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
. inverse (1 / mu)
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.
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.
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
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".
