A scaled difference chi-square test statistic for moment structure analysis

Satorra, Albert; Bentler, Peter M.

doi:10.1007/bf02296192

Cited by 5,282Psychometric Methodologies and TestingStatistical Methods and Bayesian InferenceStatistical Distribution Estimation and Applications

A scaled difference chi-square test statistic for moment structure analysis

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Satorra, Albert, Bentler, Peter M.·2001·Psychometrika

DOI 10.1007/bf02296192

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Abstract

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model say M 0 implies on a less restricted one M 1 . If T 0 and T 1 denote the goodness-of-fit test statistics associated to M 0 and M 1 , respectively, then typically the difference T d = T 0 − T 1 is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models M 0 and M 1 . As in the case of the goodness-of-fit test, it is of interest to scale the statistic T d in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of models M 0 and M 1 . A Monte Carlo study is provided to illustrate the performance of the competing statistics.

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Robust fit indices were reported to account for distributional departures.
“Robust fit indices (Satorra & Bentler, 2001) were reported to account for distributional departures”
From Countering dis/misinformation sharing by hindering moral disengagement in adolescence: the role of prosocial conversational modeling (2026) · cites this paper as Satorra & Bentler, 2001

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Authors: Satorra, A., & Bentler, P. M.
Affiliations: Universitat Pompeu Fabra (ES)
University of California, Los Angeles (US)
Corresponding author: Satorra, Albert
ORCID: Satorra, Albert
Bentler, Peter M.
Venue: Psychometrika
Publisher: Cambridge University Press (CUP)
Published: 2001-12
Volume: 66
Issue: 4
Pages: 507-514
ISSN: 0033-3123 (ISSN-L)
1860-0980
DOI: 10.1007/bf02296192
OpenAlex: W3021353408
Cited by: 5,282
References: 0
Citations by year: 2026: 349 · 2025: 328 · 2024: 290 · 2023: 296 · 2022: 301 · 2021: 369 · 2020: 318 · 2019: 331 · 2018: 337 · 2017: 295 · 2016: 294 · 2015: 277 · 2014: 268 · 2013: 269 · 2012: 206
FWCI: 15.68
Type: journal article
Language: en
Open access: green
doi.org
Licence: www.cambridge.org/core/terms (unspecified, 8432 d)
Locations: Psychometrika (publishedVersion)
SSRN Electronic Journal (acceptedVersion)
Repositori digital de la UPF (Universitat Pompeu Fabra) (submittedVersion, cc-by-nc-nd)
RePEc: Research Papers in Economics (submittedVersion)
Topics: Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
Keywords: structural equation modeling, nested model comparison, non-normal data, small sample inference, Monte Carlo simulation

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