Pedro R. Peres-Neto

VARIATION PARTITIONING OF SPECIES DATA MATRICES: ESTIMATION AND COMPARISON OF FRACTIONS

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.

ANALYZING BETA DIVERSITY: PARTITIONING THE SPATIAL VARIATION OF COMMUNITY COMPOSITION DATA

Robert H. Whittaker defined beta diversity as the variation in species com- position 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 func- tions. 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 com- position 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 parti- tioning. The Mantel approach is appropriate for testing other hypotheses, such as the var- iation in beta diversity among groups of sites. Regression on distance matrices is also appropriate for fitting models to similarity decay plots.

How well do multivariate data sets match? The advantages of a Procrustean superimposition approach over the Mantel test

The Mantel test provides a means to test the association between distance matrices and has been widely used in ecological and evolutionary studies. Recently, another permutation test based on a Procrustes statistic (PROTEST) was developed to compare multivariate data sets. Our study contrasts the effectiveness, in terms of power and type I error rates, of the Mantel test and PROTEST. We illustrate the application of Procrustes superimposition to visually examine the concordance of observations for each dimension separately and how to conduct hypothesis testing in which the association between two data sets is tested while controlling for the variation related to other sources of data. Our simulation results show that PROTEST is as powerful or more powerful than the Mantel test for detecting matrix association under a variety of possible scenarios. As a result of the increased power of PROTEST and the ability to assess the match for individual observations (not available with the Mantel test), biologists now have an additional and powerful analytical tool to study ecological and evolutionary relationships.

How many principal components? stopping rules for determining the number of non-trivial axes revisited

Principal component analysis is one of the most widely applied tools in order to summarize common patterns of variation among variables. Several studies have investigated the ability of individual methods, or compared the performance of a number of methods, in determining the number of components describing common variance of simulated data sets. We identify a number of shortcomings related to these studies and conduct an extensive simulation study where we compare a larger number of rules available and develop some new methods. In total we compare 20 stopping rules and propose a two-step approach that appears to be highly effective. First, a Bartletts test is used to test the significance of the first principal component, indicating whether or not at least two variables share common variation in the entire data set. If significant, a number of different rules can be applied to estimate the number of non-trivial components to be retained. However, the relative merits of these methods depend on whether data contain strongly correlated or uncorrelated variables. We also estimate the number of non-trivial components for a number of field data sets so that we can evaluate the applicability of our conclusions based on simulated data.

GIVING MEANINGFUL INTERPRETATION TO ORDINATION AXES: ASSESSING LOADING SIGNIFICANCE IN PRINCIPAL COMPONENT ANALYSIS

Principal component analysis (PCA) is one of the most commonly used tools in the analysis of ecological data. This method reduces the effective dimensionality of a multivariate data set by producing linear combinations of the original variables (i.e., com- ponents) that summarize the predominant patterns in the data. In order to provide meaningful interpretations for principal components, it is important to determine which variables are associated with particular components. Some data analysts incorrectly test the statistical significance of the correlation between original variables and multivariate scores using standard statistical tables. Others interpret eigenvector coefficients larger than an arbitrary absolute value (e.g., 0.50). Resampling, randomization techniques, and parallel analysis have been applied in a few cases. In this study, we compared the performance of a variety of approaches for assessing the significance of eigenvector coefficients in terms of type I error rates and power. Two novel approaches based on the broken-stick model were also evaluated. We used a variety of simulated scenarios to examine the influence of the number of real dimensions in the data; unique versus complex variables; the magnitude of eigen- vector coefficients; and the number of variables associated with a particular dimension. Our results revealed that bootstrap confidence intervals and a modified bootstrap confidence interval for the broken-stick model proved to be the most reliable techniques.

Predictive Models of Fish Species Distributions: A Note on Proper Validation and Chance Predictions

Abstract The prediction of species distributions is a primary goal in the study, conservation, and management of fisheries resources. Statistical models relating patterns of species presence or absence to multiscale habitat variables play an important role in this regard. Researchers, however, have paid little attention to how improper model validation and chance predictions can result in unfounded confidence in the performance and utility of such models. Using simulated and empirical data for 40 lake and stream fish species, we demonstrate that the commonly employed resubstitution approach to model validation (in which the same data are used for both model construction and prediction) produces highly biased estimates of correct classification rates and consequently an inaccurate perception of true model performance. In contrast, a jackknife approach to validation resulted in relatively unbiased estimates of model performance. The estimated rates of model correct classification are also shown to be substa...

