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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Do you mean what implications it has for your scientific question or do you concerned about what it means fr the gamma distribution? – mdewey Nov 24 '16 at 12:25
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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 $\phi$.
Glen_b
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Andrey Kolyadin
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@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. – Andrey Kolyadin Oct 23 '17 at 07:00