I'm attempting to model how 2 predictor variables affect the relative proportions of 3 different categorical groups. I started off by running a binomial model for the proportion of each individual group, then running multinomial model for the proportions of all three together.
When running assessments of the various models however, I was confused by the disparity in results.
In particular, the McFadden's Pseudo R^2 for the multinomial model was much lower than for any of the three binomial models (about 1/3 the value of the lowest scoring binomial model), and well outside what I have found are the bounds for "good" model fit (despite the multinomial model being significantly different from the null model).
The predictions made by the multinomial model are extremely similar to the predictions made by each of the individual models, and the relationship between observed and predicted is both generally good and similar for all of the binomial models and the multinomial model.
So I'm left wondering: is McFadden's Pseudo R^2 an appropriate metric for multinomial models? Should I expect much lower values in a multinomial model than for any of the individual binomial models?
The results I've gotten would lead me to believe that the multinomial models is providing so little explanatory power as to not be worthwhile, but I'm concerned that this is an indication about problems in my binomial models that are not apparent when assessing them individually.
Everything I've read about the differences between multinomial and multiple binomial models seems to suggest that, in general, they should be very similar (e.g. Here and Here).
This has been a bit rambly, so I'll try to boil it down to more concrete questions:
Is McFadden's Pseudo R^2 and appropriate metric when assessing multinomial models and, if yes, is it expected that the value should roughly correspond to the related binomial models? If yes to both, then what kinds of issues might cause a large mismatch?
If not, what is the appropriate metric for assessing model fit for multinomial models?
and finally, given the issues that I am having, is there any reason that I shouldn't just stick to the binomial models, or is it worth trying to figure out the issues I'm having with the multinomial? AKA: is there a significant advantage to one versus the other technique? (the answers to the linked questions led me to believe that there is not)
I've tried to do as much reading as possible to figure out the issues here, but have had very little success. Any help or even links to reference materials that would be helpful would be greatly appreciated
