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Linear Structural Equations with Latent Variables

Published online by Cambridge University Press:  01 January 2025

P. M. Bentler  and
David G. Weeks
P. M. Bentler*
Affiliation:
University of California, Los Angeles
David G. Weeks
Affiliation:
Washington University
*
Requests for reprints should be sent to P. M. Bentler, Department of Psychology, University of California, Los Angeles, California 90024.
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Abstract

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.

Keywords

structural equationssimultaneous equationslinear relationscovariance structureslatent variableserrors in variablesfactor analysisstructural models
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 3 , September 1980 , pp. 289 - 308
Copyright
Copyright © 1980 The Psychometric Society

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Footnotes

This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017) and a research grant (DA01070) from the U. S. Public Health Service.

References

Reference Notes

Nel, D. G. A review of matrix differentiation in statistics, 1978, South Africa: Department of Mathematical Statistics, University of the Orange Free State.Google Scholar
Weeks, D. G. & Bentler, P. M. Estimation of three-mode factor analysis via reparameterization. Unpublished manuscript, Washington University School of Medicine, 1979.Google Scholar

References

Aigner, D. J. & Goldberger, A. S. Latent variables in socio-economic models, 1977, Amsterdam: North-Holland.Google Scholar
Bentler, P. M. Multistructure statistical model applied to factor analysis. Multivariate Behavioral Research, 1976, 11, 3–25.CrossRefGoogle ScholarPubMed
Bentler, P. M. Linear systems with multiple levels and types of latent variables. In Jöreskog, K. G. & Wold, H. (Eds.), Systems under indirect observation: Causality, structure, prediction. In press, 1980. (a)Google Scholar
Bentler, P. M. Multivariate analysis with latent variables: Causal modeling. Annual Review of Psychology, 1980, 31, 419–456 (b)CrossRefGoogle Scholar
Bentler, P. M. & Huba, G. J. Simple minitheories of love. Journal of Personality and Social Psychology, 1979, 37, 124–130.CrossRefGoogle Scholar
Bentler, P. M. & Lee, S. Y. Matrix derivatives with chain rule and rules for simple, Hadamard and Kronecker products. Journal of Mathematical Psychology, 1978, 17, 255–262.CrossRefGoogle Scholar
Bentler, P. M. & Lee, S. Y. A statistical development of three-mode factor analysis. British Journal of Mathematical and Statistical Psychology, 1979, 32, 87–104.CrossRefGoogle Scholar
Bentler, P. M. & Peeler, W. H. Models of female orgasm. Archives of Sexual Behavior, 1979, 8, 405–423.CrossRefGoogle ScholarPubMed
Bentler, P. M. & Speckart, G. Models of attitude-behavior relations. Psychological Review, 1979, 86, 452–464.CrossRefGoogle Scholar
Bentler, P. M., Weeks, D. G. Restricted multidimensional scaling models. Journal of Mathematical Psychology, 1978, 17, 138–151.CrossRefGoogle Scholar
Bentler, P. M. & Weeks, D. G. Interrelations among models for the analysis of moment structures. Multivariate Behavioral Research, 1979, 14, 169–185.CrossRefGoogle ScholarPubMed
Bentler, P. M. & Weeks, D. G. Multivariate analysis with latent variables. In Krishnaiah, P. R. and Kanal, L. (Eds.), Handbook of statistics, (Vol. 2), 1980, Amsterdam: North-Holland. In pressGoogle Scholar
Bentler, P. M. & Woodward, J. A. A Head Start re-evaluation: Positive effects are not yet demonstrable. Evaluation Quarterly, 1978, 2, 493–510.CrossRefGoogle Scholar
Bielby, W. T. & Hauser, R. M. Structural equation models. Annual Review of Sociology, 1977, 3, 137–161.CrossRefGoogle Scholar
Browne, M. W. Generalized least squares estimates in the analysis of covariance structures. South African Statistical Journal, 1974, 8, 1–24.Google Scholar
Bunch, J. R. & Rose, D. J. Sparse matrix computations, 1976, New York: Academic.Google Scholar
Burt, R. S. Interpretational confounding of unobserved variables in structural equation models. Sociological Methods and Research, 1976, 5, 3–22.CrossRefGoogle Scholar
Chechile, R. Likelihood and posterior identification: Implications for mathematical psychology. British Journal of Mathematical and Statistical Psychology, 1977, 30, 177–184.CrossRefGoogle Scholar
Dagenais, M. G. The computation of FIML estimates as iterative generalized least squares estimates in linear and nonlinear simultaneous equation models. Econometrica, 1978, 46, 1351–1362.CrossRefGoogle Scholar
Deistler, M. & Seifert, H. G. Identifiability and consistent estimability in econometric models. Econometrica, 1978, 46, 969–980.CrossRefGoogle Scholar
Gabrielsen, A. Consistency and identifiability. Journal of Econometrics, 1978, 8, 261–263.CrossRefGoogle Scholar
Geraci, V. J. Identification of simultaneous equation models with measurement error. Journal of Econometrics, 1976, 4, 262–283.CrossRefGoogle Scholar
Geraci, V. J. Estimation of simultaneous equation models with measurement error. Econometrica, 1977, 45, 1243–1255.CrossRefGoogle Scholar
Goldberger, A. S. Econometrics and psychometrics: A survey of communalities. Psychometrika, 1971, 36, 83–107.CrossRefGoogle Scholar
Goldberger, A. S. Structural equation methods in the social sciences. Econometrica, 1972, 40, 979–1001.CrossRefGoogle Scholar
