Calculating minimum discrepancy to assess the nestedness of species assemblages

Nestedness is a pattern whereby species-poor assemblages are composed of subsets of the species occurring in richer assemblages. One of the most commonly used measures of the degree of nestedness for presence-absence matrices is the ‘discrepancy’ metric. A hitherto neglected property of that metric is that it may take several values for a given site-by-species matrix in the presence of ties in the marginal totals. This complicates the quantification of nestedness for the observed presence-absence matrix, as well as the assessment of statistical significance, which is typically achieved through Monte Carlo simulations. A solution to the problem is to calculate the minimum discrepancy using a modified algorithm involving permutations of columns with tied totals.