The Model-Size Effect on Traditional and Modified Tests of Covariance Structures

Abstract

According to Kenny and McCoach (2003) Kenny, D. A. and McCoach, D. B. 2003. Effect of the number of variables on measures of fit in structural equation modeling. Structural Equation Modeling, 10: 333–351. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], chi-square tests of structural equation models produce inflated Type I error rates when the degrees of freedom increase. So far, the amount of this bias in large models has not been quantified. In a Monte Carlo study of confirmatory factor models with a range of 48 to 960 degrees of freedom it was found that the traditional maximum likelihood ratio statistic, T ML , overestimates nominal Type I error rates up to 70% under conditions of multivariate normality. Some alternative statistics for the correction of model-size effects were also investigated: the scaled Satorra–Bentler statistic, T SC ; the adjusted Satorra–Bentler statistic, T AD (Satorra & Bentler, 1988 Satorra, A. and Bentler, P. M. 1988. Scaling corrections for statistics in covariance structure analysis, Los Angeles: University of California, Department of Psychology. (UCLA Statistics Series, No. 2) [Google Scholar], 1994 Satorra, A. and Bentler, P. M. 1994. “Corrections to test statistics and standard errors in covariance structure analysis”. In Latent variable analysis: Applications for developmental research, Edited by: von Eye, A. and Clogg, C. C. 399–419. Thousand Oaks, CA: Sage. [Google Scholar]); corresponding Bartlett corrections, T MLb , T SCb , and T ADb (Bartlett, 1950 Bartlett, M. S. 1950. Tests of significance in factor analysis. British Journal of Psychology (Statistical Section), 3: 77–85. [Crossref], [Web of Science ®] , [Google Scholar]); and corresponding Swain corrections, T MLs , T SCs , and T ADs (Swain, 1975 Swain, A. J. 1975. Analysis of parametric structures for variance matrices, Australia: University of Adelaide. Unpublished doctoral dissertation [Google Scholar]). The empirical findings indicate that the model test statistic T MLs should be applied when large structural equation models are analyzed and the observed variables have (approximately) a multivariate normal distribution.