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I'm trying to get my head around GLMs.

In order to setup the GLM we need a smooth invertible function of the mean which maps $η$ into the scale of the response. I don't really understand what the "scale of the response" is. Basically, we have a function $g$, such that $g(\mu)=\eta$ but what does this scale mean?

I feel like this is a pretty basic question and my guess would be that this is the range of values the response can take, but still want to be sure.

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  • $\begingroup$ It might be basic, but it's a good question because IMHO the statement is highly misleading. The link function maps an abstract set of real numbers into a parameter of a distribution family. That parameter need not be on any scale related to the response itself. As an example, one could formulate a GLM with a negated Gamma response with a fixed mean and a shape parameter varying with a linear combination of the explanatory variables. The link must be a function of the real numbers with non-negative values even though the "scale of the response" is the negative numbers! $\endgroup$
    – whuber ♦
    Commented Apr 11, 2023 at 18:20

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This is probably best explained with examples. Let's say you have a binomial model, so the response is measured by the binomial probability $p$, so $\mu=p$. The linear predictor $\eta_i= x_i^\beta$ can take any real value, so taking $\eta=\mu=p$ can (and will) lead to impossible values for $p$.

Enter the link function. Let $F$ be a cdf (cumulative distribution function) and set $$ \mu = F(\eta) $$ which will guarantee a legal value for $\mu$. The link function is $g=F^{-1}$. So the link function translates between the probability scale and the linear scale.

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