Why the beta-function cannot be used to estimate skewness of species responses

Abstract. The beta-function (β-function) has been suggested for testing the significance of the skewness of species responses along a gradient. However, the location of the optimum and skewness are correlated so that these parameters cannot be estimated independently. The only way for an independent estimation is to let the endpoints of the response curve vary. In that case they would no longer define the range of species occurrence. However, non-linear estimation of endpoints often leads to overwhelming problems in model fitting. Therefore, the beta-function is not suitable to test the shape of species response curves. Hierarchic models proposed by Huisman et al. (1993) seem to be superior to generalized additive models or third-degree polynomials and seem to be the best alternative to study the skewness of responses.     

Species response curves along environmental gradients. A case study from SE Norwegian swamp forests

Abstract. Vegetation science has relied on untested paradigms relating to the shape of species response curves along environmental gradients. To advance in this field, we used the HOF approach to model response curves for 112 plant species along six environmental gradients and three ecoclines (as represented by DCA ordination axes) in SE Norwegian swamp forests. Response curve properties were summarized in three binary response variables: (1) model unimodal or monotonous (determinate) vs. indeterminate; (2) for determinate models, unimodal vs. monotonous and (3) for unimodal models, skewed vs. symmetric. We used logistic regression to test the influence, singly and jointly, of seven predictor variables on each of three response variables. Predictor variables included gradient type (environmental or ecocline) and length (compositional turnover); species category (vascular plant, moss, Sphagnum or hepatic), species frequency and richness, tolerance (the fraction of the gradient along which the species occurs) and position of species along each gradient. The probability for fitting a determinate model increased as the main occurrence of species approached gradient extremes and with increasing species tolerance and frequency and gradient length. Appearance of unimodal models was favoured by low species tolerance and disfavoured by closeness of species to gradient extremes. Appearance of skewed models was weakly related to predictors but was slightly favoured by species optima near gradient extremes. Contrary to the results of previous studies, species category, gradient type and variation in species richness along gradients did not contribute independently to model prediction. The overall best predictors of response curve shape were position along the gradient (relative to extremes) and tolerance; the latter also expressing gradient length in units of compositional turnover. This helps predicting species responses to gradients from gradient specific species properties. The low proportion of skewed response curves and the large variation of species response curves along all gradients indicate that skewed response curves is a smaller problem for the performance of ordination methods than often claimed. We find no evidence that DCA ordination increases the unimodality, or symmetry, of species response curves more than expected from the higher compositional turnover along ordination axes. Thus ordination axes may be appropriate proxies for ecoclines, applicable for use in species response modelling.     

Shape and evolution of the fundamental niche in marine Vibrio

Hutchinson''s fundamental niche, defined by the physical and biological environments in which an organism can thrive in the absence of inter-species interactions, is an important theoretical concept in ecology. However, little is known about the overlap between the fundamental niche and the set of conditions species inhabit in nature, and about natural variation in fundamental niche shape and its change as species adapt to their environment. Here, we develop a custom-made dual gradient apparatus to map a cross-section of the fundamental niche for several marine bacterial species within the genus Vibrio based on their temperature and salinity tolerance, and compare tolerance limits to the environment where these species commonly occur. We interpret these niche shapes in light of a conceptual model comprising five basic niche shapes. We find that the fundamental niche encompasses a much wider set of conditions than those strains typically inhabit, especially for salinity. Moreover, though the conditions that strains typically inhabit agree well with the strains'' temperature tolerance, they are negatively correlated with the strains'' salinity tolerance. Such relationships can arise when the physiological response to different stressors is coupled, and we present evidence for such a coupling between temperature and salinity tolerance. Finally, comparison with well-documented ecological range in V. vulnificus suggests that biotic interactions limit the occurrence of this species at low-temperature—high-salinity conditions. Our findings highlight the complex interplay between the ecological, physiological and evolutionary determinants of niche morphology, and caution against making inferences based on a single ecological factor.     

