Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2025-01-04T03:18:13.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Latent Variable Modeling in Heterogeneous Populations

Published online by Cambridge University Press:  01 January 2025

Bengt O. Muthén
Bengt O. Muthén*
Affiliation:
Graduate School of Education, University of California, Los Angeles
*
Requests for reprints should be addressed to Bengt O. Muthrn, Graduate School of Education, University of California, Los Angeles, California, 90024-1521.
Get access Rights & Permissions [Opens in a new window]

Abstract

Common applications of latent variable analysis fail to recognize that data may be obtained from several populations with different sets of parameter values. This article describes the problem and gives an overview of methodology that can address heterogeneity. Artificial examples of mixtures are given, where if the mixture is not recognized, strongly distorted results occur. MIMIC structural modeling is shown to be a useful method for detecting and describing heterogeneity that cannot be handled in regular multiple-group analysis. Other useful methods instead take a random effects approach, describing heterogeneity in terms of random parameter variation across groups. These random effects models connect with emerging methodology for multilevel structural equation modeling of hierarchical data. Examples are drawn from educational achievement testing, psychopathology, and sociology of education. Estimation is carried out by the LISCOMP program.

Keywords

mixturescovariance structuresmultiple-group analysisMIMICLISCOMPrandom parametersmultilevelhierarchical data
Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 4 , December 1989 , pp. 557 - 585
Copyright
Copyright © 1989 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.)

Footnotes

Presidential address delivered at the Psychometric Society meetings in Los Angeles, USA and Leuven, Belgium, July 1989. The research was supported by Grant No. SES-8821668 from the National Science Foundation and by Grant No. OERI-G-86-003 from the Office for Educational Research and Improvement, Department of Education. I thank Leigh Burstein, Mike Hollis, Linda Muthén, and Albert Satorra for helpful discussions and Tammy Tam, Jin-Wen Yang, Suk-Woo Kim, and Lynn Short for computational assistance. Designs were created by Arlette Collier, Rita Ling and Jennifer Edic-Bryant.

