Suppose data are three-level (e.g., students within classes with schools) and there is slope variation at level-2. If the data are analyzed using twolevel complex, will correct standard errors be obtained? That is, does twolevel complex adjust standard errors for intercept and slope variation over the level-3 units and for covariance of these two random effects ?
If the standard errors are correct, do you have a reference that addresses this point?
The justification for type=twolevel complex; is the same as that for type=complex. One can think of the level 2 data as one long observed vector (long2wide) conversion so that a twolevel complex model is rewritten as a single level complex model. So the answer is yes. There is no paper specific to type=twolevel complex, but if you need more info on type=complex http://statmodel.com/resrchpap.shtml http://statmodel.com/download/webnotes/mplusnote72.pdf You can also easily conduct your own simulations to verify this within Mplus.
BTW, in this story, the standard errors is never the problem (the MLR estimator SE are just the first part of the ML standard error - it just stops earlier in the formulas due to not having independence between units from the same higher level). Usually one has to worry about consistency. Once the estimates are consistent the MLR-SE will be correct. In the standard situation you describe above the estimates are consistent.