Some extensions of Johnson's hierarchical clustering algorithms

Considerable attention has been given in the psychological literature to techniques of data reduction that partition a set of objects into optimally homogeneous groups. This paper is an attempt to extend the hierarchical partitioning algorithms proposed by Johnson and to emphasize a general connection between these clustering procedures and the mathematical theory of lattices. A goodness-of-fit statistic is first proposed that is invariant under monotone increasing transformations of the basic similarity matrix. This statistic is then applied to three illustrative hierarchical clusterings: two obtained by the Johnson algorithms and one obtained by an algorithm that produces the same chain under hypermonotone increasing transformations of the similarity measures.     

Least squares estimates and optimal classification

A simple algebraic development is given showing that criterion estimates derived by usual multiple regression procedures are optimal for personnel classification. It is also shown that, for any assignment of men to jobs, the sum of the multiple regression criterion estimates will equal the sum of the actual criterion scores.The opinions expressed are those of the author and are not to be construed as reflecting official Department of the Army policy.     

Colour specificity in episodic and in perceptual object recognition with enhanced colour impact

In three experiments we investigated the perceptual specificity of explicit (old-new object recognition) and implicit memory (word-picture matching) for colour. In order to enhance the impact of colour on processing, we increased the number of colours per object and we impaired shape information. We presented multicoloured pictures (Experiment 1), blurred and partially occluded pictures (Experiment 2) and coloured line drawings in visual noise (Experiment 3). Experiments 1 and 2 had an intentional study phase; the study phase of Experiment 3 was an incidental colour or category naming task. Changing colour from study to test always had negative effects on episodic recognition although colour was irrelevant. In contrast, in the matching task old pictures were generally matched faster than new ones independent of their colour congruence. In Experiment 3, an additional small advantage of congruent colours and of semantic processing occurred. We conclude that two different memory representations contribute to these tasks. Changes of an achromatic, more abstract representation, that is used in normal object recognition, and a representation of the specific exemplar that includes colour. This latter token is used in episodic recognition as well as in unusual perceptual tasks (Experiment 3).     

Simplified regression patterns for classification

A transformation of a set of partial regression equations to a potentially simple form is shown to be permissible for personnel classification purposes when all individuals are assigned to jobs. Simplification is achieved by maximizing the number of near zero partial regression weights.     

What people believe about memory

Two representative samples of adult Norwegians (n=2000) were asked a set of general and specific questions regarding their beliefs and opinions about human memory. The results indicate that on many questions, such as time of the earliest memories, inhibiting effects of collaboration, and memory for dramatic versus ordinary events, the views of the general public concurred with current research findings, and people in general had realistic views about their own memory performance. On other questions, such as the reliability of olfactory as compared with visual and auditory memory, the memory of small children in comparison with that of adults, the likelihood of repression of adult traumatic memories, and on more general questions such as the possibility of training memory and the capacity limitations of long-term memory, a large proportion of the participants expressed views that are less supported by scientific evidence. Implications of these findings are briefly discussed.     

Self-pluralism and dissociation

The study examined the relationships among self-pluralism and dissociative experiences. 100 students were assessed using the Self-pluralism Scale and the Dissociative Experiences Scale. Scores on self-pluralism were positively correlated with others: overall Dissociation (r=.50), Absorption (r=.54), Depersonalization (r=.43) and Amnesia (r = .28).     

Evaluating the conformity of sociometric measurements

The problem of comparing two sociometric matrices, as originally discussed by Katz and Powell in the early 1950's, is reconsidered and generalized using a different inference model. In particular, the proposed indices of conformity are justified by a regression argument similar to the one used by Somers in presenting his well-known measures of asymmetric ordinal association. A permutation distribution and an associated significance test are developed for the specific hypothesis of no conformity reinterpreted as a random matching of the rows and (simultaneously) the columns of one sociometric matrix to the rows and columns of a second. The approximate significance tests that are presented and illustrated with a simple numerical example are based on the first two moments of the permutation distribution, or alternatively, on a random sample from the complete distribution.Partial support for the research of the first author was provided by the National Science Foundation through SOC 75-07860. Equal authorship is implied. The work was done when the first author was at the University of Wisconsin.     

Generalized concordance

Based on a simple nonparametric procedure for comparing two proximity matrices, a measure of concordance is introduced that is appropriate whenK independent proximity matrices are available. In addition to the development of a general concept of concordance and specific techniques for its evaluation within and between the subsets of a partition of theK matrices, several methods are also suggested for comparing and/or for fitting a particular structure to the given data. Finally, brief indications are provided as to how the well-known notion of concordance forK rank orders can be included within the more general framework.Partial support for this research was supplied by the National Science Foundation through SOC-77-28227.     

A heuristic method for the comparison of related structures

By extending a technique for testing the difference between two dependent correlations developed by Wolfe, a strategy is proposed in a more general matrix context for evaluating a variety of data analysis schemes that are supposed to clarify the structure underlying a set of proximity measures. In the applications considered, a data analysis scheme is assumed to reconstruct in matrix form the given data set (represented as a proximity matrix) based on some specific model or procedure. Thus, an evaluation of the adequacy of reconstruction can be developed by comparing matrices, one containing the original proximities and the second containing the reconstructed values. Possible applications in multidimensional scaling, clustering, and related contexts are emphasized using four broad categories: (a) Given two different reconstructions based on a single data set, does either represent the data significantly better than the other? (b) Given two reconstructions based on a single data set using two different procedures (or possibly, two distinct data sets and a common method), is either reconstruction significantly closer to a particular theoretical structure that is assumed to underlie the data (where the latter is also represented in matrix form)? (c) Given two theoretical structures and one reconstruction based on a single data set, does either represent the reconstruction better than the other? (d) Given a single reconstruction based on one data set, is the information present in the data accounted for satisfactorily by the reconstruction? In all cases, these tasks can be approached by a nonparametric procedure that assesses the similarity in pattern between two appropriately defined matrices. The latter are obtained from the original data, the reconstructions, and/or the theoretical structures. Finally, two numerical examples are given to illustrate the more general discussion.     

A set-theoretical approach to the problem of hierarchical clustering

A set-theoretical formalization is developed for the problem of generating hierarchically organized collections of subsets, or to use a phrase common in the applied substantive literature, for the problem of hierarchical clustering. A number of terms are introduced to characterize those clustering methods that attempt to limit the size of the overlap between each pair of subsets constructed at a specific “compactness” level. Several examples, motivated primarily by graph theory, are discussed briefly to illustrate the various set-theoretical concepts presented.