Abstract
A new agglomerative method is proposed for the simultaneous hierarchical clustering of row and column elements of a two-mode data matrix. The procedure yields a nested sequence of partitions of the union of two sets of entities (modes). A two-mode cluster is defined as the union of subsets of the respective modes. 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. This measure takes into account both the variance within the respective cluster and its centroid effect defined as the squared deviation of its mean from the maximum entry in the input matrix. The procedure optionally yields an overlapping cluster solution by assigning further row and/or column elements to clusters existing at a preselected hierarchical level. Applications to real data sets drawn from consumer research concerning brand-switching behavior and from personality research concerning the interaction of behaviors and situations demonstrate the efficacy of the method at revealing the underlying two-mode similarity structure.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
ARABIE, P., and HUBERT, L.J. (1990), “The Bond Energy Algorithm Revisited,”IEEE Transactions on Systems, Man, and Cybernetics, 20, 268–274.
ARABIE, P., HUBERT, L.J., and SCHLEUTERMANN, S. (1990), “Blockmodels from the Bond Energy Approach,”Social Networks, 12, 99–126.
ARABIE, P., SCHLEUTERMANN, S., DAWS, J., and HUBERT, L. (1988), “Marketing Applications of Sequencing and Partitioning of Nonsymmetric and/or Two-Mode Matrices,” inData, Expert Knowledge and Decisions, Eds. W. Gaul, and M. Schader, Berlin: Springer-Verlag, 215–224.
BASS, F.M. (1974), “The Theory of Stochastic Preference and Brand Switching,”Journal of Marketing Research, 11, 1–20.
BASS, F.M., PESSEMIER, E.A., and LEHMANN, D.R. (1972), “An Experimental Study of Relationships Between Attitudes, Brand Preference, and Choice,”Behavioral Science, 17, 532–541.
BENNETT, J.F., and HAYS, W.L. (1960), “Multidimensional Unfolding: Determining the Dimensionality of Ranked Preference Data,”Psychometrika, 25, 27–43.
BOTH, M., and GAUL, W. (1986), “Ein Vergleich zweimodaler Clusteranalyseverfahren,”Methods of Operations Research, 57, 593–605.
CARLSON, K.A. (1972), “A Method for Identifying Homogenous Classes,”Multivariate Behavioral Research, 7, 483–488.
CARROLL, J.D., and ARABIE, P. (1980), “Multidimensional Scaling,”Annual Review of Psychology, 31, 607–649.
COOMBS, C.H. (1950), “Psychological Scaling Without a Unit of Measurement,”Psychological Review, 57, 148–158.
COPPI, R., and BOLASCO, S. (Eds.) (1989),Multiway Data Analysis, New York: North-Holland.
DE BOECK, P., and ROSENBERG, S. (1988), “Hierarchical Classes: Model and Data Analysis,”Psychometrika, 53, 361–381.
DESARBO, W.S. (1982), “GENNCLUS: New Models for General Nonhierarchical Clustering Analysis,”Psychometrika, 47, 449–475.
DESARBO, W.S., and DE SOETE, G. (1984), “On the Use of Hierarchical Clustering for the Analysis of Nonsymmetric Proximities,”Journal of Consumer Research, 11, 601–610.
DESARBO, W.S., and RAO, V.R. (1984), “GENFOLD2: A Set of Models and Algorithms for the GENeral UnFOLDing Analysis of Preference/Dominance Data,”Journal of Classification, 1, 147–186.
DE SOETE, G., DESARBO, W.S., FURNAS, G.W., and CARROLL, J.D. (1984), “The Estimation of Ultrametric and Path Length Trees from Rectangular Proximity Data,”Psychometrika, 49, 289–310.
DUFFY, D.E., and QUIROZ, A.J. (1991), “A Permutation-Based Algorithm for Block Clustering,”Journal of Classification, 8, 65–91.
