Some extensions of Johnson's hierarchical clustering algorithms

Considerable attention has been given in the psychological literature to techniques of data reduction that partition a set of objects into optimally homogeneous groups. This paper is an attempt to extend the hierarchical partitioning algorithms proposed by Johnson and to emphasize a general connection between these clustering procedures and the mathematical theory of lattices. A goodness-of-fit statistic is first proposed that is invariant under monotone increasing transformations of the basic similarity matrix. This statistic is then applied to three illustrative hierarchical clusterings: two obtained by the Johnson algorithms and one obtained by an algorithm that produces the same chain under hypermonotone increasing transformations of the similarity measures.