Are probability estimates from the permutation model of Mantel's test stable?

Abstract

Matrix comparison tests (i.e., Mantel's test or quadratic assignment) are employed with increasing frequency to measure the concordance between genetic, behavioural, morphological, ecological, and geographic distances. Such tests compare an observed measure of matrix association with a null distribution derived from a randomly generated subset of all possible permutations of one of the original matrices. Typically, 500–2000 randomly permutated matrices are used to generate the statistical distribution and thereby estimate the probability of obtaining an observed association between two matrices. We demonstrate that a considerable error (i.e., 5–6%) may be associated with probability estimates based on such low numbers of permutations. To ensure the stability of the probability estimates (i.e., increase the reliability of the test), we recommend the use of a minimum of 10 000 permutations, and 100 000 permutations if the observed probability approaches a critical significance value (e.g., 0.05).