Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

By A. Satorra, Peter M. Bentler. Psychometrika, 2010

1,754 citations recorded by the source corpus.

AbstractA scaled difference test statistic $\tilde{T}{}_{d}$ that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic $\tilde{T}_{d}$ is asymptotically equivalent to the scaled difference test statistic $\bar{T}_{d}$ introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic $\tilde{T}_{d}$ 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 $\bar{T}_{d}$ that avoids negative chi-square values.

Subfields: Computer Networks and Communications, Food Science, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/73ff98d6a95bc5a0042a4d6247872490214164b0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

In Food Science

In Statistics and Probability

Citations in the corpus

Expectation, Motivation, and Attitude: A Tourist Behavioral Model (2010, methodology): “Fortunately, Satorra and Bentler (2008) proposed a new method to handle this situation, and this method was used for testing the discriminant validity between the construct of expectation and knowledge.”
Need for recovery after emotional labor: Differential effects of daily deep and surface acting (2018, methodology): “The fit of the two models to the data was compared with the Satorra‐Bentler scaled chi‐square difference test (Satorra & Bentler, 2010) that corrects for scaling issues to better approximate chi‐square under non‐normality.”
Testing Measurement Invariance in Longitudinal Data with Ordered-Categorical Measures (2016, positive): “With ML estimation, the likelihood ratio test (Bentler & Bonett, 1980) or its robust version (Satorra & Bentler, 2001, 2010) may be used for model comparison.”
Principles and Practice of Scaled Difference Chi-Square Testing (2012, methodology): “For this reason, Satorra and Bentler (2010) recently proposed an improved scaling correction procedure that precludes negative differences in chi-square values and produces results identical to those obtained when using Satorra’s (2000) complex formula.”
Are Robust Standard Errors the Best Approach for Interval Estimation With Nonnormal Data in Structural Equation Modeling? (2018, methodology): “Under nonnormal data, it is common to use a variation of the scaled difference test (Satorra, 2000; Satorra & Bentler, 2001, 2010), TL;r ¼ N ðF̂ML;0 F̂ML;1Þ=ĉ (9) where ĉ is an estimate of the scaling constant.”
Estimating and interpreting latent variable interactions: A tutorial for applying the latent moderated structural equations method (2014, methodology): “…that this uncorrected difference test, originally proposed by Klein & Moosbrugger (2000), performs better than the robust difference test (Satorra & Bentler, 2001) or the ‘‘strictly positive’’ difference value (Satorra & Bentler, 2010), which have been applied in past studies using LMS models.”
Psychosocial safety climate (PSC) and enacted PSC for workplace bullying and psychological health problem reduction (2017, methodology): “As several of the variables were highly positively skewed, as recommended in a review by Finney and DiStefano’s (2008), we used a maximum likelihood parameter estimator that corrects for non-normality and produces a Satorra-Bentler (S-C) chi-square (χ2) (Satorra & Bentler, 2010).”