Discriminant functions for linear comparisons of means when covariance matrices are unequal

Situations exist, as in the biological example of discriminant analysis for natural hybridization, cited in the text, where (a) not all populations have equal variances, and (b) comparisions based on single degrees of freedom must be planned. This paper presents a statistical methodology of estimating discriminant functions for linear comparisons among k(<2) multivariate normal populations, and of testing their significance, when these populations have unequal covariance matrices.     

Multivariate measures of niche overlap using discriminant analysis

Two measures of multivariate niche overlap defined on p resource variables are presented. By measuring the niche overlap on the discriminant variable the multivariate problem is reduced to a univariate problem while preserving the relevant multivariate information. The niche overlap is then calculated by two different techniques. The first technique uses the MacArthur-Levins (Amer. Natur.101, 377–385, 1967) measure for probabilities of joint occurrence, while the second computes the density overlap of two use curves. An illustration of the multivariate approach to actual field data is demonstrated.     

Maximum likelihood estimation of variance components for a multivariate mixed model with equal design matrices

An algorithm is described for estimating variance and covariance components by restricted maximum likelihood for a multivariate mixed two-way classification with equal design matrices. The procedure involves a transformation to canonical scale, effectively reducing a q-variate analysis to q corresponding univariate analyses. A small numerical example is given as well as a large-scale practical application.     

QUANTITATIVE GENETICS OF MORPHOLOGICAL DIFFERENTIATION IN PEROMYSCUS. I. TESTS OF THE HOMOGENEITY OF GENETIC COVARIANCE STRUCTURE AMONG SPECIES AND SUBSPECIES

Phenotypic and additive genetic covariance matrices were estimated for 15 morphometric characters in three species and subspecies of Peromyscus. Univariate and multivariate ANOVAs indicate these groups are highly diverged in all characters, P. leucopus having the largest body size, P. maniculatus bairdii the smallest, and P. maniculatus nebrascensis being intermediate. Comparing the structure of P and G within each taxon revealed significant similarities in all three cases. This proportionality was strong enough to justify using P in the place of G to analyze evolutionary processes using quantitative genetic models when G can not be estimated, as in fossil material. However, the similarity between genetic and phenotypic covariance structures is sufficiently low that estimates of the genetic parameters should be used when possible. The additive genetic covariance matrices were compared to examine the assumption that they remain constant during evolution, an assumption which underlies many applications of quantitative-genetic models. While matrix permutation tests indicated statistically significant proportionality between the genetic covariance structures of the two P. maniculatus subspecies, there is no evidence of significant genetic structural similarity between species. This result suggests that the assumption of constant genetic covariance structure may be valid only within species. (It does not, however, necessarily imply a causal relationship between speciation and heterogeneity of genetic covariance structures.) The low matrix correlation for the two P. maniculatus subspecies' genetic covariance matrices indicates G may not be functionally constant, even within species. The lack of similarity observed here may be due partly to sampling variation.     

Measuring ecological niche overlap from occurrence and spatial environmental data

Aim Concerns over how global change will influence species distributions, in conjunction with increased emphasis on understanding niche dynamics in evolutionary and community contexts, highlight the growing need for robust methods to quantify niche differences between or within taxa. We propose a statistical framework to describe and compare environmental niches from occurrence and spatial environmental data. Location Europe, North America and South America. Methods The framework applies kernel smoothers to densities of species occurrence in gridded environmental space to calculate metrics of niche overlap and test hypotheses regarding niche conservatism. We use this framework and simulated species with pre‐defined distributions and amounts of niche overlap to evaluate several ordination and species distribution modelling techniques for quantifying niche overlap. We illustrate the approach with data on two well‐studied invasive species. Results We show that niche overlap can be accurately detected with the framework when variables driving the distributions are known. The method is robust to known and previously undocumented biases related to the dependence of species occurrences on the frequency of environmental conditions that occur across geographical space. The use of a kernel smoother makes the process of moving from geographical space to multivariate environmental space independent of both sampling effort and arbitrary choice of resolution in environmental space. However, the use of ordination and species distribution model techniques for selecting, combining and weighting variables on which niche overlap is calculated provide contrasting results. Main conclusions The framework meets the increasing need for robust methods to quantify niche differences. It is appropriate for studying niche differences between species, subspecies or intra‐specific lineages that differ in their geographical distributions. Alternatively, it can be used to measure the degree to which the environmental niche of a species or intra‐specific lineage has changed over time.     

