A cophenetic correlation coefficient for Tocher's method

Abstract

-The objective of this work was to propose a way of using the Tocher's method of clustering to obtain a matrix similar to the cophenetic one obtained for hierarchical methods, which would allow the calculation of a cophenetic correlation. To illustrate the obtention of the proposed cophenetic matrix, we used two dissimilarity matrices -one obtained with the generalized squared Mahalanobis distance and the other with the Euclidean distance -between 17 garlic cultivars, based on six morphological characters. Basically, the proposal for obtaining the cophenetic matrix was to use the average distances within and between clusters, after performing the clustering. A function in R language was proposed to compute the cophenetic matrix for Tocher's method. The empirical distribution of this correlation coefficient was briefly studied. For both dissimilarity measures, the values of cophenetic correlation obtained for the Tocher's method were higher than those obtained with the hierarchical methods (Ward's algorithm and average linkage -UPGMA). Comparisons between the clustering made with the agglomerative hierarchical methods and with the Tocher's method can be performed using a criterion in common: the correlation between matrices of original and cophenetic distances.Index terms: cluster analysis, optimization methods, clustering consistency. Um coeficiente de correlação cofenética para o método de TocherResumo -O objetivo deste trabalho foi propor uma forma de uso do método de Tocher para obtenção de uma matriz análoga à matriz cofenética obtida para métodos hierárquicos, o que permitiria o cálculo de uma correlação cofenética. Para ilustrar a obtenção da matriz cofenética proposta, foram utilizadas duas matrizes de dissimilaridade -uma obtida com a distância quadrada generalizada de Mahalanobis e outra com a distância euclidiana -entre dezessete cultivares de alho, com base em seis caracteres morfológicos. Basicamente, a proposta para obtenção da matriz cofenética foi a de usar, após a realização do agrupamento, as distâncias médias intra e intergrupos. Uma função em linguagem R foi proposta para computar a matriz cofenética para o método de Tocher. A distribuição empírica desse coeficiente de correlação foi estudada de forma sucinta. Para as duas medidas de dissimilaridade, os valores do coeficiente de correlação cofenética obtidos para o método de Tocher foram superiores aos obtidos com os métodos hierárquicos (algoritmo de Ward e ligação média -UPGMA). Comparações entre agrupamentos feitos com os métodos hierárquicos aglomerativos e com o método de Tocher podem ser realizadas com o uso de um critério em comum: o da correlação entre matrizes de distâncias cofenéticas e originais.Termos para indexação: análise de agrupamento, métodos de otimização, consistência do agrupamento.

Agro-morphological characterization and genetic divergence assessment in bush snap bean genotypes1

“…Using the squared Mahalanobis distance (D2ii'), the 29 genotypes were grouped into four clusters, either by the Tocher optimization method (Table 6) or the UPGMA method (Figure 1). This equivalence in clustering genotypes by the methods is an evidence of a high consistency in the results, also supported by the high estimates of the cophenetic correlation coefficient (Tocher's method: 0.83; UPGMA: 0.72), both significant (p < 0.001) by (Silva & Dias 2013). These results indicate that there is a high genetic divergence between some of the genotypes assessed, because, although the great majority of the genotypes are grouped with less than 50 units of distance (e.g., clusters I), the clusters diverged from each other in distances of about 100 units or more.…”

Self-organized map: the new aproach for study of genetic divergence in kale

“…The representation of the matrix by the dendrogram (Fig 3) also showed a high co-optic correlation, which shows high agreement in the grouping performed and the dissimilarity matrix [24]. This reinforces the reliability of subsequent crossing to be established based on the composition of the clusters using the RNAs.…”

Genetic dissimilarity and definition of recombination clusters among green corn half-sib progenies

“…The cophenetic matrix of the Tocher's method was obtained using the methodology developed by Silva and Dias (2013), who propose to acquire a matrix similar to the cophenetic one from the hierarchical methods, through the intra and intergroup average distances. The cophenetic correlation coefficients were obtained from the original distances and cophenetic matrices.…”