Paradigm for link function switching in ordinal response models

specification

#1

I’m trying to find the best framework for dealing with the following problem. With ordinal Y there is no link function that works the majority of times even within a specific subject matter field. I often compare goodness of fit for a proportional odds ordinal logistic model with a log-log link ordinal model (proportional hazards) because they are so different. I sense that it’s best to allow “link function uncertainty”, but I don’t seek a final result that mixes the two links but instead one that results in a single effect measure I can communicate to the researcher, e.g., I’d like to get a posterior distribution for a certain odds ratio or hazards ratio, and be able to communicate exactly which one this is.

Is there a way to do this that will reflect in the final posterior distribution the uncertainty in choosing that model? Are there other ways to think about this?

As a side note I don’t typically use the probit ordinal model because of the lack of an easy interpretation for the model parameters.


#2

I think my answer would be.

  1. Compute quantity of interest for various link functions, using the posterior predictive distribution under counterfactuals when it is not computable directly from the posterior distribution
  2. Weight the quantity of interest using stacking weights (M-open) or model weights (M-closed)
  3. Report the weighted distribution of the quantity of interest

The idea of using the posterior predictive distribution to compute things like this has come up before


but it probably has the highest ratio of usefulness to utilization. I guess with an ordinal model you are interested in the ratio of the probability that the outcome is less than or equal to y divided by the probability that the outcome is greater than y?