The Robustness of Test Statistics to Nonnormality and Specification Error in Confirmatory Factor Analysis

Monte Carlo computer simulations were used to investigate the performance of three χ² test statistics in confirmatory factor analysis (CFA). Normal theory maximum likelihood χ² (ML), Browne's asymptotic distribution free χ² (ADF), and the Satorra-Bentler rescaled χ² (SB) were examined under varying conditions of sample size, model specification, and multivariate distribution. For properly specified models, ML and SB showed no evidence of bias under normal distributions across all sample sizes, whereas ADF was biased at all but the largest sample sizes. ML was increasingly overestimated with increasing nonnormality, but both SB (at all sample sizes) and ADF (only at large sample sizes) showed no evidence of bias. For misspecified models, ML was again inflated with increasing nonnormality, but both SB and ADF were underestimated with increasing nonnormality. It appears that the power of the SB and ADF test statistics to detect a model misspecification is attenuated given nonnormally distributed data.