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

Abstract

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 ML approximately 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 ML rejects the correct model too often when p is not too small. Various corrections to T ML have been proposed, but they are mostly heuristic. Following the principle of the Bartlett correction, this paper proposes an empirical approach to correct T ML so 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 ML as reported in the literature, and they perform well.

The dark core of personality.

“…Moreover, as the models comprised a very large number of observed variables, in turn leading to inflated test-statistics (Herzog, Boomsma, & Reinecke, 2007;Moshagen, 2012), we additionally applied the correction proposed by Yuan, Tian, and Yanagihara (2015), yielding a test-statistic corrected for both the effects of non-normality and model-size (corresponding to…”

More efficient parameter estimates for factor analysis of ordinal variables by ridge generalized least squares

“…Further work on correcting Type I errors of these statistics is needed before studying their power properties. In this direction, Yuan, Tian, and Yanagihara () recently proposed a technique, termed empirical modelling, to correct the likelihood ratio statistic T ML with normally distributed data, where the multiplier n = N –1 in the composition of T ML is replaced by a formula calibrated using Monte Carlo simulation. The same idea can be used to correct the rescaled and adjusted statistics.…”

Small Sample Size Solutions

“…Variations of this correction exist, but the one most studied ; see also Fouladi, 2000;Savalei, 2010) is a simplified correction proposed by Bartlett (1950) in the context of exploratory factor analysis. A more general correction is called the Swain correction (Swain, 1975), and even more corrections were described by Yuan, Tian, and Yanagihara (2015). Herzog, Boomsma, and Reinecke (2007) and Herzog and Boomsma (2009) compared these corrections and concluded that the Swain correction worked best; however, they only looked at complete and normal data.…”

“…Herzog, Boomsma, and Reinecke (2007) and Herzog and Boomsma (2009) compared these corrections and concluded that the Swain correction worked best; however, they only looked at complete and normal data. Shi, Lee, and Terry (2018) also compared the various corrections in more realistic settings and concluded that the so-called "empirically" corrected test statistic proposed by Yuan et al (2015) generally yielded the best performance-particularly when fitting large structural equation models with many observed variables. Still, they warn that when the number of variables (P) in a model increases, the sample size (n) also needs to increase in order to control Type I error.…”