Measuring the structural similarity of two communities composed of the same species

Summary The primary purpose of this paper is to propose empirical measures of the structural differences between two communities of plants or animals composed of the same species. Structure is defined to consist of; 1) the species in the community, 2) the pattern of interactions as represented by the covariance or correlation matrix of successive observations on each species, and 3) the mean abundances of each species in each of the two communities. Statistical tests are proposed for testing whether the covariance matrices and the vectors of mean densities for each community are equal and empirical measures of the differences between the covariance matrices and mean vectors are proposed. Given unequal covariance or correlation matrices the factor analysis model is used to derive empirical measures of the degree to which each variable of the ecosystem is responsible for the observed defferences in the pattern of interactions in each community. These tests and measures were applied to data gathered byHunter (1966) on the abundances of six species ofDrosophila censused monthly over a period of approximately two and a half years in two adjacent, but different habitats near Bogota, colombia. The two covariance matrices were significantly different indicating different patterns of interactions in the twoDrosophila communities.     

Genetically informed ecological niche models improve climate change predictions

We examined the hypothesis that ecological niche models (ENMs) more accurately predict species distributions when they incorporate information on population genetic structure, and concomitantly, local adaptation. Local adaptation is common in species that span a range of environmental gradients (e.g., soils and climate). Moreover, common garden studies have demonstrated a covariance between neutral markers and functional traits associated with a species’ ability to adapt to environmental change. We therefore predicted that genetically distinct populations would respond differently to climate change, resulting in predicted distributions with little overlap. To test whether genetic information improves our ability to predict a species’ niche space, we created genetically informed ecological niche models (gENMs) using Populus fremontii (Salicaceae), a widespread tree species in which prior common garden experiments demonstrate strong evidence for local adaptation. Four major findings emerged: (i) gENMs predicted population occurrences with up to 12‐fold greater accuracy than models without genetic information; (ii) tests of niche similarity revealed that three ecotypes, identified on the basis of neutral genetic markers and locally adapted populations, are associated with differences in climate; (iii) our forecasts indicate that ongoing climate change will likely shift these ecotypes further apart in geographic space, resulting in greater niche divergence; (iv) ecotypes that currently exhibit the largest geographic distribution and niche breadth appear to be buffered the most from climate change. As diverse agents of selection shape genetic variability and structure within species, we argue that gENMs will lead to more accurate predictions of species distributions under climate change.     

生态位有关术语的定义及计算公式评述

生态位(niche)理论在种间关系、群落结构、种的多样性及种群进化的研究中已被广泛应用。但对生态位及有关术语诸如生态位宽度、生态位重叠、生态位大小的定义至今还比较混乱。对于它们的计测,虽已提出了许多公式,但对其在生态学上的合理性仍有争议,本文试就这方面的问题作一评述。     

Evaluating interspecific niche overlaps in environmental and geographic spaces to assess the value of umbrella species

The concept of umbrella species assumes that concentrating resources on the protection of a single species contributes to the conservation of a suite of species and ecological processes belonging to the same ecosystem. The environmental requirements and geographical distribution of the umbrella species should thus overlap those of the group of targeted species. In western France, the conservation of several large grassland floodplains relies on agri‐environmental schemes targeting one single bird species, the corncrake Crex crex. It is considered as an umbrella species but no real assessment of its effectiveness has been carried out so far. We used a two‐step methodology to assess the potential of the corncrake to act as an umbrella species by estimating niche overlap in the environmental and geographic space between the main ground‐nesting species of the bird community in these grasslands, including the corncrake and four passerines. The five species showed substantial differences in their ecological niches so that their distributions did not perfectly overlap. Overlaps in predicted distributions between pairs of species depended on the threshold used to convert suitability to binary maps. Moreover, the number of species that could be protected by a candidate umbrella species was affected by the overlap criterion of success. Although the corncrake may be used as an umbrella species, it would be outperformed by several passerine species. Our study highlights the potential of using niche overlap to select umbrella species. It also reveals the importance of analysing the sensitivity of outputs when changing thresholds and overlap criteria.     

Distance-Based Metrics for Comparing Conformational Ensembles of Intrinsically Disordered Proteins

Intrinsically disordered proteins are proteins whose native functional states represent ensembles of highly diverse conformations. Such ensembles are a challenge for quantitative structure comparisons because their conformational diversity precludes optimal superimposition of the atomic coordinates necessary for deriving common similarity measures such as the root mean-square deviation of these coordinates. Here, we introduce superimposition-free metrics that are based on computing matrices of the Cα-Cα distance distributions within ensembles and comparing these matrices between ensembles. Differences between two matrices yield information on the similarity between specific regions of the polypeptide, whereas the global structural similarity is captured by the root mean-square difference between the medians of the Cα-Cα distance distributions of two ensembles. Together, our metrics enable rigorous investigations of structure-function relationships in conformational ensembles of intrinsically disordered proteins derived using experimental restraints or by molecular simulations and for proteins containing both structured and disordered regions.     

