Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-30T09:25:48.624Z Has data issue: false hasContentIssue false
  • English
  • Français

Structural Equation Modeling with Heavy Tailed Distributions

Published online by Cambridge University Press:  01 January 2025

Ke-Hai Yuan ,
Peter M. Bentler  and
Wai Chan
Ke-Hai Yuan*
Affiliation:
University of Notre Dame
Peter M. Bentler
Affiliation:
University of California, Los Angeles
Wai Chan
Affiliation:
The Chinese University of Hong Kong
*
Correspondence concerning this article should be addressed to Ke-Hal Yuan, Department of Psychology, University of Notre Dame, Notre Dame, IN 46556, USA. Email: kyuan@nd.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

Data in social and behavioral sciences typically possess heavy tails. Structural equation modeling is commonly used in analyzing interrelations among variables of such data. Classical methods for structural equation modeling fit a proposed model to the sample covariance matrix, which can lead to very inefficient parameter estimates. By fitting a structural model to a robust covariance matrix for data with heavy tails, one generally gets more efficient parameter estimates. Because many robust procedures are available, we propose using the empirical efficiency of a set of invariant parameter estimates in identifying an optimal robust procedure. Within the class of elliptical distributions, analytical results show that the robust procedure leading to the most efficient parameter estimates also yields a most powerful test statistic. Examples illustrate the merit of the proposed procedure. The relevance of this procedure to data analysis in a broader context is noted.

Keywords

structural modelkurtosisrobust methodsefficiencypower

Information

Type
Theory and Methods
Information
Psychometrika , Volume 69 , Issue 3 , September 2004 , pp. 421 - 436
Copyright
Copyright © 2004 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

The authors thank the editor, an associate editor and four referees for their constructive comments, which led to an improved version of the paper.

This project was supported by Grant DA01070 from the National Institute on Drug Abuse and a Direct Grant for Research from The Chinese University of Hong Kong.

