Null models for species co-occurrence patterns: assessing bias and minimum iteration number for the sequential swap

The analysis of co-occurrence matrices is a common practice to evaluate community structure. The observed data are compared with a "null model", a randomised co-occurrence matrix derived from the observation by using a statistic, e.g. the C-score, sensitive to the pattern investigated. The most frequently used algorithm, "sequential swap", has been criticised for not sampling with equal frequencies thereby calling into question the results of earlier analysis. The bias of the "sequential swap" algorithm when used with the C-score was assessed by analysing 291 published presence-absence matrices. In 152 cases, the true p-value differed by > 5% from the p-value generated by an uncorrected "sequential swap". However, the absolute value of the difference was rather small. Out of the 291 matrices, there were only 5 cases in which an incorrect statistical decision would have been reached by using the uncorrected p-value (3 at the p < 0.05 and 2 at the p < 0.01 level), and in all 5 of these cases, the true p-value was close to the significance level. Our results confirm analytical studies of Miklos and Podani which show that the uncorrected swap gives slightly conservative results in tests for competitive segregation. However, the bias is very small and should not distort the ecological interpretation. We also estimated the number of iterations needed for the "sequential swap" to generate accurate p-values. While most authors do not exceed a number of 10(4) iterations, the suggested minimum number of swaps for 29 out of the 291 tested matrices is greater than 10(4). We recommend to use 30 000 "sequential swaps" if the required sample size is not assessed otherwise.