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I am building a gamma GLM regression model with a log link function. The model I fit is below:
I understand that I can calculate the marginal effect of x on log(y) by taking the derivative with respect to x. How can I calculate the marginal effect of x on y?
$$ \begin{split} \frac{dy}{dx} & = \frac{dy}{d\log y} \frac{d\log y}{dx} \\ & = \left(\frac{d\log y}{dy}\right)^{-1} \frac{d\log y}{dx} \\ & = y \cdot \frac{d\log y}{dx} \end{split} $$ In practice this means that the marginal effect of $x$ on $y$ *depends on the value of $y$, so you'd have to pick reference values of $x$ and $z$ (typically the mean values) to do the calculation.
Alternatively, you can frame the marginal effect on a log-response as a proportional change. If you compute the marginal effect of $x$ on $\log y$ and exponentiate it, you can frame it as a proportional effect (e.g. "a one-unit change in $x$ is associated with a ()-fold change in $y$")
