# Modelling the effect of a 2 by 4 mixed design on a three-level nominal dependent variable

A colleague just asked me this question:

### Context:

• 2 groups of participants (between subjects)
• 4 contexts (within subjects))
• each participant provided a response in each of the four contexts
• there were three categorically distinct response options (lets call them $A$, $B$, and $C$)

### Question

• What is an appropriate statistical model for analysing the effect of group and context on response?
• What would you tell someone who analysed this data as a 2 by 4 by 3 log-linear model? (e.g., problems with assuming independence of observations)
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Depending on your colleague's modeling goals, another approach might be latent class regression, which estimates class probabilities and class-conditional responses probabilities for k latent classes. If you expect strong clustering in subjects' responses, this might be a particularly nice approach because you get regression estimates for each of $k$ fuzzy classes, which might have meaningful psychological labels. Identifiability is an issue here because of the large number of parameters. See poLCA in R and the PDF write-up here.
drm in R is another package which is supposed to be able to model clustered categorical responses, but I have not tried it.