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Mixtures of Conditional Mean- and Covariance-Structure Models

Published online by Cambridge University Press:  01 January 2025

Gerhard Arminger ,
Petra Stein  and
Jörg Wittenberg
Gerhard Arminger*
Affiliation:
Department of Economics, Bergische Universität
Petra Stein
Affiliation:
Department of Social Sciences, Gerhard Mercator Universität
Jörg Wittenberg
Affiliation:
Department of Economics, Bergische Universität
*
Requests for reprints should be sent to Gerhard Arminger, Department of Economics, Bergische Universität, FB6, D-42097 Wuppertal, GERMANY.
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Abstract

Models and parameters of finite mixtures of multivariate normal densities conditional on regressor variables are specified and estimated. We consider mixtures of multivariate normals where the expected value for each component depends on possibly nonnormal regressor variables. The expected values and covariance matrices of the mixture components are parameterized using conditional mean- and covariance-structures. We discuss the construction of the likelihood function, estimation of the mixture model with regressors using three different EM algorithms, estimation of the asymptotic covariance matrix of parameters and testing for the number of mixture components. In addition to simulation studies, data on food preferences are analyzed.

Keywords

conditional mean- and covariance-structure modelsconditional multivariate normalsEM-algorithmfinite mixturesLISRELminimum-distance-estimation

Information

Type
Original Paper
Information
Psychometrika , Volume 64 , Issue 4 , December 1999 , pp. 475 - 494
Copyright
Copyright © 1999 The Psychometric Society

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Footnotes

The authors are grateful to Donald B. Rubin and Michael E. Sobel for critical reading of a first draft of this paper and to three anonymous reviewers of Psychometrika for their helpful comments and suggestions. The research of the first and the third author was supported by a grant from the Deutsche Forschungsgemeinschaft.

References

Aitkin, M., Anderson, D., & Hinde, J. (1981). Statistical modelling of data on teaching styles. Journal of the Royal Statistical Society, Series A, 144, 419–461.CrossRefGoogle Scholar
Aitkin, M., & Rubin, D.B. (1985). Estimation and hypothesis testing in finite mixture models. Journal of the Royal Statistical Society, Series B, 47, 67–75.CrossRefGoogle Scholar
Arminger, G. (1995). Specification and estimation of mean structures: Regression models. In Arminger, G., Clogg, C.C., & Sobel, M.E. (Eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 77–183). New York: Plenum.CrossRefGoogle Scholar
Arminger, G., Wittenberg, J., & Schepers, A. (1996). MECOSA 3 User Guide, Germany: ADDITIVE GmbH.Google Scholar
Becker, M.P., Yang, I., & Lange, K. (1997). EM Algorithms without Missing Data. Statistical Methods in Medical Research, 6, 37–53.CrossRefGoogle ScholarPubMed
Böhning, D., Dietz, E., Schaub, R., Schlattmann, P., & Lindsay, B.G. (1994). The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family. Annals of the Institute of Statistical Mathematics, 46(2), 373–388.CrossRefGoogle Scholar
Browne, M.W. (1974). Generalized least-squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.Google Scholar
Browne, M.W. (1977). Generalized least-squares estimators in the analysis of covariance structures. In Aigner, D.J., & Goldberger, A.S. (Eds.), Latent variables in socio-economic models (pp. 205–226). Amsterdam: North-Holland.Google Scholar
Browne, M.W., & Arminger, G. (1995). Specification and estimation of mean- and covariance-structure models. In Arminger, G., Clogg, C.C., & Sobel, M.E. (Eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 185–249). New York: Plenum.CrossRefGoogle Scholar
Brunsø, K., & Grunert, K.G. (1993). Identifying food-related life style segments by a cross-culturally valid scaling device. MAPP Conference Paper, The Aarhus School of Business, Department of Marketing, Haslegaardsvej 10, DK-8210 Aarhus 5, Denmark.Google Scholar
Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with Discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.CrossRefGoogle Scholar
DeSarbo, W.S., & Cron, W.L. (1988). A maximum likelihood methodology for clusterwise linear regression. Journal of Classification, 5, 249–282.CrossRefGoogle Scholar
Jedidi, K., Jagpal, H.S., & DeSarbo, W.S. (1997). Finite-mixture structural equation models for response-based segmentation and unobserverd heterogeneity. Marketing Science, 16(1), 39–59.CrossRefGoogle Scholar
Jedidi, K., Ramaswamy, V., DeSarbo, W.S., & Wedel, M. (1996). On estimating finite mixtures of multivariate regression and simultaneous equation models. Structural Equation Modelling, 3, 266–289.CrossRefGoogle Scholar
Jones, P.N., & McLachlan, G.J. (1992). Fitting finite mixture models in a regression context. Australian Journal of Statistics, 32(2), 233–240.CrossRefGoogle Scholar
Jöreskog, K.G., & Sörbom, D. (1988). LISREL 7-A Guide to the Program and Applications, Chicago, IL: SPSS Publications.Google Scholar
Kiefer, N.M. (1978). Discrete parameter variation: Efficient estimation of a switching regression model. Econometrica, 46, 427–434.CrossRefGoogle Scholar
Louis, T.A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B, 44, 226–233.CrossRefGoogle Scholar
McLachlan, G.J. (1987). On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Applied Statistics, 36(3), 318–324.CrossRefGoogle Scholar
McLachlan, G.J., & Basford, K.E. (1988). Mixture models, New York: Marcel Dekker.Google ScholarPubMed
Titterington, D.M., Smith, A.F.M., & Makov, U.E. (1985). Statistical analysis of finite mixture distributions, Chiester: Wiley.Google Scholar
Wolfe, J.H. (1971). A Monte Carlo study of the sampling distribution of the likelihood ratio for mixtures of multinormal distributions, San Diego: U.S. Naval Personnel and Training Research Laboratory.Google Scholar
Yung, Y.F. (1994). Finite mixtures in confirmatory factor-analytic models. Unpublished doctoral dissertation, University of California, Los Angeles, Department of Psychology.Google Scholar
Yung, Y.F. (1997). Finite mixtures in confirmatory factor-analysis models. Psychometrika, 62, 297–330.CrossRefGoogle Scholar