# Gamma GLM: why log-link is more common than canonical link

"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?

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

## 1 Answer

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.

• 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"? – WCMC Aug 24 '20 at 16:04