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The gamma distribution has two parameters, I understand that the linear predictor predicts $\mu = g^{-1}(X\beta)$ where $g$ is the link function but how does the linear predictor specify the second parameter of the gamma distribution?

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With a GLM you're essentially fitting a Gamma model under a shape-mean parameterization for the Gamma distribution, and the linear predictor in the GLM gives a model for the mean, just as it does in linear regression.

The linear predictor doesn't give a model for the other parameter (any more than the linear predictor in regression does for a normal model).

In both cases, the other parameter is assumed to be constant across observations.

[The remaining parameter can still be estimated, of course. You can do MLE or you can estimate it from the GLM output, in effect a form of method of moments.]

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