Three-Mode Factor Analysis by Means of Candecomp/Parafac
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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.
Key wordsthree-mode factor analysis multitrait-multimethod Candecomp Parafac minimum rank factor analysis
Research is supported by the Dutch Organisation for Scientific Research (NWO), VIDI grant 452-08-001.
The authors would like to thank the anonymous reviewers for their helpful comments, and Katherine Stroebe (Department of Social Psychology, University of Groningen) for providing the belief in a just world dataset.
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