Hello – what family should one use if your outcome variable is the average of 3, 5 point likert scale items?
The decision to treat the average as meaningful implies that outcomes of [3, 3, 3], [1,3,5], [3,1,5], [4,2,3] etc are all in some sense equivalent. The implicit assumption underlying this decision is that the ordinal outcome isn’t just ordinal, but actually is metric. So I would think about starting with continuous families, and only worrying about the discretization (there are only so many possible averages) if it seems important after some initial model fitting. An appropriate choice of continuous family depends substantially on the data that came out. Particularly if there are no responses of [1,1,1] (average = 1) and no responses of [5,5,5] (average = 5), I might consider scaling the response to be between 0 and 1 (by subtracting 1 and then dividing by 4), and then treating the outcome as beta-distributed.