I have a question on MSEM with random slopes (I analyzed a 1-1-1 mediation model following the examples of Preacher, Zyphur, & Zhang, 2010) regarding model equivalence.
For single-level mediation, there are at least six statistically equivalent models when permuting all paths. (Statistically equivalent insofar as they have the same general fit to the data; Lee & Hershberger, 1990; Stelzl, 1986). When computing the six models as multi-level mediation models with fixed slopes (PPZ example I) I got for all models the same fit indices including AIC and BIC, indicating that these models are statistically equivalent. When computing the "same" six models with random slopes instead (PPZ example J), the information criteria AIC & BIC differ.
Reading several postings, I found that it is not possible to calculate other fit indices, because there is not a single population covariance matrix (see http://www.statmodel.com/discussion/messages/12/7143.html?1302722459). Am I right supposing that for this very same reason the information criteria differ between the models? Moreover, does it follow that MSEM-models with random slopes are not statistically equivalent despite the fact that the "same" models with fixed slopes are?