Temporal variability of ecological niches: a study on intertidal macrobenthic fauna

The determination of temporal niche dynamics under field conditions is an important component of a species’ ecology. Recent developments in niche mapping, and the possibility to account for spatial autocorrelation in species distributions, hold promise for the statistical approach explored here. Using species counts from a landscape‐scale benthic monitoring programme in the western Dutch Wadden Sea during 1997–2005 in combination with sediment characteristics and tidal height as explanatory variables, we statistically derive realised niches for two bivalves, two crustaceans and three polychaetes, encompassing predators, suspension and bottom feeding functional groups. Unsurprisingly, realized niches varied considerably between species. Intraspecific temporal variation was assessed as overlap between the year‐specific niche and the overall mean niche, and this analysis revealed considerable variation between years. The main functional groups represented by these species showed idiosyncratic and wide variability through the study period. There were no strong associations between niche characteristics and mean abundance or body size. Our assessment of intraspecific niche variability has ramifications for species distribution models in general and offers advances from previous methods. 1) By assessing species’ realized niches in the multivariate environmental space, analyses are independent from the relative availability of particular environments. Predicted realized niches present differences between years, rather than annual differences in environmental conditions. 2) Using spatially explicit models to predict species habitat preferences provide more precise and unbiased estimates of species–environment relationships. 3) Current niche models assume constant niches, whereas we illustrate how much these can vary over only a few generations. This emphasizes the potentially limited scope of global change studies with forecasts based on single‐time species distribution snapshots.     

Simulated responses to hypothetical fundamental niches

Abstract. Two graphical models of plant competition, the ‘ghost of competition past’ and the ‘hierarchical’ model are compared using a greatly simplified individual-based forest dynamics simulation. Assumed fundamental niche shapes are used to determine the basic growth responses of the species, but competition alters the realized niche. Differences in the two models, amount of niche overlap, disturbance, and removal and invasion of species are examined in simulation experiments. Without disturbance, the realized niche responses reveal abrupt boundaries between species in all cases, and thus the responses are generally platykurtic to rectangular with little overlap. In some cases overlap through the extension of abundance of weaker competitors into the area of greater resources — dominated by better competitors but still within their fundamental niche — creates skewed distributions, as have been observed and simulated in the past. When species are removed or invade, the abrupt boundaries and the lack of difference in final response shape indicate that past removals may be difficult to detect. This difficulty may be important because former species may be influencing the responses that are observed in common non-equilibrium distributions. Even when assumptions favor the illustration of a competitive hierarchy, actual distributions and dynamics do not differentiate between it and a model of the ghost of competition past.     

Temporal variation in spatial organization of a rodent community in the southwestern Kyzylkum desert (Middle Asia)

The distribution of niches in resource space and the niche patterns of a 14-species community of Middle Asian desert rodents were studied during two years - at low and high rodent density - using discriminant function analysis Nineteen quantitative environmental parameters (soil structure and vegetation), measured in 550 plots within 22 1 -ha grids, were considered The first three canonical axes of resource space account for 72% of the variance The first two axes represent complex environmental gradients the first axis represents a general landscape gradient from sand to clay soils, the second axis reflects a gradient of in creasing productivity The third axis reflects with in-habitat environmental variation All community parameters, as well as parameters of individual species niches, were unstable between years At the same time, different parameters vary in different extent Position of niche centroids along macro-habitat axes, as. well as macrohabitat niche breadth, were relatively stable between years, but these parameters for microhabitat axis and values of niche overlap were much more variable A strong correlation between changes m relative between-habitat niche breadth and differences in average niche overlap with relative changes in species abundances indicate density dependence of these parameters Changes in niche overlap is a consequence of between-year differences in guild patterns Guild structure was pronounced at high density when the level of niche overlap was intermediate At low density, when the level of niche overlap decreased, guild structure was incon-spicous Different levels of diversity differed in their sensitivity to density changes α-diversity was relatively constant as a result of between-year stability of niche centroid positions However level of ß-diversity varied significantly between years reflecting changes in the level of niche overlap, because a decrease in niche overlap leads to an increase in the rate of species turnover     

A Multivariate Analysis of Hybridization when Covariance Matrices are Unequal

Numerous occurrences of natural hybridization have been known in many groups of animals. It has a bearing on growth, nomenclature, speciation, genetics and wildlife management. It is well recognised that demonstration of intermediacy based on several characters makes the identification of hybrids more certain than that based on just a single character, and that differences among the hybrids and the parental populations should be analysed for variations due to the additive genetic (A) and the non-additive genetic (NA) factors separately. In the present paper (i) it is pointed out that sometimes the assumption that covariance matrices of the hybrids and the parental populations are equal, may not be valid, and (ii) a multivariate method of testing hypotheses analysing differences between the hybrids and the parents, qualified by the A and the NA factors, when covariance matrices are not equal, is submitted.     

