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Inferring species interactions from co-occurrence data is one of the most controversial tasks in community ecology. One difficulty is that a single pairwise interaction can ripple through an ecological network and produce surprising indirect consequences. For example, the negative correlation between two competing species can be reversed in the presence of a third species that outcompetes both of them. Here, I apply models from statistical physics, called Markov networks or Markov random fields, that can predict the direct and indirect consequences of any possible species interaction matrix. Interactions in these models can be estimated from observed co-occurrence rates via maximum likelihood, controlling for indirect effects. Using simulated landscapes with known interactions, I evaluated Markov networks and six existing approaches. Markov networks consistently outperformed the other methods, correctly isolating direct interactions between species pairs even when indirect interactions or abiotic factors largely overpowered them. Two computationally efficient approximations, which controlled for indirect effects with partial correlations or generalized linear models, also performed well. Null models showed no evidence of being able to control for indirect effects, and reliably yielded incorrect inferences when such effects were present.

The analysis of species co‐occurrence patterns continues to be a main pursuit of ecologists, primarily because the coexistence of species is fundamentally important in evaluating various theories, principles and concepts. Examples include community assembly, equilibrium versus non‐equilibrium organization of communities, resource partitioning and ecological character displacement, the local–regional species diversity relationship, and the metacommunity concept. Traditionally, co‐occurrence has been measured and tested at the level of an entire species presence–absence matrix wherein various algorithms are used to randomize matrices and produce statistical null distributions of metrics that quantify structure in the matrix. This approach implicitly recognizes a presence–absence matrix as having some real ecological identity (e.g. a set of species exhibiting nestedness among a set of islands) in addition to being a unit of statistical analysis. An emerging alternative is to test for non‐random co‐occurrence between paired species. The pairwise approach does not analyse matrix‐level structure and thus views a species pair as the fundamental unit of co‐occurrence. Inferring process from pattern is very difficult in analyses of co‐occurrence; however, the pairwise approach may make this task easier by simplifying the analysis and resulting inferences to associations between paired species.

Nestedness is a common biogeographic pattern in which small communities form proper subsets of large communities. However, the detection of nestedness in binary presence—absence matrices will be affected by both the metric used to quantify nestedness and the reference null distribution. In this study, we assessed the statistical performance of eight nestedness metrics and six null model algorithms. The metrics and algorithms were tested against a benchmark set of 200 random matrices and 200 nested matrices that were created by passive sampling. Many algorithms that have been used in nestedness studies are vulnerable to type I errors (falsely rejecting a true null hypothesis). The best-performing algorithm maintains fixed row and fixed column totals, but it is conservative and may not always detect nestedness when it is present. Among the eight indices, the popular matrix temperature metric did not have good statistical properties. Instead, the Brualdi and Sanderson discrepancy index and Cutler's index of unexpected presences performed best. When used with the fixed-fixed algorithm, these indices provide a conservative test for nestedness. Although previous studies have revealed a high frequency of nestedness, a reanalysis of 288 empirical matrices suggests that the true frequency of nested matrices is between 10% and 40%.

Veech (2013, Global Ecology and Biogeography, 22, 252-260) introduced a formula to calculate the probability of two species co-occurring in various sites under the assumption of statistical independence between the two distributional patterns. He presented his model as a new procedure, a 'pairwise approach', different from analyses of whole presence-absence matrices to examine patterns of co-occurrence. Here I show that: (1) Veech's method is identical to Fisher's exact test, a standard procedure for measuring the statistical association between two discrete variables; (2) in a broad sense, the pairwise approach is very similar to early analyses of spatial association, such as the one advanced by Forbes in 1907; (3) implicit in Veech's formula is a sampling scheme that is indistinguishable from well-known matrix-level null models that randomize the distribution of species among equiprobable sites; (4) pairwise co-occurrence patterns can be analysed using any matrix-level null model, so pairwise comparisons are not limited to using Veech's formula. The methodological distinction that Veech proposed between pairwise and matrix-level approaches does not in fact exist, although the conceptual distinction between the two approaches is still a debated topic.

J. M. Diamond's assembly rules model predicts that competitive interactions between species lead to nonrandom co-occurrence patterns. We conducted a meta-analysis of 96 published presence-absence matrices and used a realistic \"null model\" to generate patterns expected in the absence of species interactions. Published matrices were highly nonrandom and matched the predictions of Diamond's model: there were fewer species combinations, more checkerboard species pairs, and less co-occurrence in real matrices than expected by chance. Moreover, nonrandom structure was greater in homeotherm vs. poikilotherm matrices. Although these analyses do not confirm the mechanisms of Diamond's controversial assembly rules model, they do establish that observed co-occurrence in most natural communities is usually less than expected by chance. These results contrast with previous analyses of species co-occurrence patterns and bridge the apparent gap between experimental and correlative studies in community ecology.

Baselga: [Partitioning the turnover and nestedness components of beta diversity. Global Ecology and Biogeography, 19, 134—143, 2010] proposed pairwise (β nes ) and multiple-site (β NES ) beta-diversity measures to account for the nestedness component of beta diversity. We used empirical, randomly created and idealized matrices to show that both measures are only partially related to nestedness and do not fit certain fundamental requirements for consideration as true nestedness-resultant dissimilarity measures. Both β nes and β NES are influenced by matrix size and fill, and increase or decrease even when nestedness remains constant. Additionally, we demonstrate that β NES can yield high values even for matrices with no nestedness. We conclude that βnes and βNES are not true measures of the nestedness-resultant dissimilarity between sites. Actually, they quantify how differences in species richness that are not due to species replacement contribute to patterns of beta diversity. Finally, because nestedness is a special case of dissimilarity in species composition due to ordered species loss (or gain), the extent to which differences in species composition is due to nestedness can be measured through an index of nestedness.

