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.
- Harshman, R.A. (1970). Foundations of the Parafac procedure: models and conditions for an “explanatory” multimodal factor analysis. UCLA Working Papers in Phonetics, 16, 1–84. Google Scholar
- Harshman, R.A., & Lundy, M.E. (1984). Data preprocessing and the extended Parafac model. In H.G. Law, C.W. Snyder Jr., J.A. Hattie, & R.P. McDonald (Eds.), Research methods for multimode data analysis (pp. 216–284). New York: Praeger. Google Scholar
- Harshman, R.A. (2004). The problem and nature of degenerate solutions or decompositions of 3-way arrays. In Talk at the tensor decompositions workshop, Palo Alto, USA. Google Scholar
- Hitchcock, F.L. (1927a). The expression of a tensor or a polyadic as a sum of products. Journal of Mathematics and Physics, 6, 164–189. Google Scholar
- Hitchcock, F.L. (1927b). Multiple invariants and generalized rank of a p-way matrix or tensor. Journal of Mathematics and Physics, 7, 39–70. Google Scholar
- Kroonenberg, P.M. (2008). Wiley series in probability and statistics. Applied multiway data analysis. Hoboken: Wiley. Google Scholar
- Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1989). How 3-MFA data can cause degenerate Parafac solutions, among other relationships. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 115–121). Amsterdam: North-Holland. Google Scholar
- Lorenzo-Seva, U., & Ten Berge, J.M.F. (2006). Tucker’s congruence coefficient as a meaningful index of factor similarity. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 2, 57–64. Google Scholar
- Smilde, A., Bro, R., & Geladi, P. (2004). Multi-way analysis: applications in the chemical sciences. Chichester: Wiley. Google Scholar
- Sočan, G. (2003). The incremental value of minimum rank factor analysis. Ph.D. Thesis, Department of Psychometrics & Statistics, University of Groningen, Groningen, The Netherlands. Google Scholar
- Wothke, W. (1996). Models for multitrait-multimethod matrix analysis. In G.A. Marcoulides & R.E. Schumacker (Eds.), Advanced structural equation modeling: issues and techniques (pp. 7–56). Mahwah: Erlbaum. Google Scholar