Statistical Analysis of Maximally Similar Sets in Ecological Research

Maximally similar sets (MSS) are sets of elements that share a neighborhood in a high-dimensional space defined by symmetric, reflexive similarity relation. Each element of the universe is employed as the kernel of a neighborhood of a given size (number of members), and elements are added to the neighborhood in order of similarity to the current members of the set until the desired neighborhood size is achieved. The set of neighborhoods is then reduced to the set of unique maximally similar sets by eliminating all sets that are permutations of an existing set. Subsequently, the within-MSS variability of attributes associated with the elements is compared to random sets of the same size to estimate the probability of obtaining variability as low as observed. Individual attributes can be compared for effect size by the ratio of within-MSS variability to random set variability, correcting for statistical power as necessary. The analyses performed identify constraints, as opposed to determinants, in the triangular distribution of pair-wise element similarity. In the example given here, the variability in spring temperature, summer temperature, and growing degree days of forest vegetation samples shows the greatest constraint on forest composition of a large set of candidate environmental variables