= Poisson regression margin-matching score equations
{c}
{title2=$Z^T(y-\widehat\mu)=0$}
For <independent> <Poisson distributions> with log means $Z\beta+o$, where $o$ is a known <offset>, differentiation of the <log-likelihood> gives the <score function> $Z^T(y-\mu)$. An interior <maximum-likelihood estimator> therefore satisfies the displayed equations. An intercept and categorical indicator columns make fitted count totals equal observed count totals in each represented factor level. These equations match counts, rather than unweighted averages of rates when exposures differ.
// Target: probability-and-statistics.bigb
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