Hostname: page-component-5d84bcc8dc-2qh69 Total loading time: 0 Render date: 2026-09-03T23:02:06.259Z Has data issue: false hasContentIssue false
  • English
  • Français

A Scaled Difference Chi-Square Test Statistic for Moment Structure Analysis

Published online by Cambridge University Press:  01 January 2025

Albert Satorra  and
Peter M. Bentler
Albert Satorra*
Affiliation:
Universitat Pompeu Fabra, Barcelona
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Albert Satorra, Universitat Pompeu Fabra, Ramon Trias Fargas, 25-27, 08005 Barcelona, SPAIN. E-Mail: satorra@upf.es
Get access Rights & Permissions [Opens in a new window]

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 M0 implies on a less restricted one M1. If T0 and T1 denote the goodness-of-fit test statistics associated to M0 andM1, respectively, then typically the difference Td=T0−T1 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 M0 andM1. As in the case of the goodness-of-fit test, it is of interest to scale the statistic Td 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 M0 and M1. A Monte Carlo study is provided to illustrate the performance of the competing statistics.

Keywords

moment-structuresgoodness-of-fit testchi-square difference test statisticchi-square distributionnonnormality

Information

Type
Articles
Information
Psychometrika , Volume 66 , Issue 4 , December 2001 , pp. 507 - 514
Copyright
Copyright © 2001 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This research was supported by the Spanish grants PB96-0300 and BEC2000-0983, and USPHS grants DA00017 and DA01070.

References

Bentler, P.M. (1995). EQS structural equations program manual. Encino, CA: Multivariate Software.Google Scholar
Bentler, P.M., & Dudgeon, P. (2001). Covariance structure analysis: Statistical practice, theory, and directions. Annual Review of Psychology, 47, 541–570.Google Scholar
Bentler, P.M., & Yuan, K.-H. (1999). Structural equation modeling with small samples: Test statistics. Multivariate Behavioral Research, 34, 183–199.10.1207/S15327906Mb340203CrossRefGoogle ScholarPubMed
Bollen, K.A. (1989). Structural equations with latent variables. New York, NY: Wiley.10.1002/9781118619179CrossRefGoogle Scholar
Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.10.1111/j.2044-8317.1984.tb00789.xCrossRefGoogle ScholarPubMed
Byrne, B.M., & Campbell, T.L. (1999). Cross-cultural comparisons and the presumption of equivalent measurement and theoretical structure: A look beneath the surface. Journal of Cross-Cultural Psychology, 30, 557–576.10.1177/0022022199030005001CrossRefGoogle Scholar
Chou, C.-P., Bentler, P.M., & Satorra, A. (1991). Scaled test statistics and robust standard errors for nonnormal data in covariance structure analysis: A Monte Carlo study. British Journal of Mathematical and Statistical Psychology, 44, 347–357.10.1111/j.2044-8317.1991.tb00966.xCrossRefGoogle ScholarPubMed
Curran, P.J., West, S.G., & Finch, J.F. (2001). The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis. Psychological Methods, 1, 16–29.10.1037/1082-989X.1.1.16CrossRefGoogle Scholar
Fuller, W.A. (1987). Measurement error models. New York, NY: Wiley.10.1002/9780470316665CrossRefGoogle Scholar
Hu, L., Bentler, P.M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted?. Psychological Bulletin, 112, 351–362.10.1037/0033-2909.112.2.351CrossRefGoogle ScholarPubMed
Jöreskog, K., & Sörbom, D. (1994). LISREL 8 user's reference guide. Mooresville, IN: Scientific Software.Google Scholar
Kano, Y. (1992). Robust statistics for test-of-independence and related structural models. Statistics and Probability Letters, 15, 21–26.10.1016/0167-7152(92)90279-ECrossRefGoogle Scholar
Magnus, J., & Neudecker, H. (1999). Matrix differential calculus with applications in statistics and econometrics rev. ed., New York, NY: Wiley.Google Scholar
Muthén, B. (1993). Goodness of fit test with categorical and other nonnormal variables. In Bollen, K.A., & Long, J.S. (Eds.), Testing structural equation models (pp. 205–234). Newbury Park, CA: Sage Publications.Google Scholar
Rao, C.R. (1973). Linear statistical inference and its applications 2nd. ed., New York, NY: Wiley.10.1002/9780470316436CrossRefGoogle Scholar
Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131–151.10.1007/BF02294453CrossRefGoogle Scholar
Satorra, A. (1992). Asymptotic robust inferences in the analysis of mean and covariance structures. Sociological Methodology, 22, 249–278.10.2307/270998CrossRefGoogle Scholar
Satorra, A. (2000). Scaled and adjusted restricted tests in multisample analysis of moment structures. In Heijmans, D.D.H., Pollock, D.S.G., & Satorra, A. (Eds.), Innovations in multivariate statistical analysis: A Festschrift for Heinz Neudecker (pp. 233–247). Dordrecht, The Netherlands: Kluwer Academic Publishers.10.1007/978-1-4615-4603-0_17CrossRefGoogle Scholar
Satorra, A., & Bentler, P.M. (1986). Some robustness properties of goodness of fit statistics in covariance structure analysis. 1986 ASA Proceedings of the Business and Economic Statistics Section (549–554). Alexandria, VA: American Statistical Association.Google Scholar
Satorra, A., & Bentler, P.M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis. ASA 1988 Proceedings of the Business and Economic Statistics Section (308–313). Alexandria, VA: American Statistical Association.Google Scholar
Satorra, A., & Bentler, P.M. (1988). Scaling corrections for statistics in covariance structure analysis. Los Angeles, CA: University of California.Google Scholar
Satorra, A., & Bentler, P.M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In von Eye, A., & Clogg, C.C. (Eds.), Latent variables analysis: Applications for developmental research (pp. 399–419). Thousand Oaks, CA: Sage.Google Scholar
Yuan, K.-H., & Bentler, P.M. (1997). Mean and covariance structure analysis: Theoretical and practical improvements. Journal of The American Statistical Association, 92, 767–774.10.1080/01621459.1997.10474029CrossRefGoogle Scholar
Yuan, K.-H., & Bentler, P.M. (1998). Normal theory based test statistics in structural equation modelling. British Journal of Mathematical and Statistical Psychology, 51, 289–309.10.1111/j.2044-8317.1998.tb00682.xCrossRefGoogle ScholarPubMed