Two Equivalent Discrepancy Functions for Maximum Likelihood Estimation

Do Their Test Statistics Follow a Non-Central Chi-Square Distribution under Model Misspecification

Bibliographic Data

ID2330441
AuthorsUlf Olsson (0000-0002-5972-1764), Ulf Henning Olsson, Tron Fo, Tron Foss (Norwegian School of Management BI), Einar Breivik (Norwegian School of Economics and Business Administration)
Year2004
Volume32
Issue4
Pages453-500
Publication date2004-05-01
Open AccessYes
TypeARTICLE
VenueSociological Methods & Research (JOURNAL)
Journal identifiersISSN: 0049-1241 • E-ISSN: 1552-8294
PublisherSAGE Publications Inc (PUBLISHER)
DOI10.1177/0049124103258131
OpenAlexW2067708868
LanguageEN
Citations received11
References cited18

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

Chi-square test · Confidence interval · Empirical distribution function · Empirical likelihood · Likelihood-ratio test · Pearson's chi-squared test · Square (algebra) · Statistic · Statistical hypothesis testing · Statistics · Test statistic · Advanced Statistical Methods and Models · Advanced Statistical Modeling Techniques · Applied Mathematics · Mathematics · Statistical Methods and Applications

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