References

Aitkin, M., & Longford, N. (1986). Statistical modeling issues in school effectiveness studies. Journal of the Royal Statistical Society, 149, 1–43.CrossRefGoogle Scholar
Bentler, P. M. (1980). Multivariate analysis with latent variables: Causal modeling. Annual Review of Psychology, 31, 19–456.CrossRefGoogle Scholar
Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: Specification and estimation of moment structures. Psychometrika, 48, 493–518.CrossRefGoogle Scholar
Bock, R. D. (1983). The discrete Bayesian. In Wainer, H. & Messick, S. (Eds.), Principals of modern psychological measurement (pp. 103–115). Hillsdale, NJ: Erlbaum.Google Scholar
Bock, R. D. (1989). Multilevel analysis of educational data, San Diego, CA: Academic Press.Google Scholar
Bock, R. D. (1989). Measurement of human variation: A two-stage model. In Bock, R. D. (Eds.), Multilevel analysis of educational data (pp. 319–340). San Diego, CA: Academic Press.Google Scholar
Braun, H., Jones, D., & Rubin, D., Thayer, D. (1983). Empirical Bayes estimation of coefficients in the general linear model from data of deficient rank. Psychometrika, 48, 171–181.CrossRefGoogle Scholar
Browne, M. W. (1982). Covariance structures. In Hawkins, D. M. (Eds.), Topics in applied multivariate analysis, Cambridge, MA: Cambridge University Press.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.CrossRefGoogle ScholarPubMed
Burstein, L. (1980). The analysis of multilevel data in educational research and evaluation. Review of Research in Education, 8, 158–233.Google Scholar
Burstein, L., Kim, K. S., & Delandshere, G. (1988). Multilevel investigation of systematically varying slopes: Issues, alternatives, and consequences. In Bock, R. D. (Eds.), Multilevel analysis of educational data (pp. 235–276). San Diego, CA: Academic Press.Google Scholar
Converse, P. E. (1964). The nature of belief systems in mass publics. In Apter, D. (Eds.), Ideology and discontent, London, England: The Free Press.Google Scholar
Cronbach, L. J. (1976). Research on classrooms and schools: Formulation of questions, design, and analysis. Unpublished manuscript, Stanford University, Stanford Evaluation Consortium, School of Education.Google Scholar
Crosswhite, F. J., Dossey, J. A., Swafford, J. O., McKnight, C. C., & Cooney, T. J. (1985). Second international mathematics study: Summary report for the United States, Champaign, IL: Stipes.Google Scholar
de Leeuw, J. (1985). Path models with random coefficients, Leiden, The Netherlands: University of Leiden, Department of Data Theory.Google Scholar
de Leeuw, J., & Kreft, I. (1986). Random coefficient models for multilevel analysis. Journal of Educational Statistics, 11, 57–85.CrossRefGoogle Scholar
Eaton, W., & Bohrnstedt, G. (1989). Introduction. Latent Variable Models for Dichotomous Outcomes: Analysis of data from the Epidemiological Catchment Area Program. Sociological Methods & Research, 18, 4–18.CrossRefGoogle Scholar
Everitt, B., & Hand, D. J. (1981). Finite Mixture Distributions, New York: Chapman and Hall.CrossRefGoogle Scholar
Gibbons, R., & Bock, R. (1987). Trend in correlated proportions. Psychometrika, 52, 113–124.CrossRefGoogle Scholar
Goldstein, H. I. (1986). Multilevel mixed linear model analysis using iterative generalized least squares. Biometrika, 73, 43–56.CrossRefGoogle Scholar
Goldstein, H. I. (1987). Multilevel Models in Educational and Social Research, London: Oxford University Press.Google Scholar
Goldstein, H., & McDonald, R. P. (1988). A general model for the analysis of multilevel data. Psychometrika, 53, 455–467.CrossRefGoogle Scholar
Gustafsson, J. E. (1988). Hierarchical models of individual differences in cognitive abilities. In Sternberg, R. J. (Eds.), Advances in the psychology of human intelligence, Vol. 4 (pp. 35–71). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Gustafsson, J. E. (in press). Broad and narrow abilities in research on learning and instruction. In Kanfer, R., Ackerman, P. L. & Cudeck, R. (Eds.), Abilities, motivation, and methodology: The Minnesota symposium on learning and individual differences. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Härnquist, K. (1978). Primary mental abilites of collective and individual levels. Journal of Educational Psychology, 70, 706–716.CrossRefGoogle Scholar
Hauser, R. M., & Goldberger, A. S. (1971). The treatment of unobservable variables in path analysis. In Costner, H. L. (Eds.), Sociological Methodology 1971 (pp. 81–177). San Francisco: Jossey-Bass.Google Scholar
Hedeker, D., Gibbons, R. D., & Waternaux, C. (1988, June). Random regression models for longitudinal psychiatric data. Paper presented at the Annual Meeting of the Psychometric Society, Los Angeles.Google Scholar
