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I'd be very grateful if someone could help me understand the idea behind link functions:

I know that the idea is that we want to map the mean to the $\eta$-vector. Also, I know that the canonical link is the one that makes $\eta=\theta$, where $\theta$ is the parameter.

But why/when do we want to choose the canonical link? When do we have to consider other alternatives?

Thanks a lot!

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  • $\begingroup$ I think you will find the information you need in the linked thread. Please read it. If it isn't what you want / you still have a question afterwards, come back here & edit your question to state what you learned & what you still need to know. Then we can provide the information you need without just duplicating material elsewhere that already didn't help you. $\endgroup$ – gung - Reinstate Monica Apr 24 '18 at 12:59
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The point of the link functions is to force the modelled means into sensible ranges. For count data one uses the log-link to obtain a positive mean. A count cannot be negative, and after back-transformation the mean ends up in the domain of positive real numbers. For binary data we use a logit-link to ensure a expected value (modelled success probability) between 0 and 1.

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