I am quite impressed by the possibilities of multilevel SEM as describe by Preacher and colleagues (“A General Multilevel SEM Framework for Assessing Multilevel Mediation”). But I have trouble with the interpretation the effects of intrinsically within-level variables that are specified at the between-level. Let me explain my point with a simple example.
Lets say that I want to examine the effects of student gender on academic achievement for data where students are clustered within schools. I can specify a within model ‘gender --> achievement’ and quite straightforward interpret the estimates: if I have a significant effect, that means that individual gender background of a student is related to his/her achievement within schools.
But at the same time, I can also specify a between model ‘gender --> achievement’ (at the school level). Here, I have trouble understanding and interpreting the estimates provided by Mplus. What does the estimates mean when gender (which is an intrinsically individual-level variable) has an effect on academic achievement at the school level? Does it ever make sense to specify the effect of a within-level variable on a within-level outcome at the between-level?
In addition, while I understand that a within-level variable has both a within and a between component, I have trouble with the interpretation of the between-effect. For instance, in what way is the interpretation different than the 'usual' aggregate effect?