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I am studying GLM at the moment and have a few questions regarding link functions.

  1. Why are the conditions of the link function to be smooth monotonic function? What properties are preserved by having such conditions?

  2. Furthermore, a lot of material that I am reading always takes the inverse of the link function $g$, but since the conditions are monotonic, an inverse is not guaranteed. So what happens if the link function does not have an inverse?

Any clarification will be much appreciated!

Ramiro Ramirez
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    A smooth strictly monotinic function always have an inverse, and that is what is intended. Without an inverse the glm theory will not go through – kjetil b halvorsen May 07 '22 at 18:43
  • @kjetilbhalvorsen you can probably make it an answer – Tim May 07 '22 at 19:37
  • At https://stats.stackexchange.com/a/64039/919 I proposed a nonmonotonic, non-invertible link and provided some code to implement it. cc@kjetil – whuber May 07 '22 at 21:47
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    @whuber: I revised (again) that interesting example, but I cannot see that it has a non-monotone link function. It is really a logistic regression with standard, logistic link, but the usual linear predictor $\eta$ is replaced by a non-linear predictor, so necessitating an extra step for the optimization. This is sometimes called a generalized non-linear model. There is an R package (CRAN) gnm referenced at https://stats.stackexchange.com/questions/403747/iterative-optimization-of-alternative-glm-family/403758#403758 and otherwise. – kjetil b halvorsen May 07 '22 at 22:19
  • @kjetilbhalvorsen ok, I understand. Since invertibility implies strict monotonic and vice versa. However there are some sources that I seen that does not use the strict monotonic definition. For example in “An Introduction to Generalized Linear Models Annette J. Dobson, Adrian G Barnett” the authors mention the link function can be “flat” (E.g a constant function) which means no inverse exists and mentions also differentiable . What is the purposes of the differentiable condition? And – Ramiro Ramirez May 08 '22 at 00:53
  • Can you give an exact quote? I cannot find that, and it does not make much sense! A constant link function would seem to say that the linear predictor must be that constant value? – kjetil b halvorsen May 08 '22 at 02:05
  • @kjetilbhalvorsen It’s on page 57 – Ramiro Ramirez May 08 '22 at 02:23
  • The exact quote is : “ $g$ is a monotone, differentiable function called the link function; that is, it is flat, or increasing or decreasing with μi, but it cannot be increasing for some values of μi and decreasing for other values.” – Ramiro Ramirez May 08 '22 at 02:35
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    That quote is very strange, but nevertheless, it does not make sense that it can be flat! – kjetil b halvorsen May 08 '22 at 02:53
  • Why not? If a model suggests that, it's mathematically permissible. If you want to view non-monotonic links as non-linear models, that's fine: but the equivalence of the two approaches indicates that most of your claims about links in this thread are not fully correct. – whuber May 08 '22 at 12:58

1 Answers1

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Why are the conditions of the link function to be smooth monotonic function? What properties are preserved by having such conditions?

For a glm the mean $\mu$ is linked to the linear predictor $\eta=\beta^T x$ by the link function $g(\mu)=\eta$. Assuming the link function has an inverse, $g^{-1}=m$, we find that $$ m(g(\mu))=g^{-1}(g(\mu))=\mu$$ or $$ \mu =m(\eta) $$ so we call $m$ the mean function. If $g$ where not strictly monotone, so it had no inverse, then the equation $$ g(\mu)=\eta$$ would have multiple solutions, and so the linear predictor $\eta$ would not specify a unique mean!

How woud you then relate your glm model to data? How wpuld you estimate the parameters $\beta$ if the same observed mean could be equally explained by different parameter values? It would not work.

Furthermore, a lot of material that I am reading always takes the inverse of the link function , but since the conditions are monotonic, an inverse is not guaranteed. So what happens if the link function does not have an inverse?

What is meant is strictly monotonic, and then an inverse always exists. The above paragraph explains what will happen if an inverse does not exist. Therefore, existence of an inverse is part of the definition of an glm.

As for your new question in comments:

What is the purposes of the differentiable condition?

That condition is maybe not as fundamental as is monotonicity, but as soon as you start to develop the likelihood theory for glm's, first and second derivatives start to appear. Maybe one could do without it, but the theory would be more difficult, and the need does not seem to be there. Also, numerical optimization of non-differentiable functions is more difficult. But if you see some application for a non-diff link function, just try!

kjetil b halvorsen
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  • This logic does not pan out, because "$g^{-1}$" can define a unique mean without being invertible. The parameters can be estimated as the sunflowers example shows. – whuber May 07 '22 at 21:49
  • So I understand much better the strict monotonic and differentiable properties needed to make the theory work. I am still confused about the quote from the book that mentions the non strict monotonic property as mentioned the comments above. – Ramiro Ramirez May 08 '22 at 02:44
  • Hi @kjetil , in case it's helpful to say explicitly: in terms of flatness, you're talking about $g$, whereas whuber is talking about $m$. It's OK that $m$ is flat, but it's not OK that $g$ is flat. – Ben May 11 '22 at 15:46
  • @Ben, interesting, can you elaborate? – kjetil b halvorsen May 11 '22 at 22:58
  • @kjetilbhalvorsen Sorry, could you please elaborate on what you'd like me to expand on? I was just stating the discrepancy between what you and whuber were saying. – Ben May 12 '22 at 17:33
  • @Ben: Since $g, m$ are defined as inverses of each other, I do not understand how one can be flat and the other no! – kjetil b halvorsen May 20 '22 at 11:43
  • @kjetilbhalvorsen In the case that $m$ has flat regions, $g$ doesn't even exist! – Ben May 20 '22 at 16:22