Poisson regression margin-matching score equations ($Z^T(y-\hat{\mu })=0$)

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.

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