Multilevel models with group-level predictors

Kari Lock Morgan writes:

I’m writing now though with a multilevel modeling question that has been nagging me for quite some time now. In your book with Jennifer Hill, you include a group-level predictor (for example, 12.15 on page 266), but then end up fitting this as an individual-level predictor with lmer. How can this be okay? It seems as if lmer can’t really still be fitting the model specified in 12.15? In particular, I’m worried about analyzing a cluster randomized experiment where the treatment is applied at the cluster level and outcomes are at the individual level – intuitively, of course it should matter that the treatment was applied at the cluster level, not the individual level, and therefore somehow this should enter into how the model is fit? However, I can’t seem to grasp how lmer would know this, unless it is implicitly looking at the covariates to see if they vary within groups or not, which I’m guessing it’s not? In your book you act as if fitting the model with county-level uranium as an individual predictor is the same as fitting it as a group-level predictor, which makes me think perhaps I am missing something obvious?

My reply: It indeed is annoying that lmer (and, for that matter, stan_lmer in its current incarnation) only allows individual-level predictors, so that any group-level predictors need to be expanded to the individual level (for example, u_full <- u[group]). But from the standpoint of fitting the statistical model, it doesn't matter. Regarding the question, how does the model "know" that, in this case, u_full is actually an expanded group-level predictor: The answer is that it "figures it out" based on the dependence between u_full and the error terms. It all works out.

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