The limited value of cophenetic correlation as a clustering criterion

doi:10.1016/0031-3203(78)90038-9

Pattern Recognition

Volume 10, Issue 4, 1978, Pages 287-295

https://doi.org/10.1016/0031-3203(78)90038-9 Get rights and content

Abstract

A common procedure for evaluating hierarchical cluster techniques is to compare the input data, in terms of for example a matrix of similarities or dissimilarities, with the output hierarchy expressed in matrix form. If an ordinary product-moment correlation is used for this comparison, the technique is known as that of cophenetic correlations, frequently used by numerical taxonomists. A high correlation between the input similarities and the output dendrogram has been regarded as a criterion of a successful classification. This paper contains a Monte Carlo study of the characteristics of the cophenetic correlation and a related measure of agreement which have been both interpreted in terms of generalized variance for some different hierarchical cluster algorithms. The generalized variance criterion chosen for this study is Wilk's lambda, whose sampling distribution under the null hypothesis of identical group centroids is used in this context to define the degree of separation between clusters. Thus, a probabilistic approach is introduced into the evaluation procedure. With the above definition of presence of clusters, use of the cophenetic correlation and related measures of agreement as criteria of goodness-of-fit is shown to be quite misleading in most cases. This is due to their large variability for low separation of clusters.

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The cophenetic correlation coefficient c based on Pearson's correlation (Legendre and Legendre, 1998; Sokal and Rohlf, 1962) is currently the most commonly used approach (e.g. Blackburn et al., 2005; Kreft and Jetz, 2010; May, 1999) for assessing the most faithful dendrogram (i.e. the dendrogram that preserves most faithfully D). However, it is known that c may not always be a reliable measure of distortion generated by algorithms (Farris, 1969; Holgersson, 1978; Mérigot et al., 2010). For instance, Mérigot et al. (2010) raised three criticisms on the reliability of information given by c measure: (i) it is only a measure of intensity of the monotonic linear relationship between D and U; (ii) it is sensitive to extreme values; and (iii) a c close to 1 indicates a perfect correspondence of D and U whereas the match between the two matrices can in fact be weak.
Investigation of patterns in beta diversity has received increased attention over the last years particularly in light of new ecological theories such as the metapopulation paradigm and metacommunity theory. Traditionally, beta diversity patterns can be described by cluster analysis (i.e. dendrograms) that enables the classification of samples. Clustering algorithms define the structure of dendrograms, consequently assessing their performance is crucial. A common, although not always appropriate approach for assessing algorithm suitability is the cophenetic correlation coefficient c. Alternatively the 2-norm has been recently proposed as an increasingly informative method for evaluating the distortion engendered by clustering algorithms. In the present work, the 2-norm is applied for the first time on field data and is compared with the cophenetic correlation coefficient using a set of 105 pairwise combinations of 7 clustering methods (e.g. UPGMA) and 15 (dis)similarity/distance indices (e.g. Jaccard index). In contrast to the 2-norm, cophenetic correlation coefficient does not provide a clear indication on the efficiency of the clustering algorithms for all combinations. The two approaches were not always in agreement in the choice of the most faithful algorithm. Additionally, the 2-norm revealed that UPGMA is the most efficient clustering algorithm and Ward's the least. The present results suggest that goodness-of-fit measures such as the 2-norm should be applied prior to clustering analyses for reliable beta diversity measures.
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Average linkage is known to be more robust than other standard methods such as single, complete linkage or Ward's method (Everitt, Landau, & Leese, 2001, pp. 59–64), and it was successfully used by Huckvale (2007) in a study whose principles are very close to our own, although Huckvale concentrated on clustering accents only, not vowels. Incidentally, among standard agglomeration methods, average linkage is also known to yield the highest cophenetic correlation coefficients (Holgersson, 1978, p. 291). The cophenetic correlation coefficient measures the correlation (the closer to 1, the stronger the positive correlation) between the original distance matrix and the corresponding distances as they appear in the tree.
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The authors provide a discussion of CA that seeks to introduce the various methods, issues, and considerations to the researcher who is largely unfamiliar with CA. A conceptual understanding of CA is guided through breaking down CA into a series of steps and issues to consider including composition of the dataset, selection of variables, decisions about standardizing variables, selecting a measure of association, selecting a clustering method, determining the number of clusters, and interpretation.
Because the range of CA methods is diverse, and because the steps within each method are so varied, an attempt to offer a complete “how-to” process in a single article is imprudent. Rather, the novice reader will be able to use this article as a starting point for conducting his or her own particular CA study.
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In essence, CPCC appears ineffective in suggesting optimal partitioning scheme for the Indiana dataset. Detailed discussion on the performance of CPCC can be found in Holgersson (1978). The average silhouette width (ASW), which has a feasible range from −1 to +1, varied generally within a narrow range of 0.31–0.46 for the Indiana data over the variety of clustering options considered.
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Lattice-based similarity measures between ordered trees
1982, Journal of Mathematical Psychology
A clustering algorithm has recently been developed by Reitman and Rueter to express both the structure of chunking in multi-trial free recall and the order of chunk production. The resulting ordered trees differ from ordinary rooted trees in that the elements of a chunk, at any level, may be restricted to a specific ordering. In order to make comparisons of long-term memory structures between subjects, a measure of the similarity between trees is needed. Previously developed similarity measures are shown to be inadequate for ordered trees. Lattice theory is used to generate new similarity measures suited to these richer structures. First, ordered trees are shown to form a nonmodular, graded lattice. Then, moves through this lattice are defined and used to produce several distance measures. These new measures are compared both to each other, and to existing measures, by examining the properties of each measure, and through application to hypothetical trees. The lattice-based measures prove to be theoretically superior, but lack computational ease. The general problem of describing paths in a nonmodular lattice is discussed.

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The limited value of cophenetic correlation as a clustering criterion

Abstract

Cluster Analysis for Applications

Finding natural cluster

Syst. Zool.

Pattern Classification and Scene Analysis

On the cophenetic correlation coefficient

Syst. Zool.

On grouping for maximal homogenity

J. Am. Statist. Assoc.

On some invariant criteria for grouping data

J. Am. Statist. Assoc.

Statistical methods for comparing different multivariate analyses of the same data

Classification problems

Bull. Int. Statist. Inst.

Goodness-of-fit criteria in cluster analysis and their empirical distributions

Representation of similarity matrices by trees

J. Am. Statist. Assoc.

An interpretation of two measures of agreement between the dissimilarity matrix and the resulting hierarchy in terms of generalized variance for some different cluster algorithms