Skip to main content
SpringerLink
Log in

Latent variable modeling in heterogeneous populations

  • Published:
Psychometrika Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aitkin, M., & Longford, N. (1986). Statistical modeling issues in school effectiveness studies.Journal of the Royal Statistical Society, Series A149, 1–43.

    Google Scholar 

  • Bentler, P. M. (1980). Multivariate analysis with latent variables: Causal modeling.Annual Review of Psychology, 31, 19–456.

    Article  Google Scholar 

  • Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: Specification and estimation of moment structures.Psychometrika, 48, 493–518.

    Google Scholar 

  • Bock, R. D. (1983). The discrete Bayesian. In H. Wainer & S. Messick (Eds.),Principals of modern psychological measurement (pp. 103–115). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Bock, R. D. (1989a).Multilevel analysis of educational data. San Diego, CA: Academic Press.

    Google Scholar 

  • Bock, R. D. (1989b). Measurement of human variation: A two-stage model. In R. D. Bock (Ed.),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.

    Article  Google Scholar 

  • Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.)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.

    PubMed  Google Scholar 

  • 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 R. D. Bock (Ed.),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 D. Apter (Ed.),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.

  • 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.

    Google 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.

    Google Scholar 

  • Everitt, B., & Hand, D. J. (1981).Finite Mixture Distributions. New York: Chapman and Hall.

    Google Scholar 

  • Gibbons, R., & Bock, R. (1987). Trend in correlated proportions.Psychometrika, 52, 113–124.

    Article  Google Scholar 

  • Goldstein, H. I. (1986). Multilevel mixed linear model analysis using iterative generalized least squares.Biometrika, 73, 43–56.

    Google 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.

    Article  Google Scholar 

  • Gustafsson, J. E. (1988). Hierarchical models of individual differences in cognitive abilities. In R. J. Sternberg (Ed.),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 R. Kanfer, P. L. Ackerman, & R. Cudeck (Eds.),Abilities, motivation, and methodology: The Minnesota symposium on learning and individual differences. Hillsdale, NJ: Lawrence Erlbaum.

  • Härnquist, K. (1978). Primary mental abilites of collective and individual levels.Journal of Educational Psychology, 70, 706–716.

    Google Scholar 

  • Hauser, R. M., & Goldberger, A. S. (1971). The treatment of unobservable variables in path analysis. In H. L. Costner (Ed.),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.

  • 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.

  • 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.

    Google Scholar 

  • Jöreskog, K. G. (1973). A general method for estimating a linear structural equation system. In A. S. Goldberger & O. D. Duncan (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.

    Article  Google 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.

    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.

  • Lindley, D. V., & Smith, A. F. M. (1972). Bayes estimates for the linear model.Journal of the Royal Statistical Society, Series B,34, 1–41.

    Google Scholar 

  • Longford, N. T. (1987). A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested effects.Biometrika, 74, 817–827.

    Google 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.

    Google 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.

  • Meredith, W. (1964). Notes on factorial invariance.Psychometrika, 29, 177–185.

    Google Scholar 

  • Mislevy, R. J. (1975). Estimation of latent group effects.Journal of the American Statistical Association, 80, 993–997.

    Google Scholar 

  • Mislevy, R. J. (1987). Exploiting auxiliary information about examinees in the estimation of item parameters.Applied Psychological Measurement, 11, 81–91.

    Google Scholar 

  • Mundlak, Y. (1978). Models with variable coefficients: Integration and extension.Annales de'l INSEE, 30–31, 483–509.

    Google Scholar 

  • Muthén, B. (1978). Contributions to factor analysis of dichotomous variables.Psychometrika, 43, 551–560.

    Article  Google 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.

    Google Scholar 

  • Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators.Psychometrika, 49, 115–132.

    Article  Google Scholar 

  • Muthén, B. (1985, July).Tobit factor analysis. Paper presented at the Fourth European Meeting of the Psychometric Society, Cambridge, England. (Forthcoming inThe British Journal of Mathematical and Statistical Psychology.

  • Muthén, B. (1987).LISCOMP: Analysis of linear structural equations with a comprehensive measurement model [User's Guide]. Mooresville, IN: Scientific Software.

