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Properties of and Algorithms for Fitting Three-Way Component Models with Offset Terms

Published online by Cambridge University Press:  01 January 2025

Henk A. L. Kiers
Henk A. L. Kiers*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Henk A.L. Kiers, Heymans Institute, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands. E-mail: h.a.l.kiers@rug.nl
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Abstract

Prior to a three-way component analysis of a three-way data set, it is customary to preprocess the data by centering and/or rescaling them. Harshman and Lundy (1984) considered that three-way data actually consist of a three-way model part, which in fact pertains to ratio scale measurements, as well as additive “offset” terms that turn the ratio scale measurements into interval scale measurements. They mentioned that such offset terms might be estimated by incorporating additional components in the model, but discarded this idea in favor of an approach to remove such terms from the model by means of centering. Then estimates for the three-way component model parameters are obtained by analyzing the centered data. In the present paper, the possibility of actually estimating the offset terms is taken up again. First, it is mentioned in which cases such offset terms can be estimated uniquely. Next, procedures are offered for estimating model parameters and offset parameters simultaneously, as well as successively (i.e., providing offset term estimates after the three-way model parameters have been estimated in the traditional way on the basis of the centered data). These procedures are provided for both the CANDECOMP/PARAFAC model and the Tucker3 model extended with offset terms. The successive and the simultaneous approaches for estimating model and offset parameters have been compared on the basis of simulated data. It was found that both procedures perform well when the fitted model captures at least all offset terms actually underlying the data. The simultaneous procedures performed slightly better than the successive procedures. If fewer offset terms are fitted than there are underlying the model, the results are considerably poorer, but in these cases the successive procedures performed better than the simultaneous ones. All in all, it can be concluded that the traditional approach for estimating model parameters can hardly be improved upon, and that offset terms can sufficiently well be estimated by the proposed successive approach, which is a simple extension of the traditional approach.

Keywords

three-way component modelspreprocessingadditive terms
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 2 , June 2006 , pp. 231 - 256
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

The author is obliged to Jos M.F. ten Berge and Marieke Timmerman for helpful comments on an earlier version of this paper. The author is obliged to Iven van Mechelen for making available the data set used in Section 6.

References

Andersson, C.A., & Bro, R. (1998). Improving the speed of multiway algorithms: Part I. Tucker3. Chemometrics and Intelligent Laboratory Systems, 42, 93–103.CrossRefGoogle Scholar
Carroll, J.D., & Chang, J.-J. (1970). Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart—Young” decomposition. Psychometrika, 35, 283–319.CrossRefGoogle Scholar
Efron, B., & Tibshirani, R.J. (1993). An introduction to the bootstrap. New York: Chapman & Hall.CrossRefGoogle Scholar
Harshman, R.A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimode factor analysis. University of California at Los Angeles. 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 Law, H.G., Snyder, C.W., Hattie, J.A., & McDonald, R.P. (Eds.), Research methods for multimode data analysis (pp. 216–284). New York: Praeger.Google Scholar
6 Harshman, R.A., & Lundy, M.E. (1997). An Extended Parafac Model Incorporating Singly-Subscripted Constants: Theory and Application. Poster presented at the TRICAP Meeting, Lake Chelan, 1997, May.Google Scholar
Kiers, H.A.L. (2000). Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics, 14, 105–122.3.0.CO;2-I>CrossRefGoogle Scholar
8 Kiers, H.A.L. (2003). Uniqueness properties of three-way component models with offset terms. In: H. Yanai, A. Okada, K. Shigemasu, Y. Kano, & J.J. Meulman (Eds.), New developments in psychometrics: Proceedings of the international meeting of the Psychometric Society IMPS2001, pp. 354–362.CrossRefGoogle Scholar
Kiers, H.A.L. (2004). Bootstrap confidence intervals for three-way methods. Journal of Chemometrics, 18, 22–36.CrossRefGoogle Scholar
Kiers, H.A.L., & van Mechelen, I. (2001). Three-way component analysis: Principles and illustrative application. Psychological Methods, 6, 84–110.CrossRefGoogle ScholarPubMed
Kroonenberg, P.M., & De Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45, 69–97.CrossRefGoogle Scholar
Kroonenberg, P.M., & Heiser, W.J.et al. (1998). Parallel factor analysis with constraints on the configurations: An overview. In Hayashi, C.et al. (Eds.), Data science, classification and related methods (pp. 563–574). Tokyo: Springer-Verlag.Google Scholar
Kruskal, J.B. (1977). Some least squares theorems for matrices and N-way arrays. Unpublished Bell Labs memorandum. Murray Hill, NJ: Bell Laboratories.Google Scholar
Paatero, P., & Andersson, C.A. (1999). Further improvements of the speed of the Tucker3 three-way algorithm. Chemometrics and Intelligent Laboratory Systems, 47, 17–20.CrossRefGoogle Scholar
Penrose, R. (1956). On best approximate solutions of linear matrix equations. Proceedings of the Cambridge Philosophical Society, 52, 17–19.CrossRefGoogle Scholar
The Mathworks (1998). Matlab. Natick: The Mathworks.Google Scholar
Timmerman, M.E., & Kiers, H.A.L. (2002). Three-way component analysis with smoothness constraints. Computational Statistics and Data Analysis, 40, 447–470.CrossRefGoogle Scholar
Tucker, L.R. (1951). A method for synthesis of factor analysis studies, (Personnel Research Section Report, No. 984). Washington, DC: Department of the Army.CrossRefGoogle Scholar
Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed
20 Van Eeuwijk, F.A., & Kroonenberg, P.M. (1998). Multiplicative models for interaction in three-way ANOVA, with applications to plant breeding. Biometrics, 1315–1333.CrossRefGoogle Scholar
Van Mechelen, I., & Kiers, H.A.L. (1999). Individual differences in anxiety responses to stressful situations: A three-way component analysis model. European Journal of Personality, 13, 409–428.3.0.CO;2-I>CrossRefGoogle Scholar