So, when fitting multilevel models with level 1 predictors, it is customary, and recommended to cluster-mean center these predictors to disentagle between-cluster and within-cluster variance in the estimates (i.e., getting regression coefficient estimates that represent just level 1->level 1 relationships).
However, what if we want standardized predictors so that we get standardized regression coefficients from the model?
I have occasionally seen it suggested (can't find those suggestions now though) that one can first standardize the level 1 predictor across all observations (by subtracting the variable grand mean from each observation and divide by overall SD of the variable, ignoring clusters at this point), and then center the resulting standardized variable.
However, it is unclear to me whether this is OK in the first place and specifically
how is the subsequent centering done - do you subtract the standardized variable's cluster mean from each standardized observation, or do you subtract the raw cluster mean from each standardized observation? Or something else?
if I use this kind of standardized-centered variable in a multilevel regression model, what does the regression coefficient represent? Do I get a standardized coefficient that I can compare to other standardized coefficients from the same model directly? Or even to standardized coefficients from other models?