Hollis, M., & Muthén, B. (1988). Incorporating “Item-specific” information into covariance structure models of political attitudes and policy preferences. Paper prepared for delivery at the 1988 Annual Meeting of the American Political Science Association.Google Scholar
Johnson, N., & Kotz, S. (1972). Distribution in statistics: Continuous multivariate distributions, New York: John Wiley & Sons.Google Scholar
Jöreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409–426.CrossRefGoogle Scholar
Jöreskog, K. G. (1973). A general method for estimating a linear structural equation system. In Goldberger, A. S. & Duncan, O. D. (Eds.), Structural equation models in the social sciences (pp. 85–112). New York: Seminar Press.Google Scholar
Jöreskog, K. G. (1978). Structural analysis of covariance and correlation matrices. Psychometrika, 43, 443–477.CrossRefGoogle Scholar
Jöreskog, K. G., & Goldberger, A. S. (1975). Estimation of a model with multiple indicators and multiple causes of a single latent vaviable. Journal of the American Statistical Association, 70, 351–351.Google Scholar
Keesling, J. W., & Wiley, D. E. (1974, March). Regression models of hierarchical data. Paper presented at the Annual Meeting of the Psychometric Society, Palo Alto, CA.Google Scholar
Lindley, D. V., & Smith, A. F. M. (1972). Bayes estimates for the linear model. Journal of the Royal Statistical Society, 34, 1–41.CrossRefGoogle Scholar
Longford, N. T. (1987). A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested effects. Biometrika, 74, 817–827.CrossRefGoogle Scholar
Longford, N. T. (1988). A quasilikelihood adaption for variance component analysis, Princeton, NJ: Educational Testing Service.Google Scholar
Long, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Maddala, G. S. (1977). Econometrics, New York: McGraw-Hill.Google Scholar
Maddala, G. S. (1983). Limited-dependent and qualitative variables in econometrics, Cambridge, MA: Cambridge University Press.CrossRefGoogle Scholar
Mason, W. M., & Wong, G. Y., Entwistle, B. (1984). Contextual analysis through the multi-level linear model. Sociological Methodology, San Francisco, CA: Jossey-Bass.Google Scholar
McDonald, R. P., & Goldstein, H. (1988, June). Balanced versus unbalanced designs for linear structural relations in two-level data. Paper presented at the Annual meeting of the Psychometric Society, Los Angeles.Google Scholar
Meredith, W. (1964). Notes on factorial invariance. Psychometrika, 29, 177–185.CrossRefGoogle Scholar
Mislevy, R. J. (1975). Estimation of latent group effects. Journal of the American Statistical Association, 80, 993–997.CrossRefGoogle Scholar
Mislevy, R. J. (1987). Exploiting auxiliary information about examinees in the estimation of item parameters. Applied Psychological Measurement, 11, 81–91.CrossRefGoogle Scholar
Mundlak, Y. (1978). Models with variable coefficients: Integration and extension. Annales de'l INSEE, 30–31, 483–509.CrossRefGoogle Scholar
Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551–560.CrossRefGoogle Scholar
Muthén, B. (1979). A structural probit model with latent variables. Journal of the American Statistical Association, 74, 807–811.Google Scholar
Muthén, B. (1983). Latent variable structural equation modeling with categorical data. Journal of Econometrics, 22, 43–65.CrossRefGoogle Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
Muthén, B. (1985, July). Tobit factor analysis. Paper presented at the Fourth European Meeting of the Psychometric Society, Cambridge, England. (Forthcoming in The British Journal of Mathematical and Statistical Psychology.Google Scholar
Muthén, B. (1987). LISCOMP: Analysis of linear structural equations with a comprehensive measurement model, Mooresville, IN: Scientific Software.Google Scholar
Muthén, B. (1988). Some uses of structural equation modeling in validity studies: Extending IRT to external variables. In Wainer, H. & Braun, H. (Eds.), Test Validity (pp. 213–238). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Muthén, B. (1988). Instructionally sensitive psychometrics: Applications to the Second International Mathematics Study, Los Angeles: University of California, Graduate School of Education.Google Scholar
Muthén, B. (1989). Using item-specific instructional information in achievement modeling. Psychometrika, 54, 385–396.CrossRefGoogle Scholar
Muthén, B. (1989). Covariance structure modeling in heterogeneous populations: Mean-adjusted analysis, Los Angeles: University of California, Graduate School of Education.Google Scholar
Muthén, B. (1989). Dichotomous factor analysis of symptom data. Latent Variable Models for Dichotomous Outcomes: Analysis of Data from the Epidemiological Catchment Area Program, 18, 19–65.Google Scholar
