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

## Author Info

• Albert Satorra

()

• Peter M. 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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
File Function: Whole Paper

## Bibliographic Info

Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 412.

as in new window
Length:
Date of revision:
Handle: RePEc:upf:upfgen:412

Contact details of provider:
Web page: http://www.econ.upf.edu/

## Related research

Keywords: Moment-structures; goodness-of-fit test; chi-square difference test statisitc; chi-square distribution; non-normality;

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; Spatio-temporal Models

This paper has been announced in the following NEP Reports:

## References

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
as in new window
1. Albert Satorra, 1989. "Alternative test criteria in covariance structure analysis: A unified approach," Psychometrika, Springer, vol. 54(1), pages 131-151, March.
2. 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.
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.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as in new window

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," MPRA Paper 15390, University Library of Munich, Germany.
2. Robert Jennrich & Albert Satorra, 2013. "Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis," Psychometrika, Springer, vol. 78(3), pages 545-552, July.
3. 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.
4. 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.
5. 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.
6. 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).
7. 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).
8. 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.
9. 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).
10. Bruno Damásio & Juliana Pacico & Michele Poletto & Sílvia Koller, 2013. "Refinement and Psychometric Properties of the Eight-Item Brazilian Positive and Negative Affective Schedule for Children (PANAS-C8)," Journal of Happiness Studies, Springer, vol. 14(4), pages 1363-1378, August.
11. 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.
12. 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.
13. repec:ese:iserwp:2013-05 is not listed on IDEAS
14. 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).
15. Deutskens,Elisabeth & Ruyter,Ko,de & Wetzels,Martin, 2005. "An Assessment of Measurement Invariance between Online and Mail Surveys," Research Memorandum 003, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
16. Alberto Maydeu, 2003. "Random Intercept Item Factor Analysis," Working Papers Economia wp03-02, Instituto de Empresa, Area of Economic Environment.
17. 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.
18. Alberto Maydeu, 2005. "A Multidimensional Unfolding Latent Trait Model For Binary Data," Working Papers Economia wp05-11, Instituto de Empresa, Area of Economic Environment.
19. 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.
20. Longinos Marin & Salvador Ruiz & Alicia Rubio, 2009. "The Role of Identity Salience in the Effects of Corporate Social Responsibility on Consumer Behavior," Journal of Business Ethics, Springer, vol. 84(1), pages 65-78, January.
21. Albert Satorra & Peter Bentler, 2010. "Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic," Psychometrika, Springer, vol. 75(2), pages 243-248, June.
22. Yi-Chen Lee & Yi-Cheng Lin & Chin-Lan Huang & Barbara Fredrickson, 2013. "The Construct and Measurement of Peace of Mind," Journal of Happiness Studies, Springer, vol. 14(2), pages 571-590, April.
23. Fariborz Damanpour & Carlos Devece & Chao Chen & Vijay Pothukuchi, 2012. "Organizational culture and partner interaction in the management of international joint ventures in India," Asia Pacific Journal of Management, Springer, vol. 29(2), pages 453-478, June.

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

## Corrections

When requesting a correction, please mention this item's handle: RePEc:upf:upfgen:412. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ().

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.