Abstract.-We propose a new method to estimate and correct for phylogenetic inertia in comparative data analysis. The method, called phylogenetic eigenvector regression (PVR) starts by performing a principal coordinate analysis on a pairwise phylogenetic distance matrix between species. Traits under analysis are regressed on eigenvectors retained by a broken-stick model in such a way that estimated values express phylogenetic trends in data and residuals express independent evolution of each species. This partitioning is similar to that realized by the spatial autoregressive method, but the method proposed here overcomes the problem of low statistical performance that occurs with autoregressive method when phylogenetic correlation is low or when sample size is too small to detect it. Also, PVR is easier to perform with large samples because it is based on well-known techniques of multivariate and regression analyses. We evaluated the performance of PVR and compared it with the autoregressive method using real datasets and simulations. A detailed worked example using body size evolution of Carnivora mammals indicated that phylogenetic inertia in this trait is elevated and similarly estimated by both methods. In this example, Type I error at a = 0.05 of PVR was equal to 0.048, but an increase in the number of eigenvectors used in the regression increases the error. Also, similarity between PVR and the autoregressive method, defined by correlation between their residuals, decreased by overestimating the number of eigenvalues necessary to express the phylogenetic distance matrix. To evaluate the influence of c1adogram topology on the distribution of eigenvalues extracted from the double-centered phylogenetic distance matrix, we analyzed 100 randomly generated c1adograms (up to 100 species). Multiple linear regression of log transformed variables indicated that the number of eigenvalues extracted by the broken-stick model can be fully explained by c1adogram topology. Therefore, the broken-stick model is an adequate criterion for determining the correct number of eigenvectors to be used by PVR. We also simulated distinct levels of phylogenetic inertia by producing a trend across 10, 25, and 50 species arranged in "comblike" c1adograms and then adding random vectors with increased residual variances around this trend. In doing so, we provide an evaluation of the performance of both methods with data generated under different evolutionary models than tested previously. The results showed that both PVR and autoregressive method are efficient in detecting inertia in data when sample size is relatively high (more than 25 species) and when phylogenetic inertia is high. However, PVR is more efficient at smaller sample sizes and when level of phylogenetic inertia is low. These conclusions were also supported by the analysis of 10 real datasets regarding body size evolution in different animal clades. We concluded that PVR can be a useful alternative to an autoregressive method in comparative data analysis.

We propose a new method to estimate and correct for phylogenetic inertia in comparative data analysis. The method, called phylogenetic eigenvector regression (PVR) starts by performing a principal coordinate analysis on a pairwise phylogenetic distance matrix between species. Traits under analysis are regressed on eigenvectors retained by a broken-stick model in such a way that estimated values express phylogenetic trends in data and residuals express independent evolution of each species. This partitioning is similar to that realized by the spatial autoregressive method, but the method proposed here overcomes the problem of low statistical performance that occurs with autoregressive method when phylogenetic correlation is low or when sample size is too small to detect it. Also, PVR is easier to perform with large samples because it is based on well-known techniques of multivariate and regression analyses. We evaluated the performance of PVR and compared it with the autoregressive method using real datasets and simulations. A detailed worked example using body size evolution of Carnivora mammals indicated that phylogenetic inertia in this trait is elevated and similarly estimated by both methods. In this example, Type I error at a = 0.05 of PVR was equal to 0.048, but an increase in the number of eigenvectors used in the regression increases the error. Also, similarity between PVR and the autoregressive method, defined by correlation between their residuals, decreased by overestimating the number of eigenvalues necessary to express the phylogenetic distance matrix. To evaluate the influence of c1adogram topology on the distribution of eigenvalues extracted from the double-centered phylogenetic distance matrix, we analyzed 100 randomly generated c1adograms (up to 100 species). Multiple linear regression of log transformed variables indicated that the number of eigenvalues extracted by the broken-stick model can be fully explained by c1adogram topology. Therefore, the broken-stick model is an adequate criterion for determining the correct number of eigenvectors to be used by PVR. We also simulated distinct levels of phylogenetic inertia by producing a trend across 10, 25, and 50 species arranged in "comblike" c1adograms and then adding random vectors with increased residual variances around this trend. In doing so, we provide an evaluation of the performance of both methods with data generated under different evolutionary models than tested previously. The results showed that both PVR and autoregressive method are efficient in detecting inertia in data when sample size is relatively high (more than 25 species) and when phylogenetic inertia is high. However, PVR is more efficient at smaller sample sizes and when level of phylogenetic inertia is low. These conclusions were also supported by the analysis of 10 real datasets regarding body size evolution in different animal clades. We concluded that PVR can be a useful alternative to an autoregressive method in comparative data analysis.

