Effects of Estimation Methods, Number of Indicators per Factor, and Improper Solutions on Structural Equation Modeling Fit Indices

Testing fit is a critical and controversial step in structural equation modeling (SEM), and alternative fit indices have been proposed. To provide guidance, a simulation study was conducted to evaluate the effects of estimation method, number of indicators per factor (p/r ratio), sample size, and loading size on six SEM fit indices: chi-square/degree of freedom ratio (chi(2)/df), Normed Fit Index (NFI), Nonnormed Fit Index (NNFI), Centrality m index, Relative Noncentrality Index (RNI), and Comparative Fit Index (CFI). When improper solutions occurred, the effects of constraining out-of-bounds estimates to a reasonable range (the method used in EQS software) on the fit indices was evaluated. Four levels of sample size (50, 100, 200, and 500), three levels of loading size (0.50, 0.70, and 0.90), five levels of p/r ratio (2, 3, 4, 5, and 6), and two levels of estimation method (Generalized Least Squares [GLS] and Maximum Likelihood [ML]) were examined. The results of this study indicated that: (a) improper solutions occurred frequently when p/r = 2 and sample size was small or loadings were low; (b) GLS produced more improper solutions than ML in general; (c) there was no effect of improper solutions (constrained to some boundary) found for any of the fit indices but NFI (more downward bias occurred for NFI when improper solutions were present); (d) four incremental fit indices (NFI, NNFI, RNI, and CFI) were negatively affected by increasing the p/r ratio; (e) NFI was affected much more seriously than the other three fit indices and should not be used; (f) all fit indices except NNFI were found to be significantly affected by estimation method (less bias occurred for GLS than for ML); and (g) interaction effects between estimation method, p/r ratio, sample size, and loading size also occurred. Recommendations regarding selection of a fit index are made based on the findings.