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If I understand correctly, Probit model is a generalized linear model. I didn't see it listed in the table, so I was wondering if its link function is canonical for some distribution?

kjetil b halvorsen
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Tim
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  • In GLM terms probit defines a link; the most natural distribution family for that link is the binomial. – Nick Cox Jun 27 '13 at 23:01
  • @NickCox: Thanks! The link corresponding to the binomial is logistic function, and the regression is logistic regression, not probit regression? – Tim Jun 27 '13 at 23:03
  • Say rather that the binomial and the logit hang together, but other links make sense for the binomial too. See e.g. http://www.stata.com/manuals13/rglm.pdf Table on p.7. – Nick Cox Jun 27 '13 at 23:09
  • @NickCox: So the link function in Probit is not canonical? – Tim Jun 27 '13 at 23:31
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    Correct, I believe. See http://stats.stackexchange.com/questions/40876/difference-between-link-function-and-canonical-link-function-for-glm – Nick Cox Jun 27 '13 at 23:37
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    I had some fun with this just trying to simplify the implied variance function assuming there were some exponential family distribution for which the probit link would be natural/canonical. – Glen_b Jun 28 '13 at 06:09

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As answered in comments, the canonical link function in the binomial model is the logit. So the probit link is not canonical.

That should not stop you from using it, however. A link function being canonical is a purely mathematical concept without much statistical impact. You should use the link function which are appropriate for the context you are modelling. I once had extensive bioassay data (goal was to model LD50) where it was actually possible to see that the probit link fitted the data (marginally) better than the logit link. The difference was not large enough to have any practical significance, though. Probably the difference came from the probit "kissing" its asymptotes much faster than the logit, which is natural for some bioassay data. Simply, when the dose gets too high, all the animals die!

kjetil b halvorsen
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