I am attempting to run a multilevel CFA with 5-point ordinal data. I first ran the model specifying the outcomes as continuous and I was able to obtain estimates of the within and between residual variances. When I specify the data as categorical I can only get between residuals (after fixing the first two thresholds). I cannot obtain within residuals. Can you please clarify why this is happening? Are the within residuals fixed to a particular value, do I have identification issues, or is it something unique to the software?
With categorical outcomes, residual variances are not free parameters to be estimated but are either standardized to one (like in probit regression), or the full variance of the latent response variable is standardized to one. This is an identification matter. On the Between level, however, the residuals concern continuous variables (the random intercepts), so residual variances are parameters that can be estimated.