Peter Orlik

An error variance approach to two-mode hierarchical clustering

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.

TRIPAT: a Model for Analyzing Three-Mode Binary Data

A discrete, categorical model is presented for three-mode (conditions by objects by attributes) data arrays with binary entries x ijk ∈ {0, 1}. Basically, the model attempts a simultaneous classification of the entities or elements of the three modes in a number of common clusters. Clusters are defined by three-mode submatrices of maximum size with entries x ijk = 1. In performing a discrete representation of the data structure, the model may be classified as a non-hierarchical clustering procedure. It involves a reorganization of the data array such that the final clustering solution is interpreted directly on the data, and it allows for overlapping as well as nonoverlapping clusters. The method is similar to three-mode component models such as CANDECOMP and SUMMAX in the model function to predict the data. An application concerning recall data in a study of social perception is provided.

An Agglomerative Method for Two-Mode Hierarchical Clustering

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 (bi-cluster) is defined as the union of subsets of the respective modes. At each step of the agglomerative process, the algorithm merges two bi-clusters whose fusion results in the minimum increase in an internal heterogeneity measure. This measure takes into account both the variance within a bi-cluster and its elevation defined as the squared deviation of its mean from the maximum entry in the original matrix. Two applications concerning brand-switching data and gender subtype-situation matching data are discussed.

Three-Mode Hierarchical Cluster Analysis of Three-Way Three-Mode Data

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.

A Regression Analytic Modification of Ward's Method: A Contribution to the Relation between Cluster Analysis and Factor Analysis

A regression analytic modification of the minimum variance method (Ward’s method) is outlined. In the proposed method the within-cluster sums of squares are partitioned into the proportion accounted for by the cluster centers and the residual variation. The procedure consists of fusing the two clusters that minimize the residual variation not predicted by the centers. The method allows for a combination of clustering and factor analysis in order to determine the kind of properties that govern the relationships between the clusters.