The effects of pollen and seed migration on nuclear-dicytoplasmic systems. II. A new method for estimating plant gene flow from joint nuclear-cytoplasmic data

A new maximum-likelihood method is developed for estimating unidirectional pollen and seed flow in mixed-mating plant populations from counts of joint nuclear-cytoplasmic genotypes. Data may include multiple unlinked nuclear markers with a single maternally or paternally inherited cytoplasmic marker, or with two cytoplasmic markers inherited through opposite parents, as in many conifer species. Migration rate estimates are based on fitting the equilibrium genotype frequencies under continent-island models of plant gene flow to the data. Detailed analysis of their equilibrium structures indicates when each of the three nuclear-cytoplasmic systems allows gene flow estimation and shows that, in general, it is easier to estimate seed than pollen migration. Three-locus nuclear-dicytoplasmic data only increase the conditions allowing seed migration estimates; however, the additional dicytonuclear disequilibria allow more accurate estimates of both forms of gene flow. Estimates and their confidence limits for simulated data sets confirm that two-locus data with paternal cytoplasmic inheritance provide better estimates than those with maternal inheritance, while three-locus dicytonuclear data with three modes of inheritance generally provide the most reliable estimates for both types of gene flow. Similar results are obtained for hybrid zones receiving pollen and seed flow from two source populations. An estimation program is available upon request.     

Obtaining maximal concatenated phylogenetic data sets from large sequence databases

To improve the accuracy of tree reconstruction, phylogeneticists are extracting increasingly large multigene data sets from sequence databases. Determining whether a database contains at least k genes sampled from at least m species is an NP-complete problem. However, the skewed distribution of sequences in these databases permits all such data sets to be obtained in reasonable computing times even for large numbers of sequences. We developed an exact algorithm for obtaining the largest multigene data sets from a collection of sequences. The algorithm was then tested on a set of 100,000 protein sequences of green plants and used to identify the largest multigene ortholog data sets having at least 3 genes and 6 species. The distribution of sizes of these data sets forms a hollow curve, and the largest are surprisingly small, ranging from 62 genes by 6 species, to 3 genes by 65 species, with more symmetrical data sets of around 15 taxa by 15 genes. These upper bounds to sequence concatenation have important implications for building the tree of life from large sequence databases.     

Strengths and weaknesses of museum and national survey data sets for predicting regional species richness: comparative and combined approaches

We compiled three independent data sets of bird species occurrences in northeastern Colorado to test how predicted species richness compared to a combined analysis using all the data. The first data set was a georeferenced regional museum data set from two major repositories — the Denver Museum of Nature, and the Science and University of Colorado Museum. The two national survey data sets were the Breeding Bird Survey (summer), and the Great Backyard Bird Count (winter). Resulting analyses show that the museum data sets give richness estimates closest to the combined data set while exhibiting a skewed abundance distribution, whereas survey data sets do not accurately estimate overall richness even though they contain far more records. The combined data set allows the strengths of one data set to augment weaknesses in others. It is likely some museum data sets display skewed abundance distributions due to collectors’ potentially self‐selecting under‐represented species over common ones.     

Bayesian QTL mapping using skewed Student-t distributions

In most QTL mapping studies, phenotypes are assumed to follow normal distributions. Deviations from this assumption may lead to detection of false positive QTL. To improve the robustness of Bayesian QTL mapping methods, the normal distribution for residuals is replaced with a skewed Student-t distribution. The latter distribution is able to account for both heavy tails and skewness, and both components are each controlled by a single parameter. The Bayesian QTL mapping method using a skewed Student-t distribution is evaluated with simulated data sets under five different scenarios of residual error distributions and QTL effects.     

Normalization of high dimensional genomics data where the distribution of the altered variables is skewed

Genome-wide analysis of gene expression or protein binding patterns using different array or sequencing based technologies is now routinely performed to compare different populations, such as treatment and reference groups. It is often necessary to normalize the data obtained to remove technical variation introduced in the course of conducting experimental work, but standard normalization techniques are not capable of eliminating technical bias in cases where the distribution of the truly altered variables is skewed, i.e. when a large fraction of the variables are either positively or negatively affected by the treatment. However, several experiments are likely to generate such skewed distributions, including ChIP-chip experiments for the study of chromatin, gene expression experiments for the study of apoptosis, and SNP-studies of copy number variation in normal and tumour tissues. A preliminary study using spike-in array data established that the capacity of an experiment to identify altered variables and generate unbiased estimates of the fold change decreases as the fraction of altered variables and the skewness increases. We propose the following work-flow for analyzing high-dimensional experiments with regions of altered variables: (1) Pre-process raw data using one of the standard normalization techniques. (2) Investigate if the distribution of the altered variables is skewed. (3) If the distribution is not believed to be skewed, no additional normalization is needed. Otherwise, re-normalize the data using a novel HMM-assisted normalization procedure. (4) Perform downstream analysis. Here, ChIP-chip data and simulated data were used to evaluate the performance of the work-flow. It was found that skewed distributions can be detected by using the novel DSE-test (Detection of Skewed Experiments). Furthermore, applying the HMM-assisted normalization to experiments where the distribution of the truly altered variables is skewed results in considerably higher sensitivity and lower bias than can be attained using standard and invariant normalization methods.     