A modified pseudolikelihood approach for analysis of longitudinal data

We consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters. In general, inconsistent estimators of the covariance (variance/correlation) parameters will be produced when the "working" correlation matrix is misspecified, which may result in great loss of efficiency of the mean parameter estimators (albeit the consistency is preserved). We consider using different "working" correlation models for the variance and the mean parameters. In particular, we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified. The designated "working" correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters. Simulation studies indicate that the proposed algorithm performs very well. We also applied different estimation procedures to a data set from a clinical trial for illustration.     

On measures of association among genetic variables

Systems involving many variables are important in population and quantitative genetics, for example, in multi-trait prediction of breeding values and in exploration of multi-locus associations. We studied departures of the joint distribution of sets of genetic variables from independence. New measures of association based on notions of statistical distance between distributions are presented. These are more general than correlations, which are pairwise measures, and lack a clear interpretation beyond the bivariate normal distribution. Our measures are based on logarithmic (Kullback-Leibler) and on relative 'distances' between distributions. Indexes of association are developed and illustrated for quantitative genetics settings in which the joint distribution of the variables is either multivariate normal or multivariate-t, and we show how the indexes can be used to study linkage disequilibrium in a two-locus system with multiple alleles and present applications to systems of correlated beta distributions. Two multivariate beta and multivariate beta-binomial processes are examined, and new distributions are introduced: the GMS-Sarmanov multivariate beta and its beta-binomial counterpart.     

A comprehensive framework for the study of species co‐occurrences,nestedness and turnover

Binary presence–absence matrices (rows = species, columns = sites) are often used to quantify patterns of species co‐occurrence, and to infer possible biotic interactions from these patterns. Previous classifications of co‐occurrence patterns as nested, segregated, or modular have led to contradictory results and conclusions. These analyses usually do not incorporate the functional traits of the species or the environmental characteristics of the sites, even though the outcomes of species interactions often depend on trait expression and site quality. Here we address this shortcoming by developing a method that incorporates realized functional and environmental niches, and relates them to species co‐occurrence patterns. These niches are defined from n‐dimensional ellipsoids, and calculated from the n eigenvectors and eigenvalues of the variance–covariance matrix of measured environmental or trait variables. Average niche overlap among species and the spatial distribution of niches define a triangle plot with vertices of species segregation (low niche overlap), nestedness (high niche overlap), and modular co‐occurrence (clusters of overlapping niches). Applying this framework to temperate understorey plant communities in southwest Poland, we found a consistent modular structure of species occurrences, a pattern not detected by conventional presence–absence analysis. These results suggest that, in our case study, habitat filtering is the most important process structuring understorey plant communities. Furthermore, they demonstrate how incorporating trait and environmental data into co‐occurrence analysis improves pattern detection and provides a stronger theoretical framework for understanding community structure.     

Improved covariance matrix estimators for weighted analysis of microarray data.

Empirical Bayes models have been shown to be powerful tools for identifying differentially expressed genes from gene expression microarray data. An example is the WAME model, where a global covariance matrix accounts for array-to-array correlations as well as differing variances between arrays. However, the existing method for estimating the covariance matrix is very computationally intensive and the estimator is biased when data contains many regulated genes. In this paper, two new methods for estimating the covariance matrix are proposed. The first method is a direct application of the EM algorithm for fitting the multivariate t-distribution of the WAME model. In the second method, a prior distribution for the log fold-change is added to the WAME model, and a discrete approximation is used for this prior. Both methods are evaluated using simulated and real data. The first method shows equal performance compared to the existing method in terms of bias and variability, but is superior in terms of computer time. For large data sets (>15 arrays), the second method also shows superior computer run time. Moreover, for simulated data with regulated genes the second method greatly reduces the bias. With the proposed methods it is possible to apply the WAME model to large data sets with reasonable computer run times. The second method shows a small bias for simulated data, but appears to have a larger bias for real data with many regulated genes.     

Is the composition of phytophagous insects and parasitic fungi among trees predictable?

We explore the relationship between the pairwise similarity of assemblages of exploiters (phytophagous insects and parasitic fungi) and pairwise genetic distance, range overlap, niche overlap as well as habitat overlap of host trees. Presence of exploiters was extracted from published literature for 23 tree genera occurring in central Europe (6164 host records of phytophagous insects and 860 host records of parasitic fungi). Across all pairs of tree genera, we found a strong negative correlation between the pairwise similarity of assemblages and genetic distances of hosts. This close correlation is due to deep differences in the composition of assemblages on coniferous and deciduous tree genera. Range, niche and habitat overlap were always of much less importance than genetic distance to explain the variation of pairwise similarity of assemblages of exploiters, although some correlations were significant. Therefore in general host switches of exploiters between related hosts are more important that host switches between hosts co-occurring in the same habitat. We found a robust relationship of the pairwise similarity of assemblages of insects and the pairwise similarity of assemblages of fungi which points to the possibility that insects are vectors for parasitic fungi which promotes correlated switches of insects and fungi.