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Some Contributions to Efficient Statistics in Structural Models: Specification and Estimation of Moment Structures

Published online by Cambridge University Press:  01 January 2025

P. M. Bentler
P. M. Bentler*
Affiliation:
University of California, Los Angeles
*
Address reprint requests to P. M. Bentler, Department of Psychology, Franz Hall, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90024, USA.
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Abstract

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.

Keywords

structural equationscovariance structuresmoment structureserrors in variablesfactor analysisminimum chi squaregeneralized least squareselliptical distributionsasymptotically distribution free
Type
Original Paper
Information
Psychometrika , Volume 48 , Issue 4 , December 1983 , pp. 493 - 517
Copyright
Copyright © 1983 The Psychometric Society

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Footnotes

Presidential Address to the Psychometric Society, delivered at the annual meeting in Los Angeles, June 1983, and at the third European meetings in Jouy-en-Josas, France, July 1983. The research reported here was supported in part by USPHS grants DA00017 and DA01070. The ideas developed in this paper were favorably influenced by a number of visitors to UCLA, especially M. W. Browne, T. Dijkstra, and A. Mooijaart, by colleagues at UCLA, especially M. L. Brecht, G. J. Huba, R. I. Jennrich, and J. A. Woodward, and by graduate students in mathematics, engineering, and psychology, especially M. Berkane, D. G. Bonett, P. Leung, A. Mouawad, and J. S. Tanaka. The assistance of S. Luong in manuscript production is also acknowledged.

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