On goodness-of-fit measure for dendrogram-based analyses

Clustering methods are widely used tools in many aspects of science, such as ecology, medicine, or even market research, that commonly deal with dendrogram-based analyses. In such analyses, for a given initial dissimilarity matrix, the resulting dendrogram may strongly vary according to the selected clustering methods. However, numerous dendrogram-based analyses require adequate measurement for assessing of which of the clustering methods preserves most faithfully the initial dissimilarity matrix. While cophenetic correlation coefficient-based measures have been widely used for this purpose, we emphasize here that it is not always a suitable approach. We thus propose a measure based on a matrix norm, the 2-norm, to adequately check which of the resulting ultrametric distance matrices related to the dendrograms is the closest to the initial dissimilarity matrix. In addition, we also propose an objective way to define a benchmark value (threshold value) in order to assess whether the degree of conformity between the ultrametric distance matrix selected and the initial dissimilarity matrix is satisfactory. Our proposal may notably be incorporated within a recently proposed approach that involves the use of clustering methods in environmental science and beyond. In ecology, various functional diversity indices based on clustering species from their functional dissimilarities may benefit from this overall approach.