A method for testing the distinctness of clusters: A test of the disjunction of two clusters in Euclidean space as measured by their overlap

A method is described for testing the distinctness of two clusters in Euclidean space. One first calculates the projections, q,of the N₁ and N₂ members of the clusters onto the line joining the cluster centroids. From the distributions of qan index of disjunction, W,is calculated, which corresponds to an index of overlap, V_G.The quantity W√(N₁+N₂)is distributed as noncentral tsubject to assumptions on the multivariate normal distribution of the clusters. This allows a test of whether the observed disjunction is significantly greater than a chosen figure, which is equivalent to testing whether the overlap of the clusters is significantly less than a corresponding value of V_G.Two clusters that appear distinct may be produced simply by the partitioning of a homogeneous swarm into two contiguous regions. Provided that the clusters form a dichotomy in a dendrogram, and that the clustering method yields geometrically convex clusters, a conservative test of this situation can be derived by determining the excess of Wover the value expected for a rectangular distribution.