Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-07T18:33:06.612Z Has data issue: false hasContentIssue false
  • English
  • Français

Using Phantom and Imaginary Latent Variables to Parameterize Constraints in Linear Structural Models

Published online by Cambridge University Press:  01 January 2025

David Rindskopf
David Rindskopf*
Affiliation:
City University of New York Graduate Center
*
Requests for reprints should be sent to David Rindskopf, Educational Psychology, CUNY Graduate Center, 33 West 42nd Street, New York, N.Y. 10036.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Computer ProgramEquation ModelPublic PolicyLatent VariableFactor Loading
Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 1 , March 1984 , pp. 37 - 47
Copyright
Copyright © 1984 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

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.

References

Reference Notes

Rindskopf, D. (1982). Parameterizing equality constraints in factor analysis and structural modeling. Unpublished manuscript.Google Scholar

References

Bentler, P. M. (1976). Multistructure statistical model applied to factor analysis. Multivariate Behavioral Research, 11, 3–25.CrossRefGoogle ScholarPubMed
Bentler, P. M., & Lee, S. Y. (1983). Structural models with polynomial constraints. Applied Psychological Measurement, in press.Google Scholar
Bentler, P. M., & Weeks, D. G. (1980). Linear structural equations with latent variables. Psychometrika, 45, 289–308.CrossRefGoogle Scholar
Jöreskog, K. G. (1977). Structural equation models in the social sciences: Specification, estimation and testing. In Krishnaiah, P. R. (Eds.), Applications of statistics, Amsterdam: North-Holland.Google Scholar
Jöreskog, K. G., & Sörbom, D. (1978). LISREL IV: Analysis of linear structural relationships by the method of maximum likelihood, Chicago: National Educational Resources.Google Scholar
Jöreskog, K. G., & Sörbom, D. (1981). LISREL V: Analysis of linear structural relationships by maximum likelihood and least squares methods, Uppsala, Sweden: University of Uppsala, Department of Statistics.Google Scholar
Lee, S. Y. (1980). Estimation of covariance structure models with parameters subject to functional restraints. Psychometrika, 45, 309–324.CrossRefGoogle Scholar
Lee, S. Y., Tsui, K.-L. (1982). Covariance structure analysis in several populations. Psychometrika, 47, 297–308.CrossRefGoogle Scholar
McDonald, R. P. (1978). A simple comprehensive model for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 31, 59–72.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Rindskopf, D. (1983). Parameterizing inequality constraints on unique variances in linear structural models. Psychometrika, 48, 73–83.CrossRefGoogle Scholar
Wismer, D. A., & Chattergy, R. (1978). Introduction to nonlinear optimization, New York: North Holland.Google Scholar