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The working score function and its derivative areAt an interior working maximizer, $U(\hat{\beta })=0$. A Taylor expansion at ${\beta }_0$ yieldsfor an intermediate value ${\beta }^{\ast }$. When the last term is negligible relative to the linear one, this givesThis is the Poisson score linearization under proportional variance. Correct specification of the mean alone implies $EU({\beta }_0)=0$. The stipulated independence and variance relation giveTaking the mean and variance of the linear approximation therefore provesThese are first-order conclusions, not exact finite-sample identities. The mean and variance assumptions alone do not guarantee that the approximation is accurate: an interior consistent solution, increasing information, control of leverage, and a suitable central limit condition are needed for normal inference.

The quasi-likelihood score for variance $\varphi {\mu }_i$ is $U/\varphi$, so its zero is unchanged by the unknown common dispersion parameter. Thus the working Poisson estimating equation remains appropriate for the mean parameter, but its unadjusted Poisson variance estimate is wrong when $\varphi \ne 1$. A usual Pearson dispersion estimator for this one-parameter model isIn particular overdispersion multiplies coefficient standard errors by approximately $\sqrt{\varphi }$, even though it leaves the coefficient estimating equation unchanged.