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Statistical Aspects of a Three-Mode Factor Analysis Model

Published online by Cambridge University Press:  01 January 2025

P. M. Bentler  and
Sik-Yum Lee
P. M. Bentler*
Affiliation:
University of California, Los Angeles
Sik-Yum Lee
Affiliation:
The Chinese University of Hong Kong
*
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

A special case of Bloxom's version of Tucker's three-mode model is developed statistically. A distinction is made between modes in terms of whether they are fixed or random. Parameter matrices are associated with the fixed modes, while no parameters are associated with the mode representing random observation vectors. The identification problem is discussed, and unknown parameters of the model are estimated by a weighted least squares method based upon a Gauss-Newton algorithm. A goodness-of-fit statistic is presented. An example based upon self-report and peer-report measures of personality shows that the model is applicable to real data. The model represents a generalization of Thurstonian factor analysis; weighted least squares estimators and maximum likelihood estimators of the factor model can be obtained using the proposed theory.

Keywords

factor analysisthree-mode models
Type
Original Paper
Information
Psychometrika , Volume 43 , Issue 3 , September 1978 , pp. 343 - 352
Copyright
Copyright © 1978 The Psychometric Society

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Footnotes

This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and a research grant (DA01070) from the U. S. Public Health Service. The very helpful comments of several anonymous reviewers are gratefully acknowledged.

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