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Three-Mode Factor Analysis by Means of Candecomp/Parafac

Published online by Cambridge University Press:  01 January 2025

Alwin Stegeman  and
Tam T. T. Lam
Alwin Stegeman*
Affiliation:
Heymans Institute for Psychological Research, University of Groningen
Tam T. T. Lam
Affiliation:
Heymans Institute for Psychological Research, University of Groningen
*
Requests for reprints should be sent to Alwin Stegeman, Heymans Institute for Psychological Research, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands. E-mail: a.w.stegeman@rug.nl
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Abstract

A three-mode covariance matrix contains covariances of N observations (e.g., subject scores) on J variables for K different occasions or conditions. We model such an JK×JK covariance matrix as the sum of a (common) covariance matrix having Candecomp/Parafac form, and a diagonal matrix of unique variances. The Candecomp/Parafac form is a generalization of the two-mode case under the assumption of parallel factors. We estimate the unique variances by Minimum Rank Factor Analysis. The factors can be chosen oblique or orthogonal. Our approach yields a model that is easy to estimate and easy to interpret. Moreover, the unique variances, the factor covariance matrix, and the communalities are guaranteed to be proper, a percentage of explained common variance can be obtained for each variable-condition combination, and the estimated model is rotationally unique under mild conditions. We apply our model to several datasets in the literature, and demonstrate our estimation procedure in a simulation study.

Keywords

three-mode factor analysismultitrait-multimethodCandecompParafacminimum rank factor analysis
Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 3 , July 2014 , pp. 426 - 443
Copyright
Copyright © 2013 The Psychometric Society

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