= Gamma regression with canonical link
{c}
{title2=$\eta=-1/\mu,\quad\mu=-1/\eta$}
For a <Gamma distribution> with fixed shape $\alpha$, choose dispersion $\phi=1/\alpha$, canonical parameter $\theta=-1/\mu$, and cumulant $b(\theta)=-\log(-\theta)$. The <canonical link function> is the displayed negative reciprocal; reversing its sign is an equivalent regression convention. For independent observations, the <score function> is $\phi^{-1}X^T(y-\mu)$ and the <Fisher information matrix> is $\phi^{-1}X^T\operatorname{diag}(\mu_i^2)X$. The <deviance> is $2\sum_i[y_i/\widehat\mu_i-1-\log(y_i/\widehat\mu_i)]$, with twice the log-likelihood ratio equal to this divided by $\phi$. The fitted mean is the inverse link applied to $X\widehat\beta$, not $X\widehat\beta$ itself.
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