I received the following email:

I am interested in comparing the coefficients of two separate sets of multilevel results (identical models) conducted on two independent groups: male and female (a strategy similar to the z-test for equality of coefficients used in single level analyses). As I understand it, 2 groups are insufficient to constitute a level. Do you have any suggestions for how I might test for significant differences in predictor effects between these two groups (e.g. is there a test analogous to the single level z-test)? Any insight or information you could offer would be greatly appreciated.

My reply: A reasonable way to start is to just make the comparisons that you want, and then compute the standard error of the comparison based on the se’s from the two separate multilevel analyses. Thus, estimate is b.1-b.2, and se is sqrt(se.1^2 + se.2^2). You might be looking at lots of different comparisons, so I’d suggest plotting these (with their uncertainties).

The next step would be to model the two groups together using main effects and interactions, with the main effects and the interactions being allowed to vary by group (with groups as defined in your original multilevel model).