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Technical Aspects of Muthén's Liscomp Approach to Estimation of Latent Variable Relations with a Comprehensive Measurement Model

Published online by Cambridge University Press:  01 January 2025

Bengt O. Muthén  and
Albert Satorra
Bengt O. Muthén*
Affiliation:
Graduate School of Education & Information Studies, UCLA
Albert Satorra
Affiliation:
Department of Economics, Universitat Pompeu Fabra
*
Requests for reprints should be sent to Bengt O. Muthén, Graduate School of Education, University of California, Los Angeles, CA 90024-1521.
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Abstract

Muthén (1984) formulated a general model and estimation procedure for structural equation modeling with a mixture of dichotomous, ordered categorical, and continuous measures of latent variables. A general three-stage procedure was developed to obtain estimates, standard errors, and a chi-square measure of fit for a given structural model. While the last step uses generalized least-squares estimation to fit a structural model, the first two steps involve the computation of the statistics used in this model fitting. A key component in the procedure was the development of a GLS weight matrix corresponding to the asymptotic covariance matrix of the sample statistics computed in the first two stages. This paper extends the description of the asymptotics involved and shows how the Muthén formulas can be derived. The emphasis is placed on showing the asymptotic normality of the estimates obtained in the first and second stage and the validity of the weight matrix used in the GLS estimation of the third stage.

Keywords

structural equation modelingdichotomous measuresgeneralized least-squares estimationasymptotic covariance matrix
Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 4 , December 1995 , pp. 489 - 503
Copyright
Copyright © 1995 The Psychometric Society

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Footnotes

The research of the first author was supported by grant AA 08651-01 from NIAAA for the project “Psychometric Advances for Alcohol and Depression Studies” and grant 40859 from the National Institute for Mental Health. The research of the second author was partially supported by the Spanish DGICYT grants PB91-0814 and PB93-0403.