Spatial isolation and fish communities in drainage lakes

Fifty-two drainage lakes, located in south-central Ontario, Canada, were examined to study the association of isolation- and environment-related factors with fish community composition. Eight quantitative measures of lake isolation were examined, each of which incorporated potential ecological challenges that a fish encounters when moving between lakes. A Procrustean approach was employed to assess the degree of concordance between fish assemblage structure, measures of lake isolation and environmental conditions (i.e., lake morphology and water chemistry). Our results revealed a high concordance between patterns in fish community composition and lake isolation and lake morphology at the watershed scale, suggesting that insular and habitat-related factors influence the structure of fish communities. At the scale of the individual lake, this relationship varied greatly, ranging from a strong match of community composition with both spatial and abiotic conditions to communities exhibiting weak association with these conditions. Furthermore, we showed that alternative measures of lake isolation provide additional insight into potential factors shaping patterns in fish community composition; information not provided using straight-line distances between lakes. Finally, the statistical methodology outlined in this paper provides a robust technique for assessing both the overall association between multivariate data matrices (i.e., landscape or regional scale), as well as facilitating the examination of smaller-scale relationships of individual observations (i.e., local scale).

Patterns in the co-occurrence of fish species in streams: the role of site suitability, morphology and phylogeny versus species interactions

A number of studies at large scales have pointed out that abiotic factors and recolonization dynamics appear to be more important than biotic interactions in structuring stream-fish assemblages. In contrast, experimental and field studies at small scales show the importance of competition among stream fishes. However, given the highly variable nature of stream systems over time, competition may not be intense enough to generate large-scale complementary distributions via competitive exclusion. Complementary distribution is a recurrent pattern observed in fish communities across stream gradients, though it is not clear which instances of this pattern are due to competitive interactions and which to individual species’ requirements. In this study, I introduce a series of null models developed to provide a more robust evaluation of species associations by facilitating the distinction between different processes that may shape species distributions and community assembly. These null models were applied to test whether conspicuous patterns in species co-occurrences are more consistent with their differences in habitat use, morphological features and/or phylogenetic constraints, or with species interactions in fish communities in the streams of a watershed in eastern Brazil. I concluded that patterns in species co-occurrences within the studied system are driven by common species-habitat relationships and species interactions may not play a significant role in structuring these communities. I suggest that large-scale studies, where adequate designs and robust analytical tools are applied, can contribute substantially to understanding the importance of different types of processes in structuring stream-fish communities.

Metacommunity phylogenetics: separating the roles of environmental filters and historical biogeography

Biogeographical, evolutionary and ecological processes interact to regulate patterns in metacommunities. However, as there are few quantitative methods for evaluating their joint effects, resolving this interaction is difficult. We develop a method that aims to evaluate the interaction between phylogenetic structure, historical biogeographic events and environmental filtering in driving species distributions in a large-scale metacommunity. Using freshwater zooplankton as a case study, we contrast the phylogenetic metacommunity structure of calanoid copepods and an ecologically similar but more vagile group, daphniids, in the northeastern US. We find that legacies of historical biogeographical events have strongly constrained calanoid distributions within this area, but that adaptation to different water chemistry and lake morphology drives the metacommunity structure of daphniids. Our findings show that biogeographic history and metacommunity processes jointly regulate community structure in these lakes and suggest that this also depends on factors that affect the colonization rate of different types of organisms.

The influence of swimming demand on phenotypic plasticity and morphological integration: a comparison of two polymorphic charr species

In northern freshwater lakes, several fish species have populations composed of discrete morphs, usually involving a divergence between benthic and limnetic morphs. Although it has been suggested that swimming demand plays an important role in morphological differentiation, thus influencing habitat selection, it is unclear how it affects reaction norms, patterns in character correlation, and levels of morphological integration. We examined whether swimming demand could induce morphological plasticity in the directions expected under divergent habitat selection, and evaluated its influence on the morphological integration in Arctic charr (Salvelinus alpinus) and brook charr (S. fontinalis), two congeneric species exhibiting conspicuous and subtle resource polymorphism, respectively. We found that changes in morphology were induced by differential swimming demands in both species. The length of the pectoral fin was the character that responded most strongly according to the predicted morphological expectations under divergent habitat selection. High levels of morphological plasticity, relatively low levels of integration, and differences found in the morphological correlation structure among water velocity treatments suggest that constraints on morphological change are unlikely in either species, thus allowing great potential for phenotypic flexibility in both species. The magnitude of character integration, however, was larger for Arctic charr than for brook charr. This latter result is discussed in the light of the differences in the level of polymorphism between the two species in the wild. The results of the present study indicate that swimming demand alone may not be sufficient to generate the polymorphism encountered in nature. Given that both diet and swimming demands can induce morphological changes, it would be important to conduct experiments targeting the interaction between the morphological modules related to trophic and swimming demands.