit; the second estimates the dispersion parameter, which equals one in a correctly specified Poisson regression. A standard rough calculation substitutes the residual deviance for the Pearson statistic and givesIf the reported upper-tail probability $4.908651\times 10^{-11}$ is inverted numerically, the actual Pearson statistic used by the code is about $82.93$, giving $\hat{\varphi }\simeq 5.18$. Either calculation reveals severe overdispersion.

The quasi-Poisson standard errors are the Poisson standard errors multiplied by $\sqrt{\hat{\varphi }}$, and the Pearson estimator divides the Pearson chi-squared statistic by the residual degrees of freedom $40-3=37$. Therefore the requested sum of squared Pearson residuals isup to the rounding in the printed standard errors.