    Google Scholar 

  • Muthén, B. (1988a). Some uses of structural equation modeling in validity studies: Extending IRT to external variables. In H. Wainer & H. Braun (Eds.),Test Validity (pp. 213–238). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Muthén, B. (1988b).Instructionally sensitive psychometrics: Applications to the Second International Mathematics Study. Unpublished manuscript, University of California, Graduate School of Education, Los Angeles.

    Google Scholar 

  • Muthén, B. (1989a). Using item-specific instructional information in achievement modeling.Psychometrika, 54, 385–396.

    Article  Google Scholar 

  • Muthén, B. (1989b).Covariance structure modeling in heterogeneous populations: Mean-adjusted analysis. In preparation, University of California, Graduate School of Education, Los Angeles.

    Google Scholar 

  • Muthén, B. (1989c). Dichotomous factor analysis of symptom data. In Eaton and Bohrnstedt (Eds.), Latent Variable Models for Dichotomous Outcomes: Analysis of Data from the Epidemiological Catchment Area Program.Sociological Methods & Reserch, 18, 19–65.

    Google Scholar 

  • Muthén, B. (1989d). Multiple-group factor analysis with non-normal continuous variables.British Journal of Mathematical and Statistical Psychology, 42, 55–62.

    Google Scholar 

  • Muthén, B. (1989e).Covariance structure analysis of hierarchical data. Paper presented at the American Statistical Association meeting in Washington D.C.

  • 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)

  • Muthén, B. (in press). Moments of the censored and truncated bivariate normal distribution.British Journal of Mathematical and Statistical Psychology.

  • Muthén, B., Burstein, L., Gustafsson, J.-E., Webb, N., Kim, S-W., Short, L. (1989).General and specific factors in mathematics achievement data. In preparation, University of California, Graduate School of Education, Los Angeles.

    Google Scholar 

  • Muthén, B., & Christoffersson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups.Psychometrika, 46, 485–500.

    Article  Google Scholar 

  • Muthén, B., & Jöreskog. (1983). Selectivity problems in quasi-experimental studies.Evaluation Review, 7, 139–173.

    Google 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.

  • Muthén, B., & Satorra, A. (1989). Multilevel aspects of varying parameters in structural models. In Bock(Ed.),Multilevel Analysis of Educational Data (pp. 87–99). San Diego: Academic Press. (Invited paper for the conference onMultilevel analysis of educational data, Princeton, NJ, April 1987)

    Google Scholar 

  • Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation coefficient.Psychometrika, 44, 443–460.

    Article  Google Scholar 

  • Olsson, U., Drasgow, F., & Dorans, N. J. (1982). The polyserial correlation coefficient.Psychometrika, 37, 337–347.

    Google Scholar 

  • Raudenbush, S., & Bryk, A. (1988). Methodological advances in studying effects of schools and classrooms on student learning.Review of Research in Education, 1988.

  • Robinson, W. S. (1950). Ecological correlations and the behavior of individuals.American Sociological Review, 15, 351–357.

    Google 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.

    Google Scholar 

  • Rogosa, D. R., & Willett, J. B. (1985). Understanding correlates of change by modeling individual differences in growth.Psychometrika, 50, 203–228.

    Article  Google Scholar 

  • Rubin, D. B. (1980). Using empirical Bayes techniques in the Law School Validity Studies.Journal of the American Statistical Association, 75, 801–827.

    Google Scholar 

  • Rubin, D. B. (1983). Some applications of Bayesian statistics to educational data.The Statistician, 32, 55–68.

    Google Scholar 

  • Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis.Psychometrika, 50, 83–90.

    Article  Google Scholar 

  • Schmidt, W. H. (1969).Covariance structure analysis of the multivariate random effects model. Unpublished doctoral dissertation, University of Chicago.

  • 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.

  • 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.

    Google Scholar 

  • Sörbom, D. (1982). Structural equation models with structured means. In K. G. Jöreskog & H. Wold (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.

  • 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.

    Google 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.

    Google Scholar 

  • Yamamoto, K. (1988).A model that combines IRT and latent class models. Paper presented at the 1988 American Educational Research Association Annual Meeting.

Download references

Author information

Authors and Affiliations

  1. Graduate School of Education, University of California, 90024-1521, Los Angeles, California

    Bengt O. Muthén

Authors
  1. Bengt O. Muthén
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Muthén, B.O. Latent variable modeling in heterogeneous populations. Psychometrika 54, 557–585 (1989). https://doi.org/10.1007/BF02296397

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02296397

Key words

Search

Navigation