Empirical Correction to the Likelihood Ratio Statistic for Structural Equation Modeling with Many Variables

Survey data typically contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. The most widely used statistic for evaluating the adequacy of a SEM model is T_ML, a slight modification to the likelihood ratio statistic. Under normality assumption, T_MLapproximately follows a chi-square distribution when the number of observations (N) is large and the number of items or variables (p) is small. However, in practice, p can be rather large while N is always limited due to not having enough participants. Even with a relatively large N, empirical results show that T_MLrejects the correct model too often when p is not too small. Various corrections to T_MLhave been proposed, but they are mostly heuristic. Following the principle of the Bartlett correction, this paper proposes an empirical approach to correct T_MLso that the mean of the resulting statistic approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics follow the nominal chi-square distribution much more closely than previously proposed corrections to T_ML, and they control type I errors reasonably well whenever N≥max(50,2p). The formulations of the empirically corrected statistics are further used to predict type I errors of T_MLas reported in the literature, and they perform well.