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?
Gamma distribution defined by two parameters – shape (α) and rate (β).
There is alternative parameterization through mean (μ) and shape, which is used in GLM.
We take μ=α/β and put it into place of rate (as β=α/μ), resulting in function Gamma(μ,α).
In R GLM assumes shape to be a constant (as linear regression assumes constant variance). To satisfy this assumption dispersion (ϕ) is introduced:
For the simple case
glm(x ~ 1, family = Gamma(link = 'identity)),
summary.glm gives you estimate, that is equal to μ (note that default link is ‘inverse’ and estimate = 1/μ) and dispersion is ϕ.