Abstract
Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes.
Similar content being viewed by others
References
Ali, M.M., & Joarder, A.H. (1991). Distribution of the correlation coefficient for the class of bivariate elliptical models.Canadian Journal of Statistics, 19, 447–452.
Anderson, T W., & Fang, K.-T. (1987). Cochran's theorem for elliptically contoured distributions.Sankhy = a A, 49, 305–315.
Bentler, P.M., & Yuan, K.-H. (1999). Structural equation modeling with small samples: Test statistics.Multivariate Behavioral Research, 34, 181–197.
Berkane, M., Oden, K., & Bentler, P.M. (1997). Geodesic estimation in elliptical distributions.Journal of Multivariate Analysis, 63, 35–46.
Bishop, Y.M.M., Fienberg, S.E., & Holland, P. W. (1975).Discrete multivariate analysis: Theory and practice. Cambridge: MIT Press.
Boomsma, A., & Hoogland, J.J. (2001). The robustness of LISREL modeling revisited. In R. Cudeck, S. du Toit, & D. S örbom (Eds.),Structural equation modeling: Present and future (pp. 139–168). Lincolnwood, IL: Scientific Software International.
Browne, M.W. (1984). Asymptotic distribution-free methods for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 37, 62–83.
Browne, M.W., & Shapiro, A. (1987). Adjustments for kurtosis in factor analysis with elliptically distributed errors.Journal of the Royal Statistical Society B, 49, 346–352.
Cheong, Y.F., Fotiu, R.P., & Raudenbush, S.W. (2001). Efficiency and robustness of alternative estimators for two- and three-level models: The case of NAEP.Journal of Educational and Behavioral Statistics, 26, 411–429.
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.
Devlin, S.J., Gnandesikan, R., & Kettenring, J.R. (1976). Some multivariate applications of elliptical distributions. In S. Ikeda (Ed.),Essays in Probability and Statistics (pp. 365–93). Tokyo: Shinko Tsusho.
du Toit, S., & du Toit, M. (in press). Multilevel structural equation modeling. In J. de Leeuw & I. Kreft (Eds.),Handbook of quantitative multilevel analysis. New York: Kluwer.
Fang, K.-T., Kotz, S., & Ng., K.W. (1990).Symmetric multivariate and related distributions. London: Chapman & Hall.
Fouladi, R.T. (2000). Performance of modified test statistics in covariance and correlation structure analysis under conditions of multivariate nonnormality.Structural Equation Modeling, 7, 356–410.
Goldstein, H. (1995).Multilevel statistical models, 2nd edition. London: Edward Arnold.
Goldstein, H., & McDonald, R.P. (1988). A general model for the analysis of multilevel data.Psychometrika, 53, 435–467.
Gupta, A.K., & Varga, T. (1993).Elliptically contoured models in statistics. Dordrecht: Kluwer Academic.
Hayakawa, T. (1987). Normalizing and variance stabilizing transformations of multivariate statistics under an elliptical population.Annals of the Institute of Statistical Mathematics, 39, 299–306.
Heck, R.H., & Thomas, S. L. (2000).An introduction of multilevel modeling techniques. Mahwah, NJ: Erlbaum.
Hoogland, J.J. (1999).The robustness of estimation methods for covariance structure analysis. Unpublished Ph.D. dissertation, Rijksuniversiteit Groningen.
Hox, J.J. (2002).Multilevel analysis: Techniques and applications. Mahwah, NJ: Erlbaum.
Kano, Y. (1992). Robust statistics for test-of-independence and related structural models.Statistics & Probability Letters, 15, 21–26.
Kano, Y. (1994). Consistency property of elliptical probability density functions.Journal of Multivariate Analysis, 51, 343–350.
Kano, Y., Berkane, M., & Bentler, P.M. (1990). Covariance structure analysis with heterogeneous kurtosis parameters.Biometrika, 77, 575–585.
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, 135–143.
Kreft, I., & de Leeuw, J. (1998).Introducing multilevel modeling. London: Sage.
Lee, S.-Y. (1990). Multilevel analysis of structural equation models.Biometrika, 77, 763–772.
Lee, S.Y., & Poon, W.Y. (1998). Analysis of two-level structural equation models via EM type algorithms.Statistica Sinica, 8, 749–766.
Liang, J., & Bentler, P.M. (2004). An EM algorithm for fitting two-level structural equation models.Psychometrika, 69, 101–122.
Longford, N.T. (1987). A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects.Biometrika, 74, 817–827.
Longford, N.T. (1993). Regression analysis of multilevel data with measurement error.British Journal of Mathematical and Statistical Psychology, 46, 301–311.
Longford, N.T., & Muth én, B.O. (1992). Factor analysis for clustered observations.Psychometrika, 57, 581–597.
McDonald, R.P., & Goldstein, H. (1989). Balanced versus unbalanced designs for linear structural relations in two-level data.British Journal of Mathematical and Statistical Psychology, 42, 215–232.
