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

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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.

### Key words

moment-structures goodness-of-fit test chi-square difference test statistic chi-square distribution nonnormality## Preview

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### References

- Bentler, P.M. (1995).
*EQS structural equations program manual*. Encino, CA: Multivariate Software.Google Scholar - Bentler, P.M., & Dudgeon, P. (1996). 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.CrossRefGoogle Scholar - Bollen, K.A. (1989).
*Structural equations with latent variables*. New York, NY: Wiley.Google 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.PubMedGoogle Scholar - 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.Google 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.PubMedGoogle Scholar - Curran, P.J., West, S.G., & Finch, J.F. (1996). The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis.
*Psychological Methods, 1*, 16–29.Google Scholar - Fuller, W.A. (1987).
*Measurement error models*. New York, NY: Wiley.Google Scholar - Hu, L., Bentler, P.M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted?
*Psychological Bulletin, 112*, 351–362.CrossRefPubMedGoogle Scholar - 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.CrossRefGoogle 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 K.A. Bollen & J.S. Long (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.Google Scholar - Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach.
*Psychometrika, 54*, 131–151.CrossRefGoogle Scholar - Satorra, A. (1992). Asymptotic robust inferences in the analysis of mean and covariance structures.
*Sociological Methodology, 22*, 249–278.Google Scholar - Satorra, A. (2000). Scaled and adjusted restricted tests in multisample analysis of moment structures. In D.D.H. Heijmans, D.S.G. Pollock, & A. Satorra (Eds.),
*Innovations in multivariate statistical analysis: A Festschrift for Heinz Neudecker*(pp. 233–247). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google 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. (1988a). 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. (1988b).
*Scaling corrections for statistics in covariance structure analysis*(UCLA Statistics Series #2). 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 A. von Eye & C.C. Clogg (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.Google 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.PubMedGoogle Scholar