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Hierarchical Classes Models for Three-Way Three-Mode Binary Data: Interrelations and Model Selection

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
*
Requests for reprints should be sent to Eva Ceulemans, Department of Psychology, Tiensestraat 102, B-3000 Leuven, Belgium. Email: Eva.Ceulemans@psy.kuleuven.be
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Abstract

Several hierarchical classes models can be considered for the modeling of three-way three-mode binary data, including the INDCLAS model (Leenen, Van Mechelen, De Boeck, and Rosenberg, 1999), the Tucker3-HICLAS model (Ceulemans, Van Mechelen, and Leenen, 2003), the Tucker2-HICLAS model (Ceulemans and Van Mechelen, 2004), and the Tucker1-HICLAS model that is introduced in this paper. Two questions then may be raised: (1) how are these models interrelated, and (2) given a specific data set, which of these models should be selected, and in which rank? In the present paper, we deal with these questions by (1) showing that the distinct hierarchical classes models for three-way three-mode binary data can be organized into a partially ordered hierarchy, and (2) by presenting model selection strategies based on extensions of the well-known scree test and on the Akaike information criterion. The latter strategies are evaluated by means of an extensive simulation study and are illustrated with an application to interpersonal emotion data. Finally, the presented hierarchy and model selection strategies are related to corresponding work by Kiers (1991) for principal component models for three-way three-mode real-valued data.

Keywords

three-way three-mode databinary datahierarchical classesmultiway analysismodel selection
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 3 , September 2005 , pp. 461 - 480
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/2000/02 and PDM/03/074). Furthermore, the authors are obliged to Kaatje Bollaerts and the three anonymous reviewers for useful comments on an earlier version of this paper.

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