Ward's Hierarchical Agglomerative Clustering Method: Which Algorithms Implement Ward's Criterion?

The Ward error sum of squares hierarchical clustering method has been very widely used since its first description by Ward in a 1963 publication. It has also been generalized in various ways. Two algorithms are found in the literature and software, both announcing that they implement the Ward clustering method. When applied to the same distance matrix, they produce different results. One algorithm preserves Ward's criterion, the other does not. Our survey work and case studies will be useful for all those involved in developing software for data analysis using Ward's hierarchical clustering method.