The effect of latent and error non-normality on corrections to the test statistic in structural equation modeling

In structural equation modeling, several corrections to the likelihood-ratio model test statistic have been developed to counter the effects of non-normal data. Previous robustness studies investigating the performance of these corrections typically induced non-normality in the indicator variables. However, non-normality in the indicators can originate from non-normal errors or non-normal latent factors. We conducted a Monte Carlo simulation to analyze the effect of non-normality in factors and errors on six different test statistics based on maximum likelihood estimation by evaluating the effect on empirical rejection rates and derived indices (RMSEA and CFI) for different degrees of non-normality and sample sizes. We considered the uncorrected likelihood-ratio model test statistic and the Satorra–Bentler scaled test statistic with Bartlett correction, as well as the mean and variance adjusted test statistic, a scale-shifted approach, a third moment-adjusted test statistic, and an approach drawing inferences from the relevant asymptotic chi-square mixture distribution. The results indicate that the values of the uncorrected test statistic—compared to values under normality—are associated with a severely inflated type I error rate when latent variables are non-normal, but virtually no differences occur when errors are non-normal. Although no general pattern regarding the source of non-normality for all analyzed measures of fit can be derived, the Satorra–Bentler scaled test statistic with Bartlett correction performed satisfactorily across conditions.