Weight matrix in WLS(s) and GLS
Message/Author
 Dorothee Durpoix posted on Wednesday, February 04, 2009 - 8:09 pm
Hello,

I have to admit I'm quite lost between the different terminologies I've encountered in my different readings regarding estimators for categorical data.
It seems that weighted least squares (WLS) estimator actually encompasses somewhat different methods (I found a Christopherson's (1975) WLS, Browne's (1984) WLS, besides Muthén's (1984) WLS).
Browne in his 2006 book "Confirmatory Factor Analysis for Applied Research" states that the weight matrix in WLS "is based on estimates of the variances and covariances of each element of S [observed covariance matrix], and fourth-order moments based on multivariate kurtosis".

Now, Muthen (1984) WLS is also called Generalized Least Square (GLS). Browne writes that the weight matrix in GLS is only the inverse of S. However, in Mplus both WLS and GLS are present.

Could you please help to understand the subtleties between these different methods and what weight matrix Mplus uses in WLS, GLS, WLSM and WLSMV for categorical data? I would really appreciate any insight.

Cheers,
Dorothee
 Bengt O. Muthen posted on Thursday, February 05, 2009 - 10:52 am
There is consistency here in the following way. WLS can use any weight matrix, but to be called GLS the weight matrix has to be an estimate of the asymptotic covariance matrix of the statistics you are analyzing. You should make a distinction between continuous and categorical outcomes, and for continous outcomes also distinguish between normally and non-normally distributed outcomes. With Browne's inverse of S weight matrix for GLS, he is considering continuous outcomes under normality (or a Wishart distribution for S) in which case the asympototics leads to using S inverse in the weight matrix. Fourth-order moments (so not only S) are needed when you don't have normality for these continuous outcomes. With the categorical outcomes of Christoffersson and Muthen (1984), the weight matrix is also derived as the asymptotic covariance matrix (so "GLS" like). Hope that helps.
 Bengt O. Muthen posted on Thursday, February 05, 2009 - 10:53 am

Muthén, B., du Toit, S.H.C. & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes.

which is on our web site (Papers, SEM).
 Dorothee Durpoix posted on Sunday, February 08, 2009 - 1:39 pm
Thanks a lot professor Muthén; it does help!