I wonder what happens in Mplus if a predictor is assigned to the within-level (i.e. within = X), but in the dataset X varies both on within- and between levels? I was under the impression that only the within-level variation in X was used, but then I got uncertain when discussing with an author of an article I had read who stated that this predictor would "control for" variation at the between-level anyway. I know that this is probably true in multilevel software where you don't explicitly assign predictors to a certain level, but since Mplus uses this explicit assignment I thought that a within-level variable couldn't affect outcome at the between-level.
Thank you very much for enlightening me about this.
Within=x is used when x is not decomposed into latent within and between components such as with a random slope for x. So that's the traditional multilevel situation of other software. You have to create a cluster mean of x if you want to use it on between. So, if you say within=x in Mplus then the "whole" x (the observed x) is used on within.
Ok, just to be sure I get it right: When the "whole" x is used, and there happen to be variation not just on the within- but also on the between level, then that variation in x on the between level will affect the dependent variable on the between level even though the variable is stated as a within level predictor?
Within=x means that no between-level variance parameter for x is estimated; it is zero. So that's just like traditional multilevel (software) where the parameters of the marginal distribution of x are not estimated (on within or between).
So if you want across-cluster variation in x to play a role on between you need to form a cluster-level version of x, e.g. using the Mplus option Cluster_mean(x).
My question comes from reading a study about organization effects on individual patient outcomes, in which the authors had used the pre-test as covariate on the within level (i.e. patients within organizations). My comment to the authors of the article on this was that this wouldn't control for average differences among organizations in baseline level of distress, since no between-level coefficient for the baseline level was estimated. The authors responded like this:
"In their text on Hierarchical Linear Models, Raudenbush and Bryk (2002) describe how grandmean centering level-1 covariates adjusts the level 2 means for the characteristics of the individuals in the organizations (pp. 111-112). On p. 142 they explain that under grandmean centering of level-1 covariates “the estimate of the effect of Wj [that is, the level 2 predictor, in this case organizational culture/climate] will be adjusted for differences between organizations in the mean of X, the level-1 explanatory variable …"
This seemed to me like an authoritative and credible answer to my question, but now that I read your latest response I get uncertain again. Can you help me clear up my confusion, please?
Page 142 of R&B proposes grand-mean centering of x, which you can do in your Mplus analysis. But that doesn't mean that a between-level x variable is included on Between (level-2). That page also starts with the assumption that "there is no compositional effect", that is, no between-level (contextual) effect of x.
So the author reply is correct as far as it goes, but your critique may be warranted if there is a compositional/contextual effect.