On nonequivalence of several procedures of structural equation modeling

The normal theory based maximum likelihood procedure is widely used in structural equation modeling. Three alternatives are: the normal theory based generalized least squares, the normal theory based iteratively reweighted least squares, and the asymptotically distribution-free procedure. When data are normally distributed and the model structure is correctly specified, the four procedures are asymptotically equivalent. However, this equivalence is often used when models are not correctly specified. This short paper clarifies conditions under which these procedures are not asymptotically equivalent. Analytical results indicate that, when a model is not correct, two factors contribute to the nonequivalence of the different procedures. One is that the estimated covariance matrices by different procedures are different, the other is that they use different scales to measure the distance between the sample covariance matrix and the estimated covariance matrix. The results are illustrated using real as well as simulated data. Implication of the results to model fit indices is also discussed using the comparative fit index as an example.