In simulation studies (including one I'm just finishing), estimators that use diagonal weight matrices, such as WLSMV, seem to work very well in terms of providing unbiased estimates. Essentially, estimators that use a diagonal weight matrix make the implicit assumption that the off-diagonal elements of the full weight matrix, such as that used in WLS are non-informative. My question is: why does this work? Are the off-diagonal elements simply so small that they don't make much difference in estimation? Are these off-diagonal elements typically equal to each other so that they don't result in any differential weighting? Both?
I've been puzzling over this for some time and looking at various full weight matrices trying to see a pattern in the off-diagonal elements, but am coming up empty. Any hints?
It's a good question - I don't really know. My inroad to this is that already ULS with W = I does quite well, and WLSMV probably does not do much better than that in terms of point estimates (although it is nice to give different weights to thresholds and correlations).
ULS works well because its simple r_ij - rho-hat_ij sums of squares difference does the essential job of making the estimates consistent. Adding diagonal weights is probably a small improvement, and adding off-diagonal weights probably even smaller. It would be interesting to be able to formalize how these additions make for a "tighter fit". Also, the off-diagonal elements may also be harder to get good estimates off.
Thanks, Bengt. I feel better knowing that at least I'm not missing something completely obvious. I agree that it would be nice to formalize how these elements adjust the fit. Unfortunately I probably don't have the mathematical background to work it out, but one of my colleagues may. Anyway, we are going to work on it.
I have a follow-up question. Is it possible to get the ACM either printed in the output or as an external file? From the manual it looks as though this isn't possible, but I thought I'd check anyway.
You mean the weight matrix, right? We currently don't print that, but you can analyze a model with all correlated items using WITH and request Tech3 for the estimated covariance matrix of those estimated parameters. That is the weight matrix, apart from possibly some scaling with the sample size.