Small-Sample Robust Estimators of Noncentrality-Based and Incremental Model Fit

Abstract

Traditional estimators of fit measures based on the noncentral chi–square distribution (root mean square error of approximation [RMSEA], Steiger's γ, etc.) tend to overreject acceptable models when the sample size is small. To handle this problem, it is proposed to employ Bartlett's (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], Yuan's (2005) Yuan, K.-H. 2005. Fit indices versus test statistics. Multivariate Behavioral Research, 40: 115–148. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], or Swain's (1975) Swain, A. J. 1975. Analysis of parametric structures for variance matrices., Australia: Department of Statistics, University of Adelaide. Unpublished doctoral dissertation [Google Scholar] correction of the maximum likelihood chi–square statistic for the estimation of noncentrality–based fit measures. In a Monte Carlo study, it is shown that Swain's correction especially produces reliable estimates and confidence intervals for different degrees of model misspecification (RMSEA range: 0.000–0.096) and sample sizes (50, 75, 100, 150, 200). In the second part of the article, the study is extended to incremental fit indexes (Tucker–Lewis Index, Comparative Fit Index, etc.). For their small–sample robust estimation, use of Swain's correction is recommended only for the target model, not for the independence model. The Swain–corrected estimators only require a ratio of sample size to estimated parameters of about 2:1 (sometimes even less) and are thus strongly recommended for applied research. R software is provided for convenient use.