# Dispersion parameter for Gamma family

I have ran a glm in R, and near the bottom of the summary() output, it states:

(Dispersion parameter for Gamma family taken to be 1.680014)

What does this mean/represent?

## Answer

Gamma distribution defined by two parameters – shape ($\alpha$) and rate ($\beta$).

There is alternative parameterization through mean ($\mu$) and shape, which is used in GLM.

We take $\mu = \alpha/\beta$ and put it into place of rate (as $\beta = \alpha/\mu$), resulting in function $Gamma(\mu,\alpha)$.

In R GLM assumes shape to be a constant (as linear regression assumes constant variance). To satisfy this assumption dispersion ($\phi$) is introduced:

For the simple case glm(x ~ 1, family = Gamma(link = 'identity)), summary.glm gives you $\text{estimate}$, that is equal to $\mu$ (note that default link is ‘inverse’ and estimate = $1/\mu$) and $\text{dispersion}$ is $\phi$.

Attribution
Source : Link , Question Author : Maggie Smith , Answer Author : Glen_b