Magnus, J.R., & Neudecker, H. (1999).Matrix differential calculus with applications in statistics and econometrics, revised edition. New York: Wiley.
Muirhead, R.J. (1982).Aspects of multivariate statistical theory. New York: Wiley.
Muirhead, R.J., & Waternaux, C.M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations.Biometrika, 67, 31–43.
Muth én, B. (1994). Multilevel covariance structure analysis.Sociological Methods & Research, 22, 376–398.
Muth én, B. (1997). Latent variable modeling of longitudinal and multilevel data. In A. Raftery (ed.),Sociological methodology 1997 (pp. 453–480). Boston: Blackwell Publishers.
Muth én, B., & Satorra, A. (1995). Complex sample data in structural equation modeling. In P. V. Marsden (E.),Sociological methodology 1995 (pp. 267–316). Cambridge, MA: Blackwell Publishers.
Nevitt, J. (2000).Evaluating small sample approaches for model test statistics in structural equation modeling. Unpublished Ph.D. dissertation, University of Maryland.
Poon, W.-Y., & Lee, S.-Y. (1994). A distribution free approach for analysis of two-level structural equation model.Computational Statistics & Data Analysis, 17, 265–275.
Purkayastha, S., & Srivastava, M.S. (1995). Asymptotic distributions of some test criteria for the covariance matrix in elliptical distributions under local alternatives.Journal of Multivariate Analysis, 55, 165–186.
Raudenbush, S.W. (1995). Maximum likelihood estimation for unbalanced multilevel covariance structure models via the EM algorithm.British Journal of Mathematical and Statistical Psychology, 48, 359–370.
Raudenbush, S.W., & Bryk, A.S. (2002).Hierarchical linear models, 2nd edition. Newbury Park: Sage.
Satorra, A. (2002). Asymptotic robustness in multiple group linear-latent variable models.Econometric Theory, 18, 297–312.
Satorra, A., & Bentler, P.M. (1986). Some robustness properties of goodness of fit statistics in covariance structure analysis.American Statistical Association 1986 proceedings of Business and Economics Sections, 549–554, American Statistical Association.
Satorra, A., & Bentler, P.M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis.American Statistical Association 1988 proceedings of Business and Economics Sections, 308–313, American Statistical Association.
Satorra, A., & Bentler, P.M. (1990). Model conditions for asymptotic robustness in the analysis of linear relations.Computational Statistics & Data Analysis, 10, 235–249.
Satorra, A., & Bentler, P.M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. C. Clogg (Es.),Latent variables analysis: Applications for developmental research (pp. 399–419). Thousand Oaks, CA: Sage.
Shapiro, A., & Browne, M. (1987). Analysis of covariance structures under elliptical distributions.Journal of the American Statistical Association, 82, 1092–1097.
Snijders, T., & Bosker, R. (1999).Multilevel analysis: An introduction to basic and advanced multilevel modeling. Thousand Oaks, CA: Sage.
Steyn, H.S. (1993). On the problem of more than one kurtosis parameter in multivariate analysis.Journal of Multivariate Analysis, 44, 1–22.
Steyn, H.S. (1996). The distribution of the covariance matrix for a subset of elliptical distributions with extension to two kurtosis parameters.Journal of Multivariate Analysis, 58, 96–106.
Tyler, D.E. (1982). Radial estimates and the test for sphericity.Biometrika, 69, 429–36.
Tyler, D.E. (1983). Robustness and efficiency properties of scatter matrices.Biometrika, 70, 411–420.
Wakaki, H. (1997). Asymptotic expansion of the joint distribution of sample mean vector and sample covariance matrix from an elliptical population.Hiroshima Mathematics Journal, 27, 295–305.
Yuan, K.-H., & Bentler, P.M. (1999). On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions.Statistica Sinica, 9, 831–853.
Yuan, K.-H., & Bentler, P.M. (2000). Inferences on correlation coefficients in some classes of nonnormal distributions.Journal of Multivariate Analysis, 72, 230–248.
Yuan, K.-H., & Bentler, P.M. (2002). On normal theory based inference for multilevel models with distributional violations.Psychometrika, 67, 539–561.
Yuan, K.-H., & Bentler, P.M. (2003). Eight test statistics for multilevel structural equation models.Computational Statistics & Data Analysis, 44, 89–107.
Yuan, K.-H., & Bentler, P.M. (in press). Asymptotic robustness of the normal theory likelihood ratio statistic for two-level covariance structure models.Journal of Multivariate Analysis.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors thank an associate editor and three referees for their constructive comments, which led to an improved version of the paper.
This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of Notre Dame faculty research grant.
Rights and permissions
About this article
Cite this article
Yuan, KH., Bentler, P.M. On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions. Psychometrika 69, 437–457 (2004). https://doi.org/10.1007/BF02295645
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02295645