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I would like some help understanding when the sqrt, 1 / mu^2, and inverse link functions would be useful.

Thanks!

My notes are shown below - . sqrt - mean of predicted value must be positive number inclusive of 0

  • . 1 / mu^2

    • canonical link function for inverse gaussian family
    • ????

  • . inverse (1 / mu)

    • canonical link function for gamma family
    • ????
gung - Reinstate Monica
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Zach Hill
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  • These are just *common* link functions. You can use any function you want as a link, depending on the question you are trying to answer. – AdamO Jun 06 '19 at 16:52
  • For an example of an *uncommon* link function, see https://stats.stackexchange.com/a/64039/919. Its analysis addresses your broader question concerning what considerations are involved in selecting a link function. – whuber Jun 07 '19 at 21:51

1 Answers1

2

  1. When the canonical link (and accompanying mean-variance relationship is used), estimating GLM with Newton-Raphson is equivalent to maximizing a likelihood with Fisher Scoring. Hence it is the asymptotically efficient and unbiased estimator. But who cares? a little bias isn't bad.

  2. The link affects the interpretation of the model coefficients. For instance, the linear link estimates mean differences, the log link estimates geometric mean differences, the inverse link estimates harmonic mean differences.

  3. Dovetailing with 1, sometimes the choice of appropriate link comes up when considering the probability model relating response and outcome. For example, when studying the number of sexual contacts vs. risk of contracting HIV, it's not quite right to think of the "odds" of HIV increasing with each sexual contact: with no sexual encounters the risk is (very near) 0; epidemiologic data support this, but the logistic model attributes non-zero risk to those w/o sexual contact and this creates a form of bias in estimates. See more here: https://www.jstor.org/stable/2532454?seq=1#page_scan_tab_contents

  4. Hate to say this, but sometimes a link is picked because it provides stable estimates. This isn't the worst thing: predictions, post-estimates, or marginal standardization can be used to summarize the output. We actually see lots of forms of nonparametric structural equation models ("oracle models") that use regression based on least-squares using spline encoded exposures and robust estimates of errors, then the model is boiled down to provide useful summaries and descriptions. In the words of John Tukey "Build your model as big as a house".

AdamO
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  • I'm not really asking for general info about link functions. I'm more asking for the specific attributes of the sqrt, 1 / mu^2, and inverse link functions as well as additional info on the other ones I have listed. For eaxample, the sqrt link function seems like it would cause all kinds of issues because if you square both side of sqrt(u) = B0 + B1X1 + ... BPXP, you will get u = (B0 + B1X1 + ... BPXP)^2 which after factoring out would be a very strange result and I can't think of a time when this would be useful – Zach Hill Jun 07 '19 at 12:54
  • @ZachHill have you read McCullogh and Nelder's "Generalized Linear Models"? – AdamO Jun 07 '19 at 13:35
  • No. I just looked it up and it's a 500 page book. Looking through the table of contents I don't see any specific sections on these link functions. Are there any sections that would be particularly useful? – Zach Hill Jun 07 '19 at 15:58
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    @ZachHill As a bit of hard dealt advice, it's hard to have a meaningful discussion without the appropriate background. If you can ebay a cheap copy, the definitions and examples will really help contextualize it. The right answer is, as I said, "Use the right link for your question." – AdamO Jun 07 '19 at 16:01
  • I have a good understanding of glms and link functions, but im not reading a 500 page book which I propbably already know 90%+ of the material of to learn a few properties for a few link functions... Thanks? – Zach Hill Jun 12 '19 at 15:12