I am not sure if this make sense, but I think the model in example 9.7 has random intercepts for the items and I only want random intercepts for the the factor/theta.
Even if the random intercepts for the items are necessary, I would like an intraclass correlation for the the factor, a mean and variance for the factor and if possible, cluster level estimates of the factor/theta. Is it possible to get this? Thank you.
See slides 28-31 of my recent UConn keynote address which you find on our website. I think the model you are after is the one shown on slide 28. As you see, it involves modifying UG ex 9.7 by setting the loadings equal across levels and fixing the between-level residual variances for the indicators at zero.
I could get what I am after by outputing the factor scores from the single level IRT to a dataset and using them as a dependent variable in SAS Proc Mixed. But my understanding is that this is would have problems because the factor scores are shrunken estimates.
I just want to use the factor/theta as a dependent variable in a 2 or 3 level model if the icc indicates that I should.
Would setting the loadings equal at both levels accomplish this? Will it output an icc for the factor?