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"The canonical link of Gamma GLM is $g(x)=1/x$ is often not very practical. Log-link is more appropriated in most cases." One reason I can think of is that log-link makes sure $\mu$, the mean, is always greater than zero. (please correct me if I'm wrong.) But I believe there are other considerations.

So why people often choose the log-link over the canonical one? Is it related to the fact that $\log$ transformation stabilizes data variance?

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  • $\begingroup$ who are you quoting? $\endgroup$ – Glen_b Aug 25 '20 at 6:07
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    $\begingroup$ Quoting my class instructor, Prof. Hans Muller. $\endgroup$ – WCMC Aug 25 '20 at 6:23
  • $\begingroup$ The prime reason would be that you expect that the logs of the conditional means are linear in the supplied predictors. $\endgroup$ – Glen_b Aug 25 '20 at 6:59
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A good question. I see the following reasons:

  • A log link produces a multiplicative model on the original scale and is thus easy to interprete. This is not true for the canonical link of the Gamma GLM.

  • In insurance pricing, the expected claim amount $E(L)$ is often decomposed into $E(L) = E(F)E(S)$, where $F$ is the claim frequency and $S$ the claim height. If both $E(F)$ and $E(S)$ are modeled by GLMs with log link, the coefficients of both models can be easily combined to determine the effects on the expected loss $E(L)$.

  • Not using the canonical link introduces a prediction bias on the original scale, i.e. the average prediction differs from the average response. This would be a reason against the log link. However, in my experience, this bias is small with the log link and can easily be fixed by a multiplicative correction factor.

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  • $\begingroup$ Thanks. Can you please give more detail on "decomposed into a product of a claim count regression and a claim height regression"? And what is "prediction bias"? $\endgroup$ – WCMC Aug 24 '20 at 16:04

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