Muthén, B. (1989). Multiple-group factor analysis with non-normal continuous variables. British Journal of Mathematical and Statistical Psychology, 42, 55–62.CrossRefGoogle Scholar
Muthén, B. (1989e). Covariance structure analysis of hierarchical data. Paper presented at the American Statistical Association meeting in Washington D.C.Google Scholar
Muthén, B. (1989f, October). Analysis of longitudinal data using latent variable models with varying parameters. Invited paper presented at the University of Southern California Conference on Best Methods for the Analysis of Change, Los Angeles. (Forthcoming as a chapter in a book edited by J. Horn and L. Collins)Google Scholar
Muthén, B. (in press). Moments of the censored and truncated bivariate normal distribution. British Journal of Mathematical and Statistical Psychology.Google Scholar
Muthén, B., Burstein, L., Gustafsson, J.-E., Webb, N., Kim, S-W., & Short, L. (1989). General and specific factors in mathematics achievement data, Los Angeles: University of California, Graduate School of Education.Google Scholar
Muthén, B., & Christoffersson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups. Psychometrika, 46, 485–500.CrossRefGoogle Scholar
Muthén, B., & Jöreskog, (1983). Selectivity problems in quasi-experimental studies. Evaluation Review, 7, 139–173.CrossRefGoogle Scholar
Muthén, B., Kao, Chih-fen, & Burstein, L. (in press). Instructional sensitivity in mathematics achievement test items: Applications of a new IRT-based detection technique. Journal of Educational Measurement.Google Scholar
Muthén, B., & Satorra, A. (1989). Multilevel aspects of varying parameters in structural models. In Bock, (Eds.), Multilevel Analysis of Educational Data (pp. 87–99). San Diego: Academic Press.Google Scholar
Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44, 443–460.CrossRefGoogle Scholar
Olsson, U., Drasgow, F., & Dorans, N. J. (1982). The polyserial correlation coefficient. Psychometrika, 37, 337–347.CrossRefGoogle Scholar
Raudenbush, S., & Bryk, A. (1988). Methodological advances in studying effects of schools and classrooms on student learning. Review of Research in Education, 1988.Google Scholar
Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15, 351–357.CrossRefGoogle Scholar
Rogosa, D. R. (1987). Causal models do not support scientific conclusions: A comment in support of Freedman. Journal of Educational Statistics, 12, 185–195.CrossRefGoogle Scholar
Rogosa, D. R., & Willett, J. B. (1985). Understanding correlates of change by modeling individual differences in growth. Psychometrika, 50, 203–228.CrossRefGoogle Scholar
Rubin, D. B. (1980). Using empirical Bayes techniques in the Law School Validity Studies. Journal of the American Statistical Association, 75, 801–827.CrossRefGoogle Scholar
Rubin, D. B. (1983). Some applications of Bayesian statistics to educational data. The Statistician, 32, 55–68.CrossRefGoogle Scholar
Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50, 83–90.CrossRefGoogle Scholar
Schmidt, W. H. (1969). Covariance structure analysis of the multivariate random effects model. Unpublished doctoral dissertation, University of Chicago.Google Scholar
Schmidt, W., & Wisenbaker, J. (1986). Hierarchical data analysis: An approach based on structural equations (CEPSE, No. 4., Research Series). University of Michigan, Department of Counseling Educational Psychology and Special Education.Google Scholar
Schuman, H., & Presser, S. (1981). Questions and answers in attitude surveys, New York: Academic Press.Google Scholar
Sörbom, D. (1974). A general method for studying differences in factor means and factor structure between groups. British Journal of Mathematical and Statistical Psychology, 27, 229–239.CrossRefGoogle Scholar
Sörbom, D. (1982). Structural equation models with structured means. In Jöreskog, K. G. & Wold, H. (Eds.), Systems under indirect observation: Causality, structure, prediction (pp. 183–195). Amsterdam: North-Holland.Google Scholar
Swamy, P. A. V. B. (1970). Efficient inference in a random coefficient regression model. Econometrica, 311–323.CrossRefGoogle Scholar
Tiao, G. C., & Tan, W. Y. (1965). Bayesian analysis of random-effect models in the analysis of variance. I. Posterior distribution of variance components. Biometrika, 52, 37–53.CrossRefGoogle Scholar
Wong, G. Y., & Mason, W. M. (1985). The hierarchical logistic regression model for multilevel analysis. Journal of the American Statistical Association, 80, 513–524.CrossRefGoogle Scholar
Yamamoto, K. (1988). A model that combines IRT and latent class models. Paper presented at the 1988 American Educational Research Association Annual Meeting.Google Scholar