This paper has a funny history. I’d read an article by Jan de Leeuw (see here) that was pretty critical of Bayesian multilevel modeling, and I had the thought of writing a paper with Jan where we lay out where we agree and disagree on the topic. The idea would be to give the reader some idea of our overlap, which presumably would represent some safe zone, falling between Jan’s complete skepticism and my naive faith. I told Jan I’d write a draft of an article with my perspective, then I could send to him and he could add his part. So I wrote my half, but then when I sent it to him, he said he actually agreed with what I wrote, so I should submit it as is. So I did. Perhaps I had internalized his critical view while writing the article.
Anyway, here’s the abstract:
Multilevel (hierarchical) modeling is a generalization of linear and generalized linear modeling in which regression coefficients are themselves given a model, whose parameters are also estimated from data. We illustrate the strengths and limitations of multilevel modeling through an example of the prediction of home radon levels in U.S. counties. The multilevel model is highly effective for predictions at both levels of the model, but could easily be misinterpreted for causal inference.
and here’s the article.
Andrew,
An instructive note that I enjoyed reading. I must be missing something, though. The equation after (1) describes G as an n x J matrix of dummies, when I should think it would be an n x (J-1). Having J dummies sets up perfect collinearity with the intercept, gamma_0 {f 1}. Or that not an intercept term?
Bruce
Bruce,
No, I think it's OK. In that equation, G is multiplied by u, which summarizes it as a one-dimensional predictor, hence no collinearity (unless, of course, u is collinear with x).
Hi Andrew –
Do you have a reference for Jan's original paper?
thanks,
Scott
Scott,
See here.
Is there another problem in the causal inference of these models, sometimes? when the x is correlated with county level error, you include a centered covariate, xbar, for each county. This will work if you care about the causal effect of xbar on the outcome, but when you care about the causal effect of x on the outcome (commonly the case), it seems like the coefficient for x becomes more difficult to interpret.