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I am performing ordinal regression on several datasets, I have 5 ordered response categories and only one explanatory variable X. For each dataset I run the analysis 3 times, each time using a different link function (1. probit, 2. logit, 3. comploglog) and I calculate the AIC to see which function fits my data best.

It seems that for different datasets I get different link functions significantly providing a "best" fit; for example probit is better for dataset 1 and logit is better for dataset 2 etc. I am trying to find an explanation for such difference.

So my question is, what is the "physical" meaning of each link function? For example, I understand the probit link function assumes the response scale can be related to a latent continuous, normally distributed variable but for the other 2 I have no idea.

Any insight on this would be great!

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marked as duplicate by Momo, Reinstate Monica - G. Simpson, gung - Reinstate Monica, Scortchi - Reinstate Monica, Nick Cox Oct 19 '13 at 18:43

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  • $\begingroup$ One explanation you don't seem to have ruled out is random variation. If you observed a new data set for each, might your decisions have gone the other way? You might like to consider cross-validation $\endgroup$ – Glen_b -Reinstate Monica Oct 17 '13 at 7:16
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    $\begingroup$ Each of the link functions is an actual function whose value can be computed. Try looking up the formula for each of your links (for example, the logit is clearly defined on wikipedia), and then graph each function to see its "physical" meaning. $\endgroup$ – zkurtz Oct 17 '13 at 12:15
  • $\begingroup$ Closely related: stats.stackexchange.com/questions/20523/… and $\endgroup$ – Momo Oct 19 '13 at 15:25
  • $\begingroup$ Very helpful, thanks! I hadnt seen the other thread. $\endgroup$ – Neodyme Oct 21 '13 at 2:27