# Multinomial (Categorical) Multilevel (Hierarchical) Bayesian Model in R

I have a couple of questions, so I hope it is ok that I ask them here.

Before that, here is some background information on my data:

• Outcome variable (1): categorical, 6 categories, N=168
• Predictor variable (2): both categorical, 3 and 4 categories, N=25
• Control variable (1): categorical, 25 categories, N=25
• There are levels (a hierarchy) in my data: the outcome variable is at level 1 and the predictors and the control variable are at level 2.

Goal: Is there a correlation between the predictor variables and the outcome variable, taking the control variable into account?

For example, 168 students responded to a question that had 6 possible answers (outcome var.). These 168 students are unequally distributed across 25 schools (control var.). Each of these 25 schools has 2 characteristics (predictor var.). Do students' answers correlate with the kind of school they attend, controlling for the school itself?

Questions:

1. As far as I know, there is no frequentist solution to analyzing such a data set. I think it is best to run a bayesian analysis due to the categorical outcome variable. Is this correct or is there a frequentist approach I can/should take?

2. If I take the Bayesian approach (which I am a fan of, but a total novice), would a multinomial multilevel model be the best kind of analysis to run? This is possible with the brms package, correct? (I've stumbled across MCMCglmm quite a bit as well, but I'm not sure it is suitable and it is intimidating for a beginner.)

3. Can anyone point me in the right direction as far as setting up the priors for categorical variables goes? Most of the help I've found online has been for continuous variables and/or is difficult to understand and apply to my research question. I think I understand how to set priors for continuous variables, but when there are categories, are the priors treated as though they are binary?

So something like dunif(0.5,0.5) would mean that anything between 0 and 1 is equally possible, but 1.1 is highly unlikely, correct? And if the prior is dnorm(0.5,0.5), the possibility of getting 0.5 is higher than 0.9, correct?

If, for example, I wanted to set the prior for each category of the outcome variable (e.g. category 1 is less likely than category 2), is this possible and advisable?

Thanks a lot!