References

Amemiya, T. (1978). On a two-step estimation of a multivariate logit model. The Journal of Econometrics, 8, 13–21.CrossRefGoogle Scholar
Amemiya, T. (1985). Advanced econometrics, Cambridge, MA: Harvard University Press.Google Scholar
Arminger, G., Küsters, U. (1988). Latent trait models with indicators of mixed measurement level. In Langeheine, R., Rost, J. (Eds.), Latent trit and latent class models (pp. 51–73). New York: Plenum.CrossRefGoogle Scholar
Arminger, G., Küsters, U. (1989). Construction principles for latent trait models. In Clogg, C. (Eds.), Sociological Methodology (pp. 369–393). Oxford: Basil Blackwell.Google Scholar
Bermann, G. (1993). Estimation and inference in bivariate and multivariate ordinal probit models, Stocholm, Sweden: Almqvist & Wiksell.Google Scholar
Christoffersson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5–32.CrossRefGoogle Scholar
Christoffersson, A., Gunsjö, A. (1983). Analysis of covariance structures for ordinal data, Uppsala, Sweden: University of Uppsala, Department of Statistics.Google Scholar
Gong, G., Samaniego, F. J. (1981). Pseudo maximum likelihood estimation: Theory and applications. Annals of Statistics, 9, 861–869.CrossRefGoogle Scholar
Gourieroux, C., Montfort, A. (1989). Statistique et modeles econometriques, Vol. I, II, Paris: Economica.Google Scholar
Jöreskog, K. G. (1985, July). Estimation of the polyserial correlation from summary statistics. Paper presented at the Fourth European Meeting of the Psychometric Society in Cambridge, England.Google Scholar
Jöreskog, K. G. (1991, July), Latent variable modeling with ordinal variables. Paper presented at the International Workshop on Statistical Modeling and Latent Variables in Trento, Italy.Google Scholar
Jöreskog, K. G., Sörbom, D. (1988). PRELIS—A program for multivariate data screening and data summarization. A preprocessor for LISREL 2nd ed.,, Chicago, IL: Scientific Software.Google Scholar
Jöreskog, K. G., Sörbom, D. (1989). LISREL 7—A guide to the program and applications 2nd ed.,, Chicago: SPSS Publications.Google Scholar
Küsters, U. (1987). Hierarchische Mittlewert- und Kovarianzstrukturmodelle mit Nichtmetrischen Endogenen Variablen [Hierarchical mean—and covariance structure models with nonmetric endogenous variables], Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
Küsters, U. (1990). A note on sequential ML estimates and their asymptotic covariances. Statistical Papers, 31, 131–145.CrossRefGoogle Scholar
Lee, L.-F. (1982). Health and wage: A simultaneous equation model with multiple discrete indicators. International Economic Review, 23, 199–221.CrossRefGoogle Scholar
Lee, S.-Y. (1985). Maximum-likelihood estimation of polychoric correlations in r ×s ×t contingency tables. Journal of Statistical Computation and Simulations, 23, 53–67.CrossRefGoogle Scholar
Lee, S.-Y., Poon, W.-Y. (1986). Maximum likelihood estimation of polyserial correlations. Psychometrika, 51, 113–121.CrossRefGoogle Scholar
Lee, S.-Y., Poon, W.-Y. (1987). Two-step estimation of multivariate polychoric correlations. Communications in Statistics: Theory and Methods, 16, 307–320.CrossRefGoogle Scholar
Lee, S.-Y., Poon, W.-Y., Bentler, P. M. (1989). Simultaneous analysis of multivariate polytomous variates in several groups. Psychometrika, 54, 63–73.CrossRefGoogle Scholar
Lee, S.-Y., Poon, W.-Y., Bentler, P. M. (1990). Full maximum likelihood analysis of structural equation models with polytomous variables. Statistics and Probability Letters, 9, 91–97.CrossRefGoogle Scholar
Lee, S.-Y., Poon, W.-Y., Bentler, P. M. (1990). A 3-stage estimation procedure for structural equation models with polytomous variables. Psychometrika, 55, 45–57.CrossRefGoogle Scholar
Lee, S.-Y., Poon, W.-Y., Bentler, P. M. (1992). Structural equation models with continuous and polytomous variables. Psychometrika, 57, 89–105.CrossRefGoogle Scholar
Maddala, G. S. (1983). Limited-dependent and qualitative variables in econometrics, Cambridge: Cambridge University Press.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, 48–65.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. (1987). LISCOMP. Analysis of linear structural equations with a comprehensive measurement model. Theoretical integration and user's guide, Mooresville, IN: Scientific Software.Google Scholar
Muthén, B. (1989). Multiple-group structural modeling with non-normal continuous variables. British Journal of Mathematical and Statistical Psychology, 42, 55–62.CrossRefGoogle Scholar
Muthén, B. (1989). Tobit factor analysis. British Journal of Mathematical and Statistical Psychology, 42, 241–250.CrossRefGoogle Scholar
Muthén, B. (1989). Latent variable modeling in heterogenous populations. Psychometrika, 54, 557–585.CrossRefGoogle Scholar
Muthén, B. (1990). Moments of the censored and truncated bivariate normal distribution. British Journal of Mathematical and Statistical Psychology, 43, 131–143.CrossRefGoogle Scholar
Muthén, B. (1993). Goodness of fit with categorical and other non-normal variables. In Bollen, K., Long, S. (Eds.), Testing structural equation models (pp. 205–243). Newbury Park: Sage Publications.Google Scholar
Muthén, B., Christofferson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups. Psychometrika, 46, 407–419.CrossRefGoogle Scholar
Muthén, B., Kaplan, D. (1992). A comparison of some methologies for the factor analysis of non-normal Likert variables: A Note on the Size of the Model. British Journal of Mathematical and Statistical Psychology, 45, 19–30.CrossRefGoogle Scholar
Parke, W. (1986). Pseudo maximum likelihood estimation: The asymptotic distribution. The Annals of Statistics, 14, 355–357.CrossRefGoogle Scholar
Poon, W.-Y., Lee, S.-Y. (1987). Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients. Psychometrika, 52, 409–430.CrossRefGoogle Scholar
Poon, W.-Y., Lee, S.-Y. (1988). Analyses of structural equation models with censored or truncated data, Hong Kong: The Chinese University of Hong Kong, Department of Statistics.Google Scholar
Poon, W.-Y., Lee, S.-Y. (1992). Statistical analysis of continuous and polytomous variables in several populations. British Journal of Mathematical and Statistical Psychology, 45, 139–149.CrossRefGoogle Scholar
Satorra, A. (1992). Asymptotic robust inferences in the analysis of mean and covariance structures. In Marsden, P. V. (Eds.), Sociological Methodology 1992 (pp. 249–278). New York: Plenum.Google Scholar
Schepers, A. (1991). Numerische Verfahren und Implementation der Schatzung von Mittlewert und Kovarianz-Strukturmodellen mit Nichmetrische Variablen, Ahaus: Verlag Frank Hartman.Google Scholar
Schepers, A., Arminger, G., Küsters, U. (1992). The analysis of non-metric endogenous variables in latent variable models: The MECOSA approach. In Gruber, P. (Eds.), Econometric decision models: New methods of modeling and applications. Lecture notes in mathematical systems (pp. 459–472). Heidelberg: Springer Verlag.Google Scholar
Serfling, R. J. (1980). Approximation theorems of mathematical statistics, New York: Wiley.CrossRefGoogle Scholar
Sobel, M. E., Arminger, G. (1992). Modeling household fertility decision: A non-linear simultaneous probit model. Journal of the American Statistical Association, 87, 38–47.CrossRefGoogle Scholar