Null models for species co-occurrence patterns: assessing bias and minimum iteration number for the sequential swap Veiko Lehsten and Peter Harmand Lehsten, V. and Harmand P. 2006. Null models for species co-occurrence patterns: assessing bias and minimum iteration number for the sequential swap.  Ecography 29: 786  792. 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 pB /0.05 and 2 at the pB /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. V. Lehsten (veiko.lehsten@uni-oldenburg.de), Landscape Ecology Group, Univ. of Oldenburg (Fak V), DE-26111 Oldenburg, Germany.  P. Harmand, Inst. of Mathematics, Univ. of Oldenburg, DE-26111 Oldenburg, Germany. Analysing co-occurrence data has become a common practice in ecology to study the community structure within single observations (Gotelli et al. 1987) as well as to verify general ecological theories by using meta- analysis of co-occurrence matrices (Gotelli and McCabe 2002). All these analyses require a randomisation of the observed data, i.e. (0, 1)-matrices, to which the observed pattern is compared. Although a number of different null models is used to test different ecological hypotheses (Gotelli (2000) compares nine different null models), most authors use a variant of the null model proposed by Connor and Simberloff (1979). It retains row and column sums simultaneously to incorporate site effects such as island size as well as rarity of species to account for species dependent characteristics such as niche breadth. Connor and Simberloff (1979) also used a third constraint by restricting species occurrences to those islands for which the total species richness fells within the range occupied by the species. The basic assumption of the null model methodology is, that if the observed co-occurrence matrix differs by much with respect to a certain pattern from the total set of unique matrices, Accepted 31 July 2006 Copyright # ECOGRAPHY 2006 ISSN 0906-7590 ECOGRAPHY 29: 786  792, 2006 786 ECOGRAPHY 29:5 (2006)