Abstract
The most widely-used computer programs for structural equation models analysis are the LISREL series of Jöreskog and Sörbom. The only types of constraints which may be made directly are fixing parameters at a constant value and constraining parameters to be equal. Rindskopf (1983) showed how these simple properties could be used to represent models with more complicated constraints, namely inequality constraints on unique variances. In this paper, two new concepts are introduced which enable a much wider variety of constraints to be made. The concepts, “phantom” and “imaginary” latent variables, allow fairly general equality and inequality constraints on factor loadings and structural model coefficients.
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Rindskopf, D. (1982).Parameterizing equality constraints in factor analysis and structural modeling. Unpublished manuscript.
References
Bentler, P. M. (1976). Multistructure statistical model applied to factor analysis.Multivariate Behavioral Research, 11, 3–25.
Bentler, P. M., & Lee, S. Y. (1983). Structural models with polynomial constraints.Applied Psychological Measurement, in press.
Bentler, P. M., & Weeks, D. G. (1980). Linear structural equations with latent variables.Psychometrika, 45, 289–308.
Jöreskog, K. G. (1977). Structural equation models in the social sciences: Specification, estimation and testing. In P. R. Krishnaiah (Ed.),Applications of statistics. Amsterdam: North-Holland.
Jöreskog, K. G., & Sörbom, D. (1978).LISREL IV: Analysis of linear structural relationships by the method of maximum likelihood. Chicago: National Educational Resources.
Jöreskog, K. G. & Sörbom, D. (1981).LISREL V: Analysis of linear structural relationships by maximum likelihood and least squares methods. (Research Report 81–8) Uppsala, Sweden: University of Uppsala, Department of Statistics, November.
Lee, S. Y. (1980). Estimation of covariance structure models with parameters subject to functional restraints.Psychometrika, 45, 309–324.
Lee, S. Y., & Tsui, K.-L. (1982). Covariance structure analysis in several populations.Psychometrika, 47, 297–308.
McDonald, R. P. (1978). A simple comprehensive model for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 31, 59–72.
McDonald, R. P. (1980). A simple comprehensive model for the analysis of covariance structures: Some remarks on applications.British Journal of Mathematical and Statistical Psychology, 33, 161–183.
Rindskopf, D. (1983). Parameterizing inequality constraints on unique variances in linear structural models.Psychometrika, 48, 73–83.
Wismer, D. A., & Chattergy, R. (1978).Introduction to nonlinear optimization. New York: North Holland.
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During the preparation of this article, it was discovered that another researcher, Jack McArdle, had concurrently and independently discovered some of the techniques reported here. While he has chosen not to publish his research, I wish to acknowledge his work. I would like to thank Art Woodward for telling me about “sort-of simple” structure.
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Rindskopf, D. Using phantom and imaginary latent variables to parameterize constraints in linear structural models. Psychometrika 49, 37–47 (1984). https://doi.org/10.1007/BF02294204
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DOI: https://doi.org/10.1007/BF02294204