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Unfortunately, I found only this paper on the matter: http://dx.doi.org/10.4103%2F0972-124X.75909 And this: Difference between logit and probit models.

Can you tell me when, generally, I should use logit, probit, complementary log-log or negative log-log when performing an ordinal regression? (Or link me something which I can study and unfortunately I did not find?).

(I use R and I am not very proficient in statistics).

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There is no general guidance on this question, except that if you had to pick one model without knowing anything about the fit of any of the models, you might pick the logistic link (proportional odds ordinal logistic model) because its parameters are more interpretable. In my RMS course notes I have an in-depth case study in the chapter on ordinal models for continuous $Y$. You'll see some diagnostic plots for choosing the link function (the winner in the example was log-log, i.e., the discrete proportional hazards model). The approach I took there was to fit a tentative model (an ordinary linear model) just to get a linear predictor that could be stratified on (I used 6 quantile intervals because of the available sample size) with there being little outcome heterogeneity in each stratum. Then I computed the empirical CDF within each stratum and took various transformations including logit, log-log, probit. Only one (log-log) yielded curves that were parallel. Note that ordinal semiparametric models do not assume a shape for such curves; they only assume parallelism.

When $Y$ is discrete there are other displays you can also make, as discussed in the chapter in RMS that preceeds the continuous $Y$ chapter.

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    $\begingroup$ Poor me: I did not think about simply taking a look at web pages of whom I consider one of the most inspiring people on Earth in the field of statistical modeling and its validation. Thank you Prof. Harrell, I will study your RMS course notes which you kindly provided $\endgroup$ – statisticianwannabe Sep 9 '17 at 12:28

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