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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. Henk Kiers & Yoshio Takane & Jos Berge, 1996. "The analysis of multitrait-multimethod matrices via constrained components analysis," Psychometrika, Springer;The Psychometric Society, vol. 61(4), pages 601-628, December.
    2. Pieter Kroonenberg & Jan Leeuw, 1980. "Principal component analysis of three-mode data by means of alternating least squares algorithms," Psychometrika, Springer;The Psychometric Society, vol. 45(1), pages 69-97, March.
    3. 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.
    4. J. Carroll & Jih-Jie Chang, 1970. "Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition," Psychometrika, Springer;The Psychometric Society, vol. 35(3), pages 283-319, September.
    5. 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.
    6. Michael Eid, 2000. "A multitrait-multimethod model with minimal assumptions," Psychometrika, Springer;The Psychometric Society, vol. 65(2), pages 241-261, June.
    7. Ledyard Tucker, 1966. "Some mathematical notes on three-mode factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 279-311, September.
    8. P. Bentler & Sik-Yum Lee, 1978. "Statistical aspects of a three-mode factor analysis model," Psychometrika, Springer;The Psychometric Society, vol. 43(3), pages 343-352, September.
    9. Bruce Bloxom, 1968. "A note on invariance in three-mode factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 33(3), pages 347-350, September.
    10. Henk Kiers & Pieter Kroonenberg & Jos Berge, 1992. "An efficient algorithm for TUCKALS3 on data with large numbers of observation units," Psychometrika, Springer;The Psychometric Society, vol. 57(3), pages 415-422, September.
    11. Jos Berge & Henk Kiers, 1991. "A numerical approach to the approximate and the exact minimum rank of a covariance matrix," Psychometrika, Springer;The Psychometric Society, vol. 56(2), pages 309-315, June.
    12. 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.
    13. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
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