Nonparametric multivariate analysis of ecological data using permutation tests has two main challenges: (1) to partition the variability in the data according to a complex design or model, as is often required in ecological experiments, and (2) to base the analysis on a multivariate distance measure (such as the semimetric Bray‐Curtis measure) that is reasonable for ecological data sets. Previous nonparametric methods have succeeded in one or other of these areas, but not in both. A recent contribution to Ecological Monographs by Legendre and Anderson, called distance‐based redundancy analysis (db‐RDA), does achieve both. It does this by calculating principal coordinates and subsequently correcting for negative eigenvalues, if they are present, by adding a constant to squared distances. We show here that such a correction is not necessary. Partitioning can be achieved directly from the distance matrix itself, with no corrections and no eigenanalysis, even if the distance measure used is semimetric. An ecological example is given to show the differences in these statistical methods. Empirical simulations, based on parameters estimated from real ecological species abundance data, showed that db‐RDA done on multifactorial designs (using the correction) does not have type 1 error consistent with the significance level chosen for the analysis (i.e., does not provide an exact test), whereas the direct method described and advocated here does.

Beta diversity can be defined as the variability in species composition among sampling units for a given area. We propose that it can be measured as the average dissimilarity from individual observation units to their group centroid in multivariate space, using an appropriate dissimilarity measure. Differences in beta diversity among different areas or groups of samples can be tested using this approach. The choice of transformation and dissimilarity measure has important consequences for interpreting results. For kelp holdfast assemblages from New Zealand, variation in species composition was greater in smaller holdfasts, while variation in relative abundances was greater in larger holdasts. Variation in community structure of Norwegian continental shelf macrobenthic fauna increased with increases in environmental heterogeneity, regardless of the measure used. We propose a new dissimilarity measure which allows the relative weight placed on changes in composition vs. abundance to be specified explicitly.

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Most biologists are now aware that ordinary least square regression is not appropriate when the X and Y variables are both subject to random error. When there is no information about their error variances, there is no correct unbiased solution. Although the major axis and reduced major axis (geometric mean) methods are widely recommended for this situation, they make different, equally restrictive assumptions about the error variances. By using simulated data sets that violate these assumptions, the reduced major axis method is shown to be generally more efficient and less biased than the major axis method. It is concluded that if the error rate of the X variable is thought to be more than a third of that on the Y variable, then the reduced major axis method is preferable; otherwise the least squares technique is acceptable. An analogous technique, the standard minor axis method, is described for use in place of least squares multiple regression when all of the variables are subject to error.

In population ecology, there has been a fundamental controversy about the relative importance of competition-driven (densitydependent) population regulation vs. abiotic influences such as temperature and precipitation. The same issue arises at the community level; are population sizes driven primarily by changes in the abundances of cooccurring competitors (i.e., compensatory dynamics), or do most species have a common response to environmental factors? Competitive interactions have had a central place in ecological theory, dating back to Gleason, Volterra, Hutchison and MacArthur, and, more recently, Hubbell's influential unified neutral theory of biodiversity and biogeography. If competitive interactions are important in driving year-to-year fluctuations in abundance, then changes in the abundance of one species should generally be accompanied by compensatory changes in the abundances of others. Thus, one necessary consequence of strong compensatory forces is that, on average, species within communities will covary negatively. Here we use measures of community covariance to assess the prevalence of negative covariance in 41 natural communities comprising different taxa at a range of spatial scales. We found that species in natural communities tended to covary positively rather than negatively, the opposite of what would be expected if compensatory dynamics were important. These findings suggest that abiotic factors such as temperature and precipitation are more important than competitive interactions in driving year-to-year fluctuations in species abundance within communities.biological interactions ͉ community dynamics ͉ negative covariance ͉ neutral models ͉ zero-sum