Null matrices and the analysis of species co-occurrences

Abstract

Patterns in species occurrences on islands have been analyzed by several authors. At issue is the number of non-occurring pairs of species (also known as checkerboards). Previous authors have suggested that if the number of checkerboards differs from what is expected by chance, then island communities might have been structured by competition. Investigators have pursued this problem by first generating random (or null) matrices and then testing a metric derived from the collection of null matrices against the metric calculated from the actual species co-occurrence matrix. The random matrices were constrained by requiring the number of species on each island, and the number of islands on which each species occurred to be equal to their observed values. We show that results from previous studies are generally flawed. We present a fast, efficient algorithm to generate null matrices for any set of fixed row and column sums, and propose a modification of a previously proposed metric as a test statistic. We evaluated the efficacy of our construction method for null creation and our metric using incidence matrices from the avifauna of Vanuatu (formerly New Hebrides).

Analysis of avian communities in Lake Guri, Venezuela, using multiple assembly rule models

“…The validity of "sequential swap" algorithms for generating random null matrices (as used in this study) has been challenged by several authors Manly and Sanderson 2002;Sanderson et al 1998). It is contended that swap algorithms inherently incorporate structure from the original test matrix and are therefore prone to Type II statistical errors (false negatives).…”

Species co‐occurrence patterns and dietary resource competition in primates

“…Rigorous statistical analyses to properly test whether species co-occurrence patterns are nonrandom have been improved during the past 15-20 years (Atmar and Patterson, 1993;Colwell, 1999;Gotelli and Entsminger, 2009). During this time, a number of studies have added support to Diamond's (1975) hypotheses (Stone and Roberts, 1990;Manly, 1995;Gotelli et al, 1997;Sanderson et al, 1998;Gotelli, 2000;Gotelli and McCabe, 2002), yet the mechanisms producing these patterns of species distributions are still not always easy to discern.…”

Measuring Industry Relatedness and Corporate Coherence

“…Such kind of matrices have been used in psychometry (Snijders, 1991) and spatial econometrics (Anselin, 1988) but they have been most intensively studied in bio-geography and ecology (Connor and Simberloff, 1979;Roberts and Stone, 1990;Sanderson et al, 1998;Zaman and Simberloff, 2002;Gotelli, 2001, among others). The ecological problem deals with the detection of interactions among species (e.g.…”

“…For large and sparse matrices the fire-and-place algorithm usually reaches a locked-in state. Sanderson et al (1998) propose a modification of the well-known knight's tour algorithm in order to produce a sequence of matrices in S (û,v) such that each matrix is produced once and only once. Null-matrices are generated just iterating the fire-and-place algorithm untill a locked-in state is reached.…”