ECKES, T. (1991), “Bimodale Clusteranalyse: Methoden zur Klassifikation von Elementen zweier Mengen,”Zeitschrift für Experimentelle und Angewandte Psychologie, 38, 201–225.
ECKES, T., and ORLIK, P. (1991), “An Agglomerative Method for Two-Mode Hierarchical Clustering,” inClassification, Data Analysis, and Knowledge Organization, Eds. H.H. Bock, and P. Ihm, Berlin: Springer-Verlag, 3–8.
ENDLER, N.S., and MAGNUSSON, D. (1976), “Toward an Interactional Psychology of Personality,”Psychological Bulletin, 83, 956–974.
ESPEJO, E., and GAUL, W. (1986), “Two-Mode Hierarchical Clustering as an Instrument for Marketing Research,” inClassification as a Tool of Research, Eds. W. Gaul, and M. Schader, Amsterdam: North-Holland, 121–128.
EVERITT, B.S. (1979), “Unresolved Problems in Cluster Analysis,”Biometrics, 35, 169–181.
FREDERIKSEN, N. (1972), “Toward a Taxonomy of Situations,”American Psychologist, 27, 114–123.
FURNAS, G.W. (1980), “Objects and Their Features: The Metric Representation of Two-Class Data,” Unpublished Doctoral Dissertation, Stanford University.
GREENACRE, M.J. (1984),Theory and Applications of Correspondence Analysis, London: Academic Press.
GREENACRE, M.J., and HASTIE, P. (1987), “The Geometric Interpretation of Correspondence Analysis,”Journal of the American Statistical Association, 82, 437–447.
HARTIGAN, J.A. (1975),Clustering Algorithms, New York: Wiley.
HARTIGAN, J.A. (1976), “Modal Blocks in Definition of West Coast Mammals,”Systematic Zoology, 25, 149–160.
LAW, H.G., SNYDER, C.W., HATTIE, J.A., and McDONALD, R.P. (Eds.) (1984),Research Methods for Multimode Data Analysis, New York: Praeger.
McCORMICK, W.T., SCHWEITZER, P.J., and WHITE, T.W. (1972), “Problem Decomposition and Data Reorganization by a Clustering Technique,”Operations Research, 20, 993–1009.
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.
PRICE, R.H. (1974), “The Taxonomic Classification of Behaviors and Situations and the Problem of Behavior-Environment Congruence,”Human Relations, 27, 567–585.
PRICE, R.H., and BOUFFARD, D.L. (1974), “Behavioral Appropriateness and Situational Constraint as Dimensions of Social Behavior,”Journal of Personality and Social Psychology, 30, 579–586.
SCHLEUTERMANN, S., ARABIE, P., HUBERT, L.J., and BRONSARD, F. (1990), “Some Algorithms for “Bond Energy” Data Analysis, Including Simulated Annealing,” inKnowledge, Data and Computer-Assisted Decisions, Eds. M. Schader, and W. Gaul, Berlin: Springer-Verlag, 139–152.
SHEPARD, R.N., and ARABIE, P. (1979), “Additive Clustering: Representation of Similarities as Combinations of Discrete Overlapping Properties,”Psychological Review, 86, 87–123.
SNYDER, M., and ICKES, W. (1985), “Personality and Social Behavior,” inHandbook of Social Psychology, Vol. 2, Eds. G. Lindzey, and E. Aronoson, New York: Random House, 883–947.
TUCKER, L.R. (1964), “The Extension of Factor Analysis to Three-Dimensional Matrices,” inContributions to Mathematical Psychology, Eds. N. Frederiksen, and H. Gulliksen, New York: Holt, Rinehart and Winston, 109–127.
WARD, J.H. (1963), “Hierarchical Grouping to Optimize an Objective Function,”Journal of the American Statistical Association, 58, 236–244.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eckes, T., Orlik, P. An error variance approach to two-mode hierarchical clustering. Journal of Classification 10, 51–74 (1993). https://doi.org/10.1007/BF02638453
Issue Date:
DOI: https://doi.org/10.1007/BF02638453