I have a question about interpreting a Bayesian ordered logistic model. The outcome variable is ordinal, on a scale 1 - 5, and there are two categorical predictor variables, religion (4 levels) and country (3 levels). I would prefer to use level mean coding (and not create dummies) for the categorical predictors for the reasons McElreath gives in his Statistical Rethinking, namely because in that case I'd assume there is more uncertainty about the non-reference categories as they'd include two paramaters (and hence priors).
This would give me the following simple model (loosely based on a model by McElreath in chapter 12 of the same book, and without bothering about choosing the right priors for now):
$$\displaystyle R_{i} \sim Orderedlogit(\phi_{i}, \kappa)$$
$$\displaystyle \frac{Pr(y_{i}\geq \kappa)}{1 - Pr(y_{i}\geq \kappa)} = \alpha_{\kappa} - \phi_{i}$$
$$\displaystyle \phi = \gamma_{country[i]} + \beta_{religion[i]}$$
4 priors for the cutpoints:
$$\displaystyle \kappa = Normal(0,1)$$
3 priors, one for each level, for 'country':
$$\displaystyle \gamma = Normal(0,1)$$
4 priors, one for each level, for 'religion':
$$\displaystyle \beta = Normal(0,1)$$
From reading other posts (and very good responses) here and here, I think I understand how to interpret the results for the cutpoints. So, my questions concern the interpretation of the coefficients (and their posteriors) for the non-dummy-coded categorical predictors.
I am mainly interested in the effects of the 'country' variable. However, the result for that parameter is inevitably entangled with the other parameters in the model. I could get posterior predictive samples for each of the levels of 'country', but those samples would still depend on the values of the other predictor. So, my question is: Can I get an estimate (presumably of probabilities for each of the cutpoints) for each of the levels of country, independently from the other variable, and how? In short, I want an estimate of the effect of 'country' independently from the religion of the participants.
My second question is related but more general: How to interpret the coefficients for each of the levels of country? That is, how to explain what the coefficients mean without the use of posterior predictive samples. Interpretation is easier, I suppose, if the predictors are dummy coded, but I would like to avoid that for the reason discussed above. And even if they were dummy-coded, I'm not sure they can be interpreted independently from the other predictor.
Many thanks!
rmsbpackage I use Dirichlet priors for the probabilities that are induced by the intercepts, which makes sampling faster, automatically forces ordering of intercepts, and works well for large numbers of Y categories. But I don't handle ordinal predictors explicitly. – Frank Harrell Sep 02 '22 at 11:17