Aim: The identification of compartments (i.e. clusters of overlapping species ranges across an environmental gradient) is an important methodological challenge for biogeographical studies. Recent developments in network theory offer promising perspectives on this issue using the measurement of modularity. A presence—absence matrix is modular if particular subgroups of species are mainly linked to particular subgroups of sites. Modularity is still rarely considered in biogeographical studies. Here, I compare different modularity indices to investigate which is the most appropriate for studying presence—absence matrices and similar types of networks, such as bipartite networks. Location: Evaluation was based on 279 data sets from around the world. Methods: I consider the three most commonly used modularity indices. One was developed for unipartite networks and the other two for bipartite networks. The performance of these indices (detection of a modular pattern, quality of compartment identification) is evaluated on test matrices of known compartmentalization levels with varying sizes and fills. Modularity patterns are then evaluated for 279 presence—absence matrices. Results: The three modularity measures differ mainly in the identification of the compartments, and less in the statistical significance of the observed modularity. The modularity measure Q 3 tends to perform best, whereas Q 2 usually performs less well, especially for highly diverse and highly connected networks that include a few extremely well-connected nodes. These modularity indices all reveal the presence of modular patterns in presence—absence matrices. Main conclusions: The choice of an appropriate modularity index is particularly important when we are interested in the composition of the different compartments. This analysis suggests that compartmentalized structures can be widespread in presence—absence matrices (about 40% of the matrices considered here). Modularity should thus offer interesting perspectives on the understanding of biogeographical patterns.

Aim: A great deal of information on distribution and diversity can be extracted from presence-absence matrices (PAMs), the basic analytical tool of many biogeographic studies. This paper presents numerical procedures that allow the analysis of such information by taking advantage of mathematical relationships within PAMs. In particular, we show how range-diversity (RD) plots summarize much of the information contained in the matrices by the simultaneous depiction of data on distribution and diversity. Innovation: We use matrix algebra to extract and process data from PAMs. Information on the distribution of species and on species richness of sites is computed using the traditional R (by rows) and Q (by columns) procedures, as well as the new Rq (by rows, considering the structure of columns) and Qr (by columns, considering the structure by rows) methods. Matrix notation is particularly suitable for summarizing complex calculations using PAMs, and the associated algebra allows the implementation of efficient computational programs. We show how information on distribution and species richness can be depicted simultaneously in RD plots, allowing a direct examination of the relationship between those two aspects of diversity. We explore the properties of RD plots with a simple example, and use null models to show that while parameters of central tendency are not affected by randomization, the dispersion of points in RD plots does change, showing the significance of patterns of co-occurrence of species and of similarity among sites. Main conclusion: Species richness and range size are both valid measures of diversity that can be analysed simultaneously with RD plots. A full analysis of a system requires measures of central tendency and dispersion for both distribution and species richness.

Aim: To identify consistent biogeographical modules, and examine species diversity and distribution patterns of centipede assemblages. Location: Europe, including Turkey and Macaronesia. Methods: A dataset was compiled, detailing the occurrence of 585 species of centipedes in 56 countries. Cluster analysis using UPGMA (unweighted pairgroup method with arithmetic averages) was used to identify biogeographical modules. To cope with potential issues resulting from the use of political geographical entities, the robustness of the modules was tested using two different randomization approaches. Potential centres of diversity and dispersal for the taxa were hypothesized using two different approaches, based on nestedness analysis using NODF and on investigation of species diversity gradients, respectively. Results: The Mediterranean region was found to be the most species-rich area. Cluster analysis identified four major biogeographical modules, namely Eastern Mediterranean, Western Mediterranean, Balkan Peninsula with eastern-central Europe, and north-western Europe. The robustness of these modules was supported by two randomization approaches. Both the analysis of nestedness and of species diversity gradients consistently identified the Balkan Peninsula as a potential centre of diversity for centipedes in Europe. Main conclusions: The arrangement of the centipede fauna into four biogeographical modules is consistent with European topography and environmental heterogeneity, with high mountain ranges acting as dispersal barriers, limiting the species overlap between modules. Common palaeogeographical history may explain the high degree of nestedness observed in the central and north-western European modules, whereas the high number of singletons and endemics is responsible for the low degree of nestedness in southern Europe. The identification of the Balkan Peninsula as a potential centre of diversity is in agreement with its high environmental heterogeneity and its known role as a Pleistocene glacial refugium.

It has been proposed that the study of co-occurrence of species, which is traditionally performed using full presence-absence matrices of sets of many species, could benefit from simply testing for random co-occurrence between pairs of species, and that use of a full presence-absence matrix is tantamount to regarding it as having some real ecological identity. Here I argue that although there are valid questions that can be answered using a pairwise approach, there are many others that naturally require the analysis of entire sets of species in a joint way, as provided for through the use of full presence-absence matrices. Moreover, there are theoretical and mathematical advantages to the use of presence-absence matrices, a few of which are briefly discussed in this short note.