I have some ordinal response $y$ that I modeled using both ordinal logistic regression and multinomial logistic regression (to avoid the proportional odds assumption), using two continuous variables as predictors $x_1$ and $x_2$.
I tested different models by including the predictor variables one at a time and their interactions.
$$y \sim x_1 \\ y \sim x_2 \\ y \sim x_1 + x_2 \\ y \sim x_1 + x_2 + x_1x_2 $$
In this way I obtained 8 different models (4 models using ordinal, and 4 models using multinomial logistic regression) and therefore 8 AIC values. It turn out that the best model (the difference in AIC is like 200) is the multinomial logistic with the following predictors:
$$ y \sim x_1 + x_2 + x_1x_2 $$
As far as I know, this model provides the best fit amongst all the various options. Now, how do I quantify if this model is good for the data in an absolute sense (in order for it to be published)?
I can do the regression both using frequentist glm and bayesian glm (so far I did frequentist way because it was more computationally cheap). Ideally I'd like to have the methodology that is most honest and convincing.
I'm more interested to assess model fit in terms of inference rather than prediction. In my specific problem, one of the classes is much more probable than the others on a wide range of the parameters, so prediction is unfeasible. But I'm still interested in discovering how appropriate are the estimates of the underlying probabilities.