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

## Author Info

• Albert Satorra

()

• Peter Bentler

## 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 ${\cal M}_0$ implies on a less restricted one ${\cal M}_1$. If $T_0$ and $T_1$ denote the goodness-of-fit test statistics associated to ${\cal M}_0$ and ${\cal M}_1$, 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 ${\cal M}_0$ and ${\cal M}_1$. 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, i.e., nonasymptotic and nonnormal, applications. In a recent paper, Satorra (1999) shows that the difference between two Satorra- Bentler 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 ${\cal M}_0$ and ${\cal M}_1$. A Monte Carlo study is provided to illustrate the performance of the competing statistics.

(This abstract was borrowed from another version of this item.)

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File URL: http://hdl.handle.net/10.1007/BF02296192
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## Bibliographic Info

Article provided by Springer in its journal Psychometrika.

Volume (Year): 66 (2001)
Issue (Month): 4 (December)
Pages: 507-514

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Handle: RePEc:spr:psycho:v:66:y:2001:i:4:p:507-514

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## Related research

Keywords: moment-structures; goodness-of-fit test; chi-square difference test statistic; chi-square distribution; nonnormality;

Other versions of this item:

Find related papers by JEL classification:
• C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
• C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
• C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
• C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
• C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
• C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Longitudinal Data; Spatial Time Series

## References

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1. Albert Satorra, 1999. "Scaled and adjusted restricted tests in multi-sample analysis of moment structures," Economics Working Papers 395, Department of Economics and Business, Universitat Pompeu Fabra.
2. Albert Satorra, 1989. "Alternative test criteria in covariance structure analysis: A unified approach," Psychometrika, Springer, vol. 54(1), pages 131-151, March.
3. Albert Satorra, 1991. "Asymptotic robust inferences in the analysis of mean and covariance structures," Economics Working Papers 3, Department of Economics and Business, Universitat Pompeu Fabra.
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## Citations

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Cited by:
1. Ringle, Christian M. & Götz, Oliver & Wetzels, Martin & Wilson, Bradley, 2009. "On the Use of Formative Measurement Specifications in Structural Equation Modeling: A Monte Carlo Simulation Study to Compare Covariance-Based and Partial Least Squares Model Estimation Methodologies," Research Memoranda 014, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization.
2. Piia Seppälä & Saija Mauno & Taru Feldt & Jari Hakanen & Ulla Kinnunen & Asko Tolvanen & Wilmar Schaufeli, 2009. "The Construct Validity of the Utrecht Work Engagement Scale: Multisample and Longitudinal Evidence," Journal of Happiness Studies, Springer, vol. 10(4), pages 459-481, August.
3. Alberto Maydeu, 2003. "Random Intercept Item Factor Analysis," Working Papers Economia wp03-02, Instituto de Empresa, Area of Economic Environment.
4. Juan M. Ramon-Jeronimo & M. Concepcion Alvarez-Dardet Espejo & David Naranjo-Gil, 2008. "Effects of management control systems on commitment in inter-organizational relationships," Working Papers 08.03, Universidad Pablo de Olavide, Department of Business Administration.
5. Pei-Chen Wu, 2010. "Differential Functioning of the Chinese Version of Beck Depression Inventory-II in Adolescent Gender Groups: Use of a Multiple-Group Mean and Covariance Structure Model," Social Indicators Research, Springer, vol. 96(3), pages 535-550, May.
6. Slocum, Lee Ann, 2010. "General strain theory and the development of stressors and substance use over time: An empirical examination," Journal of Criminal Justice, Elsevier, vol. 38(6), pages 1100-1112, November.
7. Jule Specht & Boris Egloff & Stefan C. Schmukle, 2012. "Everything under Control?: The Effects of Age, Gender, and Education on Trajectories of Perceived Control in a Nationally Representative German Sample," SOEPpapers on Multidisciplinary Panel Data Research 445, DIW Berlin, The German Socio-Economic Panel (SOEP).
8. Monica Pedrazza & Elena Trifiletti & Sabrina Berlanda & Gian Antonio Di Bernardo, 2013. "Self-Efficacy in Social Work: Development and Initial Validation of the Self-Efficacy Scale for Social Workers," Social Sciences, MDPI, Open Access Journal, vol. 2(3), pages 191-207, September.
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10. Peter Halpin & Conor Dolan & Raoul Grasman & Paul Boeck, 2011. "On the Relation Between the Linear Factor Model and the Latent Profile Model," Psychometrika, Springer, vol. 76(4), pages 564-583, October.
11. Deutskens,Elisabeth & Ruyter,Ko,de & Wetzels,Martin, 2005. "An Assessment of Measurement Invariance between Online and Mail Surveys," Research Memoranda 003, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization.
12. Ana B. Casado & Francisco J. Mas & Hans Kasper, 2006. "Explaining Satisfaction In Double Deviation Scenarios: The Effects Of Anger And Distributive Justice," Working Papers. Serie EC 2006-09, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
13. Barry Forer & Bruno Zumbo, 2011. "Validation of Multilevel Constructs: Validation Methods and Empirical Findings for the EDI," Social Indicators Research, Springer, vol. 103(2), pages 231-265, September.
14. Ana Villar & César Camisón & Montserrat Boronat, 2007. "Technical Strategic Alliances And Performance: The Mediating Effect Of Knowledge ¿Based Competencies," Working Papers. Serie EC 2007-11, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
15. Spiekermann, Sarah & Strobel, Martin & Temme, Dirk, 2002. "Drivers and impediments of consumer online information search: Self-controlled versus agent-based search in a high involvement context," SFB 373 Discussion Papers 2002,37, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
16. Alberto Maydeu, 2005. "A Multidimensional Unfolding Latent Trait Model For Binary Data," Working Papers Economia wp05-11, Instituto de Empresa, Area of Economic Environment.
17. repec:ese:iserwp:2013-05 is not listed on IDEAS
18. Ute Kunzmann & David Richter & Stefan C. Schmukle, 2013. "Stability and Change in Affective Experience across the Adult Life-Span: Analyses with a National Sample from Germany," SOEPpapers on Multidisciplinary Panel Data Research 584, DIW Berlin, The German Socio-Economic Panel (SOEP).

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