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.
Similar content being viewed by others
References
Akaike H. (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov B.N., Csaki F. (eds) Second International Symposium on Information Theory. Academiai Kiado, Budapest, pp 267–281
Bozdogan H. (1987) Model selection and Akaike’s information criterion (AIC): The general theory and its analytical extensions. Psychometrika 52:345–370
Bozdogan H. (2000) Akaike’s information criterion and recent developments in informational complexity. Journal of Mathematical Psychology 44:62–91
Cattell R.B. (1966) The meaning and strategic use of factor analysis. In: Cattell R.B. (ed) Handbook of Multivariate Experimental Psychology. Rand McNally, Chicago, pp 174–243
Ceulemans E., Van Mechelen I. (2004) Tucker2 hierarchical classes analysis. Psychometrika 69:413–433
Ceulemans E., Van Mechelen I., Leenen I. (2003) Tucker3 hierarchical classes analysis. Psychometrika 68:413–433
De Boeck P., Rosenberg S. (1988) Hierarchical classes: model and data analysis. Psychometrika 53:361–381
Fowlkes E.B., Freeny A.E., Landwehr J.M. (1988) Evaluating logistic models for large contingency tables. Journal of the American Statistical Association 83:611–622
Haggard E.A. (1958) Intraclass Correlation and the Analysis of Variance. Dryden, New York
Kiers H.A.L. (1991) Hierarchical relations among three-way methods. Psychometrika 56:449–470
Kiers H.A.L. (2000) Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics 14:105–122
Kim K.H. (1982) Boolean Matrix Theory. Marcel Dekker, New York
Kirk R.E. (1982) Experimental design: Procedures for the behavioral sciences (2nd edition). Brooks/Cole, Belmont, CA
Kroonenberg P.M. (1983) Three-mode Principal Component Analysis: Theory and Applications. DSWO, Leiden
Kroonenberg P.M., Oort F.J. (2003) Three-mode analysis of multimode covariance matrices. British Journal of Mathematical and Statistical Psychology 56:305–336
Kroonenberg P.M., Van der Voort T.H.A. (1987) Multiplicatieve decompositie van interacties bij oordelen over de werkelijkheidswaarde van televisiefilms [Multiplicative decomposition of interactions for judgements of realism of television films]. Kwantitatieve Methoden 8:117–144
Kuppens P., Van Mechelen I., Smits D.J.M., De Boeck P., Ceulemans E. (2005) Individual differences in appraisal and emotion: The case of anger and irritation, submitted
Leenen I., Van Mechelen I. (2001) An evaluation of two algorithms for hierarchical classes analysis. Journal of Classification 18:57–80
Leenen I., Van Mechelen I., De Boeck P., Rosenberg S. (1999) INDCLAS: A three-way hierarchical classes model. Psychometrika 64:9–24
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 53:1–16
Van Mechelen I. (1991) Symptom and diagnosis inference based on implicit theories of psychopathology: A review. Cahiers de Psychologie Cognitive 11:155–171
Van Mechelen I., De Boeck P. (1989) Implicit taxonomy in psychiatric diagnosis: A case study. Journal of Social and Clinical Psychology 8:276–287
Van Mechelen I., De Boeck P., Rosenberg S. (1995) The conjunctive model of hierarchical classes. Psychometrika 60:505–521
Vansteelandt K., Van Mechelen I. (1998) Individual differences in situation-behavior profiles: A triple typology model. Journal of Personality and Social Psychology 75:751–765
Wilks S.S. (1938) The large sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics 9:60–62
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
Ceulemans, E., Van Mechelen, I. Hierarchical classes models for three-way three-mode binary data: interrelations and model selection. Psychometrika 70, 461–480 (2005). https://doi.org/10.1007/s11336-003-1067-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11336-003-1067-3