Two Equivalent Discrepancy Functions for Maximum Likelihood Estimation: Do Their Test Statistics Follow a Non-Central Chi-Square Distribution under Model Misspecification?

Over the years several discrepancy functions have been introduced both in the literature and in the software of Structural Equation Modeling (SEM). The test statistics for the discrepancy functions associated with Maximum Likelihood (ML), Generalized Least Squares (GLS), and Normal Theory Weighted Least Squares (NWLS) are all asymptotically equivalent. These test statistics are all approximately distributed as central chi-square under correct model specification and if the observed variables are multivariate normally distributed. However, it is known that the distribution of these test statistics will not approximate a central Chi-square distribution for models containing specification error, but is more likely to follow a non-central Chi-square distribution (Browne 1984). This study investigates the empirical distributions of the ML and NWLS discrepancy functions. The study includes 13 different factor models with different types and degrees of specification error. It is found, except for small samples, that the empirical distribution of the ML-test statistic outperforms the empirical distribution of the NWLS-test statistic in terms of approximation to the theoretical non-central Chi-square distribution. Furthermore, in some cases, it turned out that the non-central Chi-square approximation was not appropriate even for models that contained minor and moderate degrees of specification error.