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Three-Way Tucker2 Component Analysis Solutions of Stimuli × Responses × Individuals Data with Simple Structure and the Fewest Core Differences

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Abstract

Multivariate stimulus-response designs can be described by a three-way array of stimuli by responses by individuals. Its underlying structure can be represented by a network based on the Tucker2 component model in which stimulus components are connected with response components by means of the links that differ between individuals. For each individual such links are represented in a slice of the extended core array. For a proper understanding of these links, it is desirable that [1] the individual core slices as well as the component matrices have simple structures and [2] the differences of core slices between individuals are as few as possible. For attaining [1] and [2] we propose a method in which both the component matrices and the core slices of a Tucker2 solution are transformed simultaneously in order that the component matrices match simple target matrices and the core slices are summarized by a simple target slice. The proposed method is evaluated in a simulation study and illustrated with a three-way data array of semantic differential ratings.

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References

  1. Adachi, K. (2007). Joint Procrustes analysis and three-way component analysis for the data array of stimuli by responses by persons. In IMPS 2007 (International Meeting of Psychometric Society 2007) Abstracts (p. 96).

  2. Adachi, K. (2009). Joint Procrustes analysis for simultaneous nonsingular transformation of component score and loading matrices. Psychometrika, 74, 667–683.

  3. Brouwer, P., & Kroonenberg, P.M. (1991). Some notes on the diagonalization of the extended three-mode core matrix. Journal of Classification, 8, 93–98.

  4. Ceulemans, E., & Kiers, H.A.L. (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. British Journal of Mathematical and Statistical Psychology, 59, 133–150.

  5. Crawford, C.B., & Ferguson, G.A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika, 35, 321–332.

  6. Harshman, R.A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “exploratory” multi-mode factor analysis. UCLA Working Papers in Phonetics, 16, 1–84.

  7. Harshman, R.A., & Lundy, M.E. (1984). Data preprocessing and the extended PARAFAC model. In Law, H.G., Snyder, C.W. Jr., Hattie, J.A., & McDonald, R.P. (Eds.), Research methods for multimode data analysis (pp. 216–284). New York: Praeger.

  8. Hendrickson, A.E., & White, P.O. (1964). PROMAX: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17, 65–70.

  9. Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200.

  10. Kiers, H.A.L. (1991). Hierarchical relations among three-way methods. Psychometrika, 56, 449–470.

  11. Kiers, H.A.L. (1994). Simplimax: Oblique rotation to an optimal target with simple structure. Psychometrika, 59, 567–579.

  12. Kiers, H.A.L. (1998a). Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure. Computational Statistics and Data Analysis, 28, 307–324.

  13. Kiers, H.A.L. (1998b). Joint orthomax rotation of the core and component matrices resulting from three-mode principal component analysis. Journal of Classification, 15, 245–263.

  14. Kiers, H.A.L., & ten Berge, J.M.F. (1992). Minimization of a class of matrix trace functions by means of refined majorization. Psychometrika, 57, 371–382.

  15. Kiers, H.A.L., & Van Mechelen, I. (2001). Three-way component analysis: Principles and illustrative application. Psychological Methods, 6, 84–110.

  16. Kroonenberg, P.M. (1983). Three-mode principal component analysis: Theory and applications. Leiden: DSWO Press.

  17. Kroonenberg, P.M. (2008). Applied multiway data analysis. New York: Wiley.

  18. 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.

  19. Murakami, T. (1983). Quasi three-model principal component analysis: A method for assessing the factor change. Behaviormetrika, No. 14, 27–48.

  20. Murakami, T., & Kroonenberg, P.M. (2003). Three-mode models and individual differences in semantic differential data. Multivariate Behavioral Research, 38, 247–283.

  21. Osgood, C.E., Suci, G.J., & Tannenbaum, P.H. (1957). The measurement of meaning. Champaign: University of Illinois Press.

  22. Seber, G.A.F. (2008). A matrix handbook for statisticians. New York: Wiley.

  23. Smilde, A., Bro, R., & Geladi, P. (2004). Multi-way analysis: Applications in the chemical sciences. New York: Wiley.

  24. Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.

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Author information

Correspondence to Kohei Adachi.

Additional information

The author would like to thank the editor and anonymous reviewers. This research was supported by grant (C)-20500256 from the Japan Society for the Promotion of Science.

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Adachi, K. Three-Way Tucker2 Component Analysis Solutions of Stimuli × Responses × Individuals Data with Simple Structure and the Fewest Core Differences. Psychometrika 76, 285–305 (2011). https://doi.org/10.1007/s11336-011-9208-6

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Keywords

  • three-way component analysis
  • Tucker2 model
  • stimuli × responses × individuals data
  • simplimax
  • promax
  • joint Procrustes analysis