Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

Abstract

A scaled difference test statistic that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (2001). The statistic is asymptotically equivalent to the scaled difference test statistic T̄d introduced in Satorra (2000), which requires more involved computations beyond standard output of SEM software. The test statistic has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic T̄d that avoids negative chi-square values.

Pairwise Likelihood Ratio Tests and Model Selection Criteria for Structural Equation Models with Ordinal Variables

“…Asparouhov & Muthén (2006) show that this statistic works well for categorical data too. The same statistic, but only adjusted in mean, has also been discussed by Satorra & Bentler (2001); Asparouhov & Muthén (2006); Satorra & Bentler (2010). However, it is well known that mean-and-variance adjusted chi-squared statistics perform better in smaller sample sizes and converge faster to their asymptotic properties than the corresponding mean-adjusted ones.…”

“…This renders FIML computationally infeasible when the number of ordinal variables is large. As a result, two-and three-stage limited-information least squares (3S-LS) estimation and testing theory have been proposed in the literature (Jöreskog, 1990(Jöreskog, , 1994Jöreskog & Sörbom, 1996;Lee et al, 1990bLee et al, , 1992Muthén, 1984;Satorra, 2000;Satorra & Bentler, 2010, 1988Asparouhov & Muthén, 2006, 2010 and implemented in software such as LISREL (Jöreskog & Sörbom, 1996), Mplus , EQS (Bentler, 2006), and the R package lavaan (Rosseel, 2012;Rosseel et al, 2012). Bayesian estimation methods of estimation, testing and model selection have also been developed (see e.g.…”

The relationship of self-compassion and depression: Cross-lagged panel analyses in depressed patients after outpatient therapy

“…To compare nested models, we calculated differences in fit according to recommendations by Satorra and Bentler (2010) for models with only continuous variables or using the Bayesian Information Criterion (BIC) for models with binary and continuous variables. Table 1 shows the means, standard deviations, internal consistencies, and correlations of the measures used at all time points of assessment.…”

Exploring the Relations between Regret, Self-agency, and the Tendency to Repair Using Experimental Methods and Structural Equation Modeling

“…If the fit of the constrained model is statistically lower than that of the unconstrained model (in which these parameters are estimated separately and thus are allowed to differ), one can conclude that the relations are statistically different. There was a trend toward a better fit for the unconstrained model, χ D (1) = 3.29, p = .0696 (scaled difference test; Satorra, 2000;Satorra & Bentler, 2010), suggesting that disappointment was a better predictor of the tendency to repair than regret.…”