References

Arminger, G., Stein, P., & Wittenberg, J. (1999). Mixtures of conditional mean- and covariance structure models. Psychometrika, 64, 475–494CrossRefGoogle Scholar
Beran, R., & Srivastava, M.S. (1985). Bootstrap tests and confidence regions for functions of a covariance matrix. Annals of Statistics, 13, 95–115CrossRefGoogle Scholar
Bollen, K.A. (1989). Structural equations with latent variables. New York: WileyCrossRefGoogle Scholar
Bollen, K.A., & Stine, R. (1993). Bootstrapping goodness of fit measures in structural equation models. In Bollen, K.A. and Long, J.S. (Eds.), Testing structural equation models. Newbury Park, CA: SageGoogle Scholar
Browne, M.W. (1984). Asymptotic distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83CrossRefGoogle ScholarPubMed
Campbell, N.A. (1980). Robust procedures in multivariate analysis I: Robust covariance estimation. Applied Statistics, 29, 231–237CrossRefGoogle Scholar
Campbell, N.A. (1982). Robust procedures in multivariate analysis II: Robust canonical variate analysis. Applied Statistics, 31, 1–8CrossRefGoogle Scholar
Davison, A.C., & Hinkley, D.V. (1997). Bootstrap methods and their application. Cambridge: Cambridge University PressCrossRefGoogle Scholar
Devlin, S.J., Gnanadesikan, R., & Kettenring, J.R. (1981). Robust estimation of dispersion matrices and principal components. Journal of the American Statistical Association, 76, 354–362CrossRefGoogle Scholar
Donoho, D., & Huber, P.J. (1983). The notion of breakdown point. In Bickel, P.J., Doksum, K.A., and Hodges, J.L. (Eds.), A Festschrift for Erich L. Lehmann (pp. 157–184). Belmont, CA: WadsworthGoogle Scholar
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Annals of Statistics, 7, 1–26CrossRefGoogle Scholar
Efron, B., & Tibshirani, R.J. (1993). An Introduction to the bootstrap. New York: Chapman & HallCrossRefGoogle Scholar
Green, S.B., Thompson, M.S., & Poirier, J. (1999). Exploratory analysis to improve model fit: Errors due to misspecification and a strategy to reduce their occurrence. Structural Equation Modeling, 6, 113–126CrossRefGoogle Scholar
Hawkins, D.M., & Olive, D.J. (1999). Improved feasible solution algorithms for high breakdown estimation. Computational Statistics & Data Analysis, 30, 1–11CrossRefGoogle Scholar
Hampel, F.R. (1974). The influence curve and its role in robust estimation. Journal of the American Statistical Association, 69, 383–393CrossRefGoogle Scholar
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., & Stahel, W.A. (1986). Robust statistics: The approach based on influence functions. New York: WileyGoogle Scholar
Hu, F., & Hu, J. (2000). A note on breakdown theory for bootstrap methods. Statistics & Probability Letters, 50, 49–53CrossRefGoogle Scholar
Huber, P.J. (1977). Robust covariances. In Gupta, S.S., Moore, D.S. (Eds.), Statistical Decision Theory and Related Topics, Vol. 2 (pp. 165–191). New York: Academic PressCrossRefGoogle Scholar
Kano, Y., Berkane, M., & Bentler, P.M. (1993). Statistical inference based on pseudo-maximum likelihood estimators in elliptical populations. Journal of the American Statistical Association, 88, 135–143CrossRefGoogle Scholar
Lehmann, E.L. (1983). Theory of point estimation. New York: WileyCrossRefGoogle Scholar
Lopuhaä, H.P. (1989). On the relation between S-estimators and M-estimators of multivariate location and covariances. Annals of Statistics, 17, 1662–1683CrossRefGoogle Scholar
Magnus, J.R., & Neudecker, H. (1999). Matrix differential calculus with applications in statistics and econometrics. New York: WileyGoogle Scholar
Mardia, K.V. (1970). Measure of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530CrossRefGoogle Scholar
Maronna, R.A. (1976). Robust M-estimators of multivariate location and scatter. Annals of Statistics, 4, 51–67CrossRefGoogle Scholar
Marshall, L.L. (1999). Effect of men's subtle and overt psychological abuse on low income women. Violence and Victims, 14, 69–88CrossRefGoogle ScholarPubMed
Muirhead, R.J., & Waternaux, C.M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. Biometrika, 67, 31–43CrossRefGoogle Scholar
Newcomb, M.D., & Bentler, P.M. (1988). Consequences of adolescent drug use: Impact on the lives of young adults. Beverly Hills: Sage PublicationsGoogle Scholar
Rousseeuw, P.J., & van Zomeren, B.C. (1990). Unmasking multivariate outliers and leverage points (with discussion). Journal of the American Statistical Association, 85, 633–651CrossRefGoogle Scholar
Ruppert, D. (1992). Computing S estimators for regression and multivariate location/dispersion. Journal of Computational and Graphical Statistics, 1, 253–270Google Scholar
Satorra, A., & Bentler, P.M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In von Eye, A. and Clogg, C.C. (Eds.), Latent variables analysis: Applications for developmental research (pp. 399–419). Thousand Oaks, CA: SageGoogle Scholar
Shapiro, A., & Browne, M.W. (1987). Analysis of covariance structure under elliptical distributions. Journal of the American Statistical Association, 82, 1092–1097CrossRefGoogle Scholar
Singh, K. (1998). Breakdown theory for bootstrap quantiles. Annals of Statistics, 26, 1719–1732CrossRefGoogle Scholar
Sörbom, D. (1989). Model modification. Psychometrika, 54, 371–384CrossRefGoogle Scholar
Stein, J.A., Newcomb, M.D., & Bentler, P.M. (1996). Initiation and maintenance of tobacco smoking: Changing personality correlates in adolescence and young adulthood. Journal of Applied Social Psychology, 26, 160–187CrossRefGoogle Scholar
Stromberg, A.J. (1997). Robust covariance estimates based on resampling. Journal of Statistical Planning and Inference, 57, 321–334CrossRefGoogle Scholar
Tyler, D.E. (1982). Radial estimates and the test for sphericity. Biometrika, 69, 429–436CrossRefGoogle Scholar
Tyler, D.E. (1983). Robustness and efficiency properties of scatter matrices. Biometrika, 70, 411–420CrossRefGoogle Scholar
Weston, R., Marshall, L., & Yuan, K.-H. (1999). Structural equation modeling of social exchange theory, violence and positive behaviors in the relationships of low income women. KY: University of LouisvilleGoogle Scholar
Woodruff, D.L., & Rocke, D.M. (1994). Computable robust estimation of multivariate location and shape in high dimension using compound estimators. Journal of the American Statistical Association, 89, 888–896CrossRefGoogle Scholar
Yuan, K.-H., & Bentler, P.M. (1998). Robust mean and covariance structure analysis. British Journal of Mathematical and Statistical Psychology, 51, 63–88CrossRefGoogle ScholarPubMed
Yuan, K.-H., & Bentler, P.M. (1998). Structural equation modeling with robust covariances. Sociological Methodology, 28, 363–396CrossRefGoogle Scholar
Yuan, K.-H., & Bentler, P.M. (1999). On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions. Statistical Sinica, 9, 831–853Google Scholar
Yuan, K.-H., Chan, W., & Bentler, P.M. (2000). Robust transformation with applications to structural equation modeling. British Journal of Mathematical and Statistical Psychology, 53, 31–50CrossRefGoogle Scholar
Yuan, K.-H., & Hayashi, K. (2003). Standard errors with misspecified covariance structure models. Tentatively accepted by British Journal of Mathematical and Statistical Psychology.Google Scholar
Yuan, K.-H., Marshall, L.L., & Bentler, P.M. (2003). Assessing the effect of model misspecifications on parameter estimates in structural equation models. Sociological Methodology, 33, 241–265CrossRefGoogle Scholar
Yung, Y.F. (1997). Finite mixtures in confirmatory factor-analysis models. Psychometrika, 62, 297–330CrossRefGoogle Scholar
Yung, Y.-F., & Bentler, P.M. (1996). Bootstrapping techniques in analysis of mean and covariance structures. In Marcoulides, G.A. and Schumacker, R.E. (Eds.), Advanced structural equation modeling: Techniques and issues (pp. 195–226). Hillsdale, NJ: Lawrence ErlbaumGoogle Scholar