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polychotomous response data

Can anyone suggest what I might do to analyze data where:

- response has more than two values (nominal)
- observations are grouped by subject
- predictors are both categorical and continuous
- some predictors are within-subject, some are between-subject

If it wasn't for the within-subject predictors, I thought this was
going to be a compositional data analysis problem, so I was looking
into the "compositions" library (which I found confusing).

However, treating each subject's data as a single composition, while
it makes sense, is not going to enable the analysis of within-subject
predictors.

Is there a way to use lme4 or another library to extend GLMM to a
multi-category response?

Sorry for the generic question,
Thanks,
Daniel