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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. Copyright The Psychometric Society 2014

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  • Alwin Stegeman & Tam Lam, 2014. "Three-Mode Factor Analysis by Means of Candecomp/Parafac," Psychometrika, Springer;The Psychometric Society, vol. 79(3), pages 426-443, July.
    • Handle: RePEc:spr:psycho:v:79:y:2014:i:3:p:426-443
    DOI: 10.1007/s11336-013-9359-8
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    1. Wim Krijnen & Theo Dijkstra & Alwin Stegeman, 2008. "On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model," Psychometrika, Springer;The Psychometric Society, vol. 73(3), pages 431-439, September.
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    3. Alwin Stegeman, 2007. "Degeneracy in Candecomp/Parafac and Indscal Explained For Several Three-Sliced Arrays With A Two-Valued Typical Rank," Psychometrika, Springer;The Psychometric Society, vol. 72(4), pages 601-619, December.
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    13. Alwin Stegeman, 2006. "Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays of rank p + 1 or higher," Psychometrika, Springer;The Psychometric Society, vol. 71(3), pages 483-501, September.
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    3. Dawn Iacobucci & Doug Grisaffe & Wayne DeSarbo, 2017. "Statistical perceptual maps: using confidence region ellipses to enhance the interpretations of brand positions in multidimensional scaling," Journal of Marketing Analytics, Palgrave Macmillan, vol. 5(3), pages 81-98, December.

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