Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-08T11:53:16.871Z Has data issue: false hasContentIssue false
  • English
  • Français

Tucker2 Hierarchical Classes Analysis

Published online by Cambridge University Press:  01 January 2025

Eva Ceulemans  and
Iven Van Mechelen
Eva Ceulemans*
Affiliation:
Katholieke Universiteit Leuven
Iven Van Mechelen
Affiliation:
Katholieke Universiteit Leuven
*
Correspondence concerning this paper should be addressed to Eva Ceulemans, Department of Psychology, Tiensestraat 102, B-3000 Leuven, Belgium. Email: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents a new hierarchical classes model, called Tucker2-HICLAS, for binary three-way three-mode data. As any three-way hierarchical classes model, the Tucker2-HICLAS model includes a representation of the association relation among the three modes and a hierarchical classification of the elements of each mode. A distinctive feature of the Tucker2-HICLAS model, being closely related to the Tucker3-HICLAS model (Ceulemans, Van Mechelen & Leenen, 2003), is that one of the three modes is minimally reduced and, hence, that the differences among the association patterns of the elements of this mode are maximally retained in the model. Moreover, as compared to Tucker3-HICLAS, Tucker2-HICLAS implies three rather than four different types of parameters and as such is simpler to interpret. Two types of Tucker2-HICLAS models are distinguished: a disjunctive and a conjunctive type. An algorithm for fitting the Tucker2-HICLAS model is described and evaluated in a simulation study. The model is illustrated with longitudinal data on interpersonal emotions.

Keywords

three-way three-mode databinary datahierarchical classesmultiway data analysisclustering
Type
Theory and Methods
Information
Psychometrika , Volume 69 , Issue 3 , September 2004 , pp. 375 - 399
Copyright
Copyright © 2004 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author is a Researcher of the Fund for Scientific Research—Flanders (Belgium). The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/2000/02). The authors are grateful to Iwin Leenen for the fruitful discussions.

References

Carroll, J.D., & Chang, J.J. (1970). Analysis of individual differences in multidimensional scaling via ann-way generalization of “Eckart-Young” decomposition. Psychometrika, 35, 283–319CrossRefGoogle Scholar
Ceulemans, E., & Van Mechelen, I. (2003). Uniqueness ofn-wayn-mode hierarchical classes models. Journal of Mathematical Psychology, 47, 259–264CrossRefGoogle Scholar
Ceulemans, E., Van Mechelen, I., & Leenen, I. (2003). Tucker3 hierarchical classes analysis. Psychometrika, 68, 413–433CrossRefGoogle Scholar
Chaturvedi, A., & Carroll, J.D. (1994). An alternating combinatorial optimization approach to fitting theindclus and generalizedindclus models. Journal of Classification, 11, 155–170CrossRefGoogle Scholar
De Boeck, P., & Rosenberg, S. (1988). Hierarchical classes: Model and data analysis. Psychometrika, 53, 361–381CrossRefGoogle Scholar
Haggard, E.A. (1958). Intraclass correlation and the analysis of variance. New York: DrydenGoogle Scholar
Harshman, R.A. (1970). Foundations of theparafac procedure: Models and conditions for an explanatory multi-modal factor analysis. UCLA Working Papers in Phonetics, 16, 1–84Google Scholar
Izard, C.E. (1977). Human emotions. New York: Plenum PressCrossRefGoogle Scholar
Kim, K.H. (1982). Boolean matrix theory. New York: Marcel DekkerGoogle Scholar
Kirk, R.E. (1982). Experimental design: Procedures for the behavioral sciences 2nd ed., Belmont, CA: Brooks/ColeGoogle Scholar
Kroonenberg, P.M. (1983). Three-mode principal component analysis: Theory and applications. Leiden: DSWOGoogle Scholar
Kroonenberg, P.M., & De Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 54, 69–97CrossRefGoogle Scholar
Leenen, I., & Van Mechelen, I. (1998). A branch-and-bound algorithm for Boolean regression. In Balderjahn, I., Mathar, R., & Schader, M. (Eds.), Data highways and information flooding, a challenge for classification and data analysis (pp. 164–171). Berlin, Germany: Springer-VerlagCrossRefGoogle Scholar
Leenen, I., & Van Mechelen, I. (2001). An evaluation of two algorithms for hierarchical classes analysis. Journal of Classification, 18, 57–80CrossRefGoogle Scholar
Leenen, I., Van Mechelen, I., & De Boeck, P. (1999). A generic disjunctive/conjunctive decomposition model forn-ary relations. Journal of Mathematical Psychology, 43, 102–122CrossRefGoogle ScholarPubMed
Leenen, I., Van Mechelen, I., & De Boeck, P. (2001). Models for ordinal hierarchical classes analysis. Psychometrika, 66, 389–404CrossRefGoogle Scholar
Leenen, I., Van Mechelen, I., De Boeck, P., & Rosenberg, S. (1999). indclas: A three-way hierarchical classes model. Psychometrika, 64, 9–24CrossRefGoogle Scholar
Plutchik, R. (1962). The emotions: Facts, theories, and a new model. New York: Random HouseGoogle Scholar
Timmerman, M.E., & Kiers, H.A.L. (2000). Three-mode principal components analysis: Choosing the numbers of components and sensitivity to local optima. British Journal of Mathematical and Statistical Psychology, 53, 1–16CrossRefGoogle ScholarPubMed
Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311CrossRefGoogle ScholarPubMed
Van Mechelen, I., De Boeck, P., & Rosenberg, S. (1995). The conjunctive model of hierarchical classes. Psychometrika, 60, 505–521CrossRefGoogle Scholar