Summary
A method is proposed for the simultaneous hierarchical clustering of row, column, and block elements of a three-way three-mode data matrix. The procedure generalizes the two-mode error-variance approach (Eckes & Orlik, 1993) to the three-mode case. At each step of the agglomerative process, the algorithm merges those clusters whose fusion results in the smallest possible increase in an internal heterogeneity measure. Optionally, the procedure yields an overlapping cluster solution by assigning further row and/or column and/or block elements to a given number of clusters. An application to a data set drawn from object perception research illustrates the approach. Finally, several indications of three-mode clustering are discussed.
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References
ARABIE, P., CARROLL, J.D., and DESARBO, W.S. (1987): Three-Way Scaling and Clustering. Sage, Newbury Park.
ARABIE, P., and HUBERT, L.J. (1992): Combinatorial Data Analysis. Annual Review of Psychology, 43, 169–203.
BEM, D.J. (1983): Constructing a Theory of the Triple Typology: Some (Second) Thoughts on Nomothetic and Idiographic Approaches to Personality. Journal of Personality, 51, 566–577.
CARROLL, J.D., and ARABIE, P. (1980): Multidimensional Scaling. Annual Review of Psychology, 31, 607–649.
CARROLL, J.D., and ARABIE, P. (1983): Indclus: An Individual Differences Generalization of the Adclus Model and the Mapclus Algorithm. Psychometrika, 48, 157–169.
CARROLL, J.D., CLARK, L.A., and DESARBO, W.S. (1984): The Representation of Three-Way Proximity Data By Single and Multiple Tree Structure Models. Journal of Classification, 1, 25–74.
CATTELL, R.B. (1946): The Description and Measurement of Personality. World Book, New York.
CATTELL, R.B. (1952): The Three Basic Factor-Analytic Research Designs—Their Interrelations and Derivatives. Psychological Bulletin, 49, 499–520.
CATTELL, R.B. (1966): The Data Box: Its Ordering of Total Resources in Terms of Possible Relational Systems. In: R.B. Cattell (ed.): Handbook of Multivariate Experimental Psychology. Rand McNally, Chicago, 67–128.
DE SOETE, G., and CARROLL, J.D. (1989): Ultrametric Tree Representations of Three-Way Three-Mode Data. In: R. Coppi and S. Bolasco (eds.): Multiway Data Analysis. North-Holland, New York, 415–426.
ECKES, T. (1993): Multimodale Clusteranalyse: Konzepte, Modelle, Anwendungen. In: L. Montada (ed.): Bericht über den 38. Kongreßder Deutschen Gesellschaft für Psychologie in Trier 1992 (Vol. 2). Hogrefe, Göttingen, 166–176.
ECKES, T., and HASSEBRAUCK, M. (1993): Multimodale Analysen in der physischen Attraktivitäts-Forschung. In: M. Hassebrauck and R. Niketta (eds.): Physische Attraktivität. Hogrefe, Göttingen, 95–121.
ECKES, T., and ORLIK, P. (1993): An Error Variance Approach to Two-Mode Hierarchical Clustering. Journal of Classification, 10, 51–74.
EVERITT, B.S. (1979): Unresolved Problems in Cluster Analysis. Biometrics, 35, 169–181.
HARSHMAN, R.A., and LUNDY, M.E. (1984): The PARAFAC Model for Three-Way Factor Analysis and Multidimensional Scaling. In: H.G. Law, C.W. Snyder, J.A. Hattie and R.P. McDonald (eds.): Research Methods for Multimode Data Analysis. Praeger, New York, 122–215.
KIERS, H.A.L. (1991): Hierarchical Relations Among Three-Way Methods. Psychome-trika, 56, 449–470.
KROLAK-SCHWERDT, S. (1991): Modelle der dreimodalen Faktorenanalyse. Lang, Frankfurt/Main.
KROONENBERG, P.M., and DE LEEUW, J. (1980): Principal Component Analysis of Three-Mode Data By Means of Alternating Least Squares Algorithms. Psychometrika, 45, 69–97.
MILLIGAN, G.W., and COOPER, M.C. (1985): An Examination of Procedures for Determining the Number of Clusters in a Data Set. Psychometrika, 50, 159–179.
MILLIGAN, G.W., and COOPER, M.C. (1988): A Study of Standardization of Variables in Cluster Analysis. Journal of Classification, 5, 181–204.
ORLIK, P. (1965): Eine Modellstudie zur Psychophysik des Polaritätsproiils. Zeitschrift für Experimentelle und Angewandte Psychologie, 12, 615–647.
OSGOOD, C.E., SUCI, G.J., and TANNENBAUM, P.H. (1957): The Measurement of Meaning. University of Illinois Press, Urbana.
SNYDER, C.W., LAW, H.G., and HATTIE, J.A. (1984): Overview of Multimode Analytic Methods. In: H.G. Law, C.W. Snyder, J.A. Hattie and R.P. McDonald (eds.): Research Methods for Multimode Data Analysis. Praeger, New York, 2–35.
TUCKER, L.R. (1964): The Extension of Factor Analysis to Three-Dimensional Matrices. In: N. Frederiksen and H. Gulliksen (eds.): Contributions to Mathematical Psychology. Holt, Rinehart and Winston, New York, 109–127.
TUCKER, L.R. (1966): Some Mathematical Notes on Three-Mode Factor Analysis. Psychometrika, 31, 279–311.
WARD, J.H. (1963): Hierarchical Grouping to Optimize an Objective Function. Journal of the American Statistical Association, 58, 236–244.
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Eckes, T., Orlik, P. (1994). Three-Mode Hierarchical Cluster Analysis of Three-Way Three-Mode Data. In: Bock, HH., Lenski, W., Richter, M.M. (eds) Information Systems and Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46808-7_19
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DOI: https://doi.org/10.1007/978-3-642-46808-7_19
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