There have been numerous claims in the ecological literature that spatial autocorrelation in the residuals of ordinary least squares (OLS) regression models results in shifts in the partial coefficients, which bias the interpretation of factors influencing geographical patterns. We evaluate the validity of these claims using gridded species richness data for the birds of North America, South America, Europe, Africa, the ex-USSR, and Australia. We used richness in 110 )110 km cells and environmental predictor variables to generate OLS and simultaneous autoregressive (SAR) multiple regression models for each region. Spatial correlograms of the residuals from each OLS model were then used to identify the minimum distance between cells necessary to avoid short-distance residual spatial autocorrelation in each data set. This distance was used to subsample cells to generate spatially independent data. The partial OLS coefficients estimated with the full dataset were then compared to the distributions of coefficients created with the subsamples. We found that OLS coefficients generated from data containing residual spatial autocorrelation were statistically indistinguishable from coefficients generated from the same data sets in which short-distance spatial autocorrelation was not present in all 22 coefficients tested. Consistent with the statistical literature on this subject, we conclude that coefficients estimated from OLS regression are not seriously affected by the presence of spatial autocorrelation in gridded geographical data. Further, shifts in coefficients that occurred when using SAR tended to be correlated with levels of uncertainty in the OLS coefficients. Thus, shifts in the relative importance of the predictors between OLS and SAR models are expected when small-scale patterns for these predictors create weaker and more unstable broad-scale coefficients. Our results indicate both that OLS regression is unbiased and that differences between spatial and nonspatial regression models should be interpreted with an explicit awareness of spatial scale.

Comparative methods have commonly been applied in macroecological research. However, few methods exist to map and analyze phylogenetic variation in geographical space. Here we develop a general analytical framework to partition the phylogenetic and ecological structures of macroecological patterns in geographic space. As an example, we apply the framework to evaluate interspecific patterns of body size geographic variation (Bergmann's rule) in European Carnivora. We model the components of variance attributable to ecological and phylogenetic effects, and to the shared influence of both factors. Spatial patterns in the ecological component are stronger than those in the original body size data. More importantly, the magnitude of intraspecific body size patterns (as measured by the correlation coefficient between body size and latitude) is significantly correlated with the ecological component across species, providing a unified interpretation for Bergmann's rule at multiple levels of biological hierarchy. This approach provides a better understanding of patterns in macroecological traits and allows improved understanding of their underlying ecological and evolutionary mechanisms.

In this paper we evaluate plasticity of fish concerning diet. We expect that sampling over a large temporal and spatial scale, including environmental changes such as impoundments, will allow us to cover most of the diet plasticity. We also evaluate the efficacy of ordination method in discriminating trophic groups based on fish species diet. Data were obtained from 17 sampling stations sampled monthly from March/96 to February/99 in the Corumbá river drainage, before and after the formation of the Corumba reservoir. Diet was determined analysing 9,177 stomach contents from 64 fish species. Trophic categories were discriminated by a non-hierarchic grouping analysis named k-means, applied to diet data. Most of the species presented great trophic plasticity, eating a great variety of food items. Resources availability, estimated from all fish stomach contents, was similar among environments, except in creeks, where it varied more with a large importance of terrestrial insects. K-means present satisfactory results, identifying nine trophic groups (detritivores, herbivore-piscivores, insectivore-herbivores, omnivores, invertivores, aquatic insectivores, piscivore-insectivores, piscivores and herbivores).Neste estudo avaliamos a plasticidade trófica em peixes. Nós esperamos que amostras obtidas com uma ampla escala temporal e espacial, incluindo mudanças ambientais como represamentos, nos permita cobrir a maior parte desta plasticidade. Foi avaliada também a eficiência do método de ordenação em discriminar os grupos tróficos baseado na dieta das espécies. As amostragens foram realizadas mensalmente de março/96 a fevereiro/99 em 17 estações de coleta na bacia do rio Corumbá antes e após a formação do reservatório de Corumbá. Foram analisados 9177 conteúdos estomacais, pertencentes a 64 espécies. As categorias tróficas foram discriminadas através de uma análise de agrupamento não hierárquica denominada K-means aplicada aos dados de dieta. A maioria das espécies apresentou elevada plasticidade trófica, consumindo uma grande variedade de itens alimentares. A disponibilidade dos recursos alimentares, estimada através de todos os conteúdos estomacais, foi similar entre os ambientes a exceção dos riachos, onde ocorreu maior heterogeneidade e os insetos terrestres tiveram grande importância. A análise K-means revelou resultados satisfatórios, identificando nove grupos tróficos (detritívoros, herbívoro-piscívoros, insetívoro-herbívoros, onívoros, invertívoros, insetívoros aquáticos, piscívoro-insetívoros, piscívoros e herbívoros).