Correcting Too Much or Too Little? The Performance of Three Chi-Square Corrections

This simulation study investigates the performance of three test statistics, T1, T2, and T3, used to evaluate structural equation model fit under non normal data conditions. T1 is the well-known mean-adjusted statistic of Satorra and Bentler. T2 is the mean-and-variance adjusted statistic of Sattertwaithe type where the degrees of freedom is manipulated. T3 is a recently proposed version of T2 that does not manipulate degrees of freedom. Discrepancies between these statistics and their nominal chi-square distribution in terms of errors of Type I and Type II are investigated. All statistics are shown to be sensitive to increasing kurtosis in the data, with Type I error rates often far off the nominal level. Under excess kurtosis true models are generally over-rejected by T1 and under-rejected by T2 and T3, which have similar performance in all conditions. Under misspecification there is a loss of power with increasing kurtosis, especially for T2 and T3. The coefficient of variation of the nonzero eigenvalues of a certain matrix is shown to be a reliable indicator for the adequacy of these statistics.