Goldberger, A. S. & Duncan, O. D. Structural equation models in the social sciences, 1973, New York: Academic.Google Scholar
Greenstadt, J. On the relative efficiencies of gradient methods. Mathematical Computation, 1967, 21, 360–367.CrossRefGoogle Scholar
Hausman, J. A. Errors in variables in simultaneous equation models. Journal of Econometrics, 1977, 5, 389–401.CrossRefGoogle Scholar
Hsiao, C. Identification and estimation of simultaneous equation models with measurement error. International Economic Review, 1976, 17, 319–339.CrossRefGoogle Scholar
Intriligator, M. D. Econometric models, techniques, and applications, 1978, Englewood Cliffs: Prentice Hall.Google Scholar
Jennrich, R. I. Rotational equivalence of factor loading matrices with specified values. Psychometrika, 1978, 43, 421–426.CrossRefGoogle Scholar
Jennrich, R. I. & Sampson, P. F. Application of stepwise regression to nonlinear estimation. Technometrics, 40, 63–72.Google Scholar
Jöreskog, K. G. Statistical analysis of sets of congeneric tests. Psychometrika, 1971, 36, 109–133.CrossRefGoogle Scholar
Jöreskog, K. G. Analysis of covariance structures. In Krishnaiah, P. R. (Eds.), Multivariate analysis-III, 1973, New York: Academic.Google Scholar
Jöreskog, K. G. Structural equation models in the social sciences: Specification, estimation and testing. In Krishnaiah, P. R. (Eds.), Applications of statistics, 1977, Amsterdam: North-Holland.Google Scholar
Jöreskog, K. G. Structural analysis of covariance and correlation matrices. Psychometrika, 1978, 43, 443–477.CrossRefGoogle Scholar
Jöreskog, K. G. & Sörbom, D. LISREL IV user's guide, 1978, Chicago: National Educational Resources.Google Scholar
Kristof, W. On the theory of a set of tests which differ only in length. Psychometrika, 1971, 36, 207–225.CrossRefGoogle Scholar
Lawley, D. N. & Maxwell, A. E. Factor analysis as a statistical method, 1971, London: Butterworths.Google Scholar
Lee, S. Y. Some algorithms for covariance structure analysis, 1977, Los Angeles: University of California.Google Scholar
Lee, S. Y. & Bentler, P. M. Asymptotic theorems for constrained generalized least squares estimation in covariance structure models. Submitted for publication, 1979.CrossRefGoogle Scholar
Lee, S. Y. & Jennrich, R. I. A study of algorithms for covariance structure analysis with specific comparisons using factor analysis. Psychometrika, 1979, 44, 99–113.CrossRefGoogle Scholar
Li, C. C. Path analysis—A primer, 1975, Pacific Grove, Calif.: Boxwood.Google Scholar
Marquardt, D. W. An algorithm for least squares estimation of nonlinear parameters. SIAM Journal, 1963, 11, 431–441.Google Scholar
McDonald, R. P. A simple comprehensive model for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 1978, 31, 59–72.CrossRefGoogle Scholar
McDonald, R. P. & Krane, W. R. A note on local identifiability and degrees of freedom in the asymptotic likelihood ratio test. British Journal of Mathematical and Statistical Psychology, 1977, 30, 198–203.CrossRefGoogle Scholar
McDonald, R. P. & Krane, W. R. A Monte Carlo study of local identifiability and degrees of freedom in the asymptotic likelihood ratio test. British Journal of Mathematical and Statistical Psychology, 1979, 32, 121–132.CrossRefGoogle Scholar
Monfort, A. First-order identification in linear models. Journal of Econometrics, 1978, 7, 333–350.CrossRefGoogle Scholar
Olsson, U., Bergman, L. R. A longitudinal factor model for studying change in ability structure. Multivariate Behavioral Research, 1977, 12, 221–241.CrossRefGoogle ScholarPubMed
Robinson, P. M. The estimation of a multivariate linear relation. Journal of Multivariate Analysis, 1977, 7, 409–423.CrossRefGoogle Scholar
Taubman, P. Kinometrics: Determinants of socioeconomic success within and between families, 1977, Amsterdam: North-Holland.Google Scholar
Theil, H. Principles of econometrics, 1971, Amsterdam: North-Holland.Google Scholar
Thurstone, L. L. & Thurstone, T. G. Factorial studies of intelligence. Psychometric Monographs, 1941, No. 2.Google Scholar
Sörbom, D. A general method for studying differences in factor means and factor structure between groups. British Journal of Mathematical and Statistical Psychology, 1974, 27, 229–239.CrossRefGoogle Scholar
Sörbom, D. An alternative to the methodology for analysis of covariance. Psychometrika, 1978, 43, 381–396.CrossRefGoogle Scholar
Weeks, D. G. Structural equation systems on latent variables within a second order measurement model, 1978, Los Angeles: University of California.Google Scholar
Weeks, D. G. A second-order longitudinal model of ability structure. Multivariate Behavioral Research, 1980, in press.CrossRefGoogle ScholarPubMed
Wiley, D. E. The identification problem for structural equation models with unmeasured variables. In Goldberger, A. S. & Duncan, O. D. (Eds.), Structural equation models in the social sciences, 1973, New York: Seminar.Google Scholar
Williams, J. S. A definition for the common-factor analysis model and the elimination of problems of factor score indeterminacy. Psychometrika, 1978, 43, 293–306.CrossRefGoogle Scholar
Wold, H. Model construction and evaluation when theoretical knowledge is scarce: An example of the use of partial least squares. In Kmenta, J. & Ramsey, J. (Eds.), Evaluation of econometric models, 1979, New York: Seminar (in press)Google Scholar