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In a GLM, if the response variable has a Gamma distribution, why is the inverse used as the link function, i.e.: $\mu = -(X\beta)^{-1}$?

In particular, why is the inverse the canonical link? Does it have to do with the natural parameters of the gamma distribution?

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    $\begingroup$ It's the canonical link, see e.g., stats.stackexchange.com/questions/40876/… $\endgroup$ – Momo Mar 19 '16 at 23:23
  • $\begingroup$ I guess I should re-phrase the question: why is the inverse the canonical link? Does it have to do with the natural parameters of the gamma distribution? $\endgroup$ – dmjn Mar 21 '16 at 19:54
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why is the inverse used as the link function, i.e.: $μ=−(Xβ)^{−1}$

That's actually the mean-function $\mu(\eta)$. The link function is $\eta(\mu)$. However, both are in the form of a negative reciprocal in this case, since the negative of the reciprocal is its own inverse-function.

In particular, why is the inverse the canonical link? Does it have to do with the natural parameters of the gamma distribution?

Yes, that's where canonical links come from.

See for example, the Wikipedia article on the Generalized Linear model, in the section on the link function:

When using a distribution function with a canonical parameter $\theta$, the canonical link function is the function that expresses $\theta$ in terms of $\mu$

('canonical parameter' being another term for natural parameter)

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  • $\begingroup$ "However, both are in the form of a negative reciprocal in this case, since the negative of the reciprocal is its own inverse-function." But so is the positive reciprocal as well, right? $\endgroup$ – Jan Rothkegel Mar 20 at 20:11
  • $\begingroup$ It's not being self-inverse that makes it the canonical link though. The point being made was that "even though the function you give there is indeed the negative of the multiplicative inverse, it's not the canonical link, but rather the inverse-function of the canonical link". [The canonical link is often written dropping the negative and then it's still a self-inverse but this property is a consequence of the link being a multiple of the multiplicative-inverse, not the other way around] $\endgroup$ – Glen_b Mar 21 at 5:43

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