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?

  • $\begingroup$ Do you mean what implications it has for your scientific question or do you concerned about what it means fr the gamma distribution? $\endgroup$ – mdewey Nov 24 '16 at 12:25

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: $$ \phi = \frac{1}{\alpha} $$

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 $\phi$.

  • $\begingroup$ Does μ changes for different value of explanatory variables? $\endgroup$ – tatami Oct 23 '17 at 4:01
  • $\begingroup$ @tatami Well, in my example $\mu$ is the mean of x. But in general effects estimates of explanatory variables would change in accordance with model. $\endgroup$ – Andrey Kolyadin Oct 23 '17 at 7:00

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