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Use the exponential dispersion family convention $f(y)=\exp \{[y\theta -b(\theta )]/\varphi +c(y,\varphi )\}$. For the Gamma distribution, takeSubstitution reproduces the gamma density exactly. Differentiating its normalization with respect to $\theta$ yields $\mathbb{E}Y=b^{\prime }(\theta )$ and $\mathrm{Var}Y=\varphi b^{\prime \prime }(\theta )$: the first identity follows from zero mean score, and differentiation again gives the variance. ThereforeThe canonical link function is $g(\mu )=\theta =-1/\mu$. Some conventions reverse the sign and call $1/\mu$ the reciprocal link; this only reverses the regression coefficients. We retain the genuine natural parameter $\theta =-1/\mu$.

In the Gamma regression with canonical link, ${\eta }_i=x_i^T\beta <0$ and ${\mu }_i=-1/{\eta }_i$. Differentiating the independent-observation log-likelihood givesHere $d{\mu }_i/d{\eta }_i={\mu }_i^2$, so the negative Hessian equals this information matrix. Fisher scoring or Newton iteration iswith the means and information recalculated at each iteration; damping can preserve $X\beta <0$ and increase the likelihood. Full column rank makes the information positive definite for positive means.

The saturated model has mean $y_i$. Subtracting fitted from saturated log-likelihood yields the unscaled devianceTwice the actual log-likelihood ratio is $D/\varphi$. In particular, the printed parenthetical $\hat{\mu }=X\hat{\beta }$ is incorrect for this nonidentity link: $X\hat{\beta }$ estimates $\eta$, and ${\hat{\mu }}_i=-1/(x_i^T\hat{\beta })$.