# How to specify Gamma parameterizations in a generalized linear model setting

I am trying to model an outcome using a generalized linear model and the Gamma distribution with a log link function using the glm() function in R. I went to Wikipedia to look at the parameterizations for the Gamma distribution. Now I would like to state the model formally with $$shape = k$$ and $$scale = \theta$$ in a manuscript. What I would like to do is something along those lines:

$$y_{i}\sim \Gamma(k,\theta_{i})$$

$$E(y_{i})=k\theta_{i}$$ and $$var(y_{i}) =k\theta_{i}^{2}$$

$$log(k\theta_{i})=\alpha +\beta_{1}X_{i}$$

My question is whether this is correct? I read that the glm() function in R only models the scale parameter $$\theta$$ as a function of the independent variables (hence the index for $$\theta$$) whereas the shape parameter $$k$$ is constant and appears as the dispersion parameter $$\phi = 1/k$$ in the glm() output.

My second question would be how can I change the variance specification ($$Var(y_{i}) =k\theta_{i}^{2}$$) when I want $$k\theta_{i} = \mu_{i}$$ so that the model would look like:

$$log(\mu_{i})=\alpha +\beta_{1}X_{i}$$

This doesn't seem correct: $$var(y_{i}) = \mu_{i}\theta_{i}$$, or does it?

• You will probably want to use the shape-mean parameterization. Quite a few posts on site discuss this parameterization. See stats.stackexchange.com/questions/136909/… for an explicit density, One reference that explicitly discusses it is the book by de Jong and Heller. Aug 25, 2020 at 5:09
• Thank you @Glen_b for the comment and the links. I think I did found my answer now in Faraway 2006, page 149, where he suggest that in a GLM setting it is more convenient to reparameterize the distribution with $\lambda = \nu/\mu$, which then results in $E(Y)=\mu$ and $var(Y)=\mu^2/\nu$ Aug 25, 2020 at 14:20
• Sure, that would be starting with $\nu$ as shape and $\lambda$ as a rate parameter (in your question's notation, these would be $k$ and $1/\theta$). You could post an answer along the lines of your comment if you wish, Aug 25, 2020 at 23:42
• See stats.stackexchange.com/questions/474326/deviance-for-gamma-glm It is specifically for the gamma deviance but, on the way, it clarifies the parametrization for gamma glms. Aug 26, 2020 at 2:09
• @GordonSmyth Thanks for the link! Aug 26, 2020 at 5:06

Using the shape ($$\nu$$) and rate ($$\lambda$$) parameterizations for the density of the Gamma distribution we get:

$$f(y)=\frac{1}{\Gamma(\nu)}\lambda^\nu y^{\nu-1}e^{-\lambda y}\qquad y>0$$

Now we can reparameterize by setting $$\lambda=\nu/\mu$$ and get:

$$f(y)=\frac{1}{\Gamma(\nu)}\left(\frac{\nu}{\mu}\right)^\nu y^{\nu-1}e^{-\left(\frac{y\nu}{\mu}\right)}\qquad y>0$$

Now: $$EY=\mu \quad \mathbb{and} \quad var\,Y=\frac{\mu^2}{\nu}$$

To translate this into your notation:

$$y_{i}\sim \Gamma(\mu_{i},\nu)$$ $$E(y_{i})=\mu_{i}\quad \mathbb{and} \quad var(y_{i})=\frac{\mu_{i}^2}{\nu}$$ $$log(\mu_{i})=\alpha +\beta_{1}X_{i}$$

Reference: Faraway (2006): Extending the linear model with R (page 149).