Ward's Hierarchical Clustering Method: Clustering Criterion and Agglomerative Algorithm

In the literature and in software packages there is confusion in regard to what is termed the Ward hierarchical clustering method. This relates to any and possibly all of the following: (i) input dissimilarities, whether squared or not; (ii) output dendrogram heights and whether or not their square root is used; and (iii) there is a subtle but important difference that we have found in the loop structure of the stepwise dissimilarity-based agglomerative algorithm. Our main objective in this work is to warn users of hierarchical clustering about this, to raise awareness about these distinctions or differences, and to urge users to check what their favorite software package is doing. In R, the function hclust of stats with the method "ward"option produces results that correspond to a Ward method (Ward