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?
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$\begingroup$ Do you mean what implications it has for your scientific question or do you concerned about what it means fr the gamma distribution? $\endgroup$– mdeweyCommented Nov 24, 2016 at 12:25
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1 Answer
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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 $\phi$.
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$\begingroup$ Does μ changes for different value of explanatory variables? $\endgroup$– tatamiCommented Oct 23, 2017 at 4:01
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$\begingroup$ @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. $\endgroup$ Commented Oct 23, 2017 at 7:00