Abstract. A method is proposed to partition the variation of species abundance data into. independe~t components: pure spatial, pure environmental, spatial component of envuonmental mfiuence, and undetermined. The new method uses pre-existing techniques and computer programs of canonical ordination. The intrinsic spatial component of community structure is partialled out of the species-environment relationship in order to see if ~he ~nvironmental control model still holds. The method is illustrated using oribatid mttes m a peat blanket, forest vegetation data, and aquatic heterotrophic bacteria. In this latter example, the new method is shown to be complementary to another approach based on partial Mantel tests..

Establishing relationships between species distributions and environmental characteristics is a major goal in the search for forces driving species distributions. Canonical ordinations such as redundancy analysis and canonical correspondence analysis are invaluable tools for modeling communities through environmental predictors. They provide the means for conducting direct explanatory analysis in which the association among species can be studied according to their common and unique relationships with the environmental variables and other sets of predictors of interest, such as spatial variables. Variation partitioning can then be used to test and determine the likelihood of these sets of predictors in explaining patterns in community structure. Although variation partitioning in canonical analysis is routinely used in ecological analysis, no effort has been reported in the literature to consider appropriate estimators so that comparisons between fractions or, eventually, between different canonical models are meaningful. In this paper, we show that variation partitioning as currently applied in canonical analysis is biased. We present appropriate unbiased estimators. In addition, we outline a statistical test to compare fractions in canonical analysis. The question addressed by the test is whether two fractions of variation are significantly different from each other. Such assessment provides an important step toward attaining an understanding of the factors patterning community structure. The test is shown to have correct Type I. error rates and good power for both redundancy analysis and canonical correspondence analysis.

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This paper proposes a new way of using forward selection of explanatory variables in regression or canonical redundancy analysis. The classical forward selection method presents two problems: a highly inflated Type I error and an overestimation of the amount of explained variance. Correcting these problems will greatly improve the performance of this very useful method in ecological modeling. To prevent the first problem, we propose a two-step procedure. First, a global test using all explanatory variables is carried out. If, and only if, the global test is significant, one can proceed with forward selection. To prevent overestimation of the explained variance, the forward selection has to be carried out with two stopping criteria: (1) the usual alpha significance level and (2) the adjusted coefficient of multiple determination (Ra(2)) calculated using all explanatory variables. When forward selection identifies a variable that brings one or the other criterion over the fixed threshold, that variable is rejected, and the procedure is stopped. This improved method is validated by simulations involving univariate and multivariate response data. An ecological example is presented using data from the Bryce Canyon National Park, Utah, U.S.A.

Robert H. Whittaker defined beta diversity as the variation in species composition among sites in a geographic area. Beta diversity is a key concept for understanding the functioning of ecosystems, for the conservation of biodiversity, and for ecosystem management. This paper explains how hypotheses about the origin of beta diversity can be tested by partitioning the spatial variation of community composition data (presence– absence or abundance data) with respect to environmental variables and spatial base functions. We compare two statistical methods to accomplish that. The sum‐of‐squares of a community composition data table, which is one possible measure of beta diversity, is correctly partitioned by canonical ordination; hence, canonical partitioning produces correct estimates of the different portions of community composition variation. In recent years, several authors interested in the variation in community composition among sites (beta diversity) have used another method, variation partitioning on distance matrices (Mantel approach). Their results led us to compare the two partitioning approaches, using simulated data generated under hypotheses about the variation of community composition among sites. The theoretical developments and simulation results led to the following observations: (1) the variance of a community composition table is a measure of beta diversity. (2) The variance of a dissimilarity matrix among sites is not the variance of the community composition table nor a measure of beta diversity; hence, partitioning on distance matrices should not be used to study the variation in community composition among sites. (3) In all of our simulations, partitioning on distance matrices underestimated the amount of variation in community composition explained by the raw‐data approach, and (4) the tests of significance had less power than the tests of canonical ordination. Hence, the proper statistical procedure for partitioning the spatial variation of community composition data among environmental and spatial components, and for testing hypotheses about the origin and maintenance of variation in community composition among sites, is canonical partitioning. The Mantel approach is appropriate for testing other hypotheses, such as the variation in beta diversity among groups of sites. Regression on distance matrices is also appropriate for fitting models to similarity decay plots.

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Spatial structures may not only result from ecological interactions, they may also play an essential functional role in organizing the interactions. Modeling spatial patterns at multiple spatial and temporal scales is thus a crucial step to understand the functioning of ecological communities. PCNM (principal coordinates of neighbor matrices) analysis achieves a spectral decomposition of the spatial relationships among the sampling sites, creating variables that correspond to all the spatial scales that can be perceived in a given data set. The analysis then finds the scales to which a data table of interest responds. The significant PCNM variables can be directly interpreted in terms of spatial scales, or included in a procedure of variation decomposition with respect to spatial and environmental components. This paper presents four applications of PCNM analysis to ecological data representing combinations of: transect or surface data, regular or irregular sampling schemes, univariate or multivariate data. The data sets include Amazonian ferns, tropical marine zooplankton, chlorophyll in a marine lagoon, and oribatid mites in a peat bog. In each case, new ecological knowledge was obtained through PCNM analysis.