Small area estimation for semicontinuous skewed spatial data: An application to the grape wine production in Tuscany

Linear‐mixed models are frequently used to obtain model‐based estimators in small area estimation (SAE) problems. Such models, however, are not suitable when the target variable exhibits a point mass at zero, a highly skewed distribution of the nonzero values and a strong spatial structure. In this paper, a SAE approach for dealing with such variables is suggested. We propose a two‐part random effects SAE model that includes a correlation structure on the area random effects that appears in the two parts and incorporates a bivariate smooth function of the geographical coordinates of units. To account for the skewness of the distribution of the positive values of the response variable, a Gamma model is adopted. To fit the model, to get small area estimates and to evaluate their precision, a hierarchical Bayesian approach is used. The study is motivated by a real SAE problem. We focus on estimation of the per‐farm average grape wine production in Tuscany, at subregional level, using the Farm Structure Survey data. Results from this real data application and those obtained by a model‐based simulation experiment show a satisfactory performance of the suggested SAE approach.     

Preference patterns for skewed gambles in rhesus monkeys

While standard models of risky choice account for the first and second statistical moments of reward outcome distributions (mean and variance, respectively), they often ignore the third moment, skewness. Determining a decision-maker''s attitude about skewness is useful because it can help constrain process models of the mental steps involved in risky choice. We measured three rhesus monkeys’ preferences for gambles whose outcome distributions had almost identical means and variances but differed in skewness. We tested five distributions of skewness: strong negative, weak negative, normal, weak positive and strong positive. Monkeys preferred positively skewed gambles to negatively skewed ones and preferred strongly skewed and normal (i.e. unskewed) gambles to weakly skewed ones. This pattern of preferences cannot be explained solely by monotonic deformations of the utility curve or any other popular single account, but can be accounted for by multiple interacting factors.     

The Binary Regression Quantile Plot: Assessing the Importance of Predictors in Binary Regression Visually

We present a graphical measure of assessing the explanatory power of regression models with a binary response. The binary regression quantile plot and an area defined by it are used for the visual comparison and ordering of nested binary response regression models. The plot shows how well various models explain the data. Two data sets are analyzed and the area representing the fit of a model is shown to agree with the usual likelihood ratio test.     

Analyzing population growth curves

Assessing animal population growth curves is an essential feature of field studies in ecology and wildlife management. We used five models to assess population growth rates with a number of sets of population growth rate data. A 'generalized' logistic curve provides a better model than do four other popular models. Use of difference equations for fitting was checked by a comparison of that method and direct fitting of the analytical (integrated) solution for three of the models. Fits to field data indicate that estimates of the asymptote, K, from the 'generalized logistic' and the ordinary logistic agree well enough to support use of estimates of K from the ordinary logistic on data that cannot be satisfactorily fitted with the generalized logistic. Akaike's information criterion is widely used, often with a small sample version AIC_c. Our study of five models indicated a bias in the AIC_c criterion, so we recommend checking results with estimates of variance about regression for fitted models. Fitting growth curves provides a valuable supplement to, and check on computer models of populations.     

Decomposing the heterogeneity of species distributions into multiple scales: a hierarchical framework for large‐scale count surveys

We introduce a novel spatially explicit framework for decomposing species distributions into multiple scales from count data. These kinds of data are usually positively skewed, have non‐normal distributions and are spatially autocorrelated. To analyse such data, we propose a hierarchical model that takes into account the observation process and explicitly deals with spatial autocorrelation. The latent variable is the product of a positive trend representing the non‐constant mean of the species distribution and of a stationary positive spatial field representing the variance of the spatial density of the species distribution. Then, the different scales of emergent structures of the distribution of the population in space are modelled from the latent density of the species distribution using multi‐scale variogram models. Multi‐scale kriging is used to map the spatial patterns previously identified by the multi‐scale models. We show how our framework yields robust and precise estimates of the relevant scales both for spatial count data simulated from well‐defined models, and in a real case‐study based on seabird count data (the common guillemot Uria aalge) provided by large‐scale aerial surveys of the Bay of Biscay (France) performed over a winter. Our stochastic simulation study provides guidelines on the expected uncertainties of the scales estimates. Our results indicate that the spatial structure of the common guillemot can be modelled as a three‐level hierarchical system composed of a very broad‐scale pattern (～ 200 km) with a stable location over time that might be environmentally controlled, a broad‐scale pattern (～ 50 km) with a variable shape and location, that might be related to shifts in prey distribution, and a fine‐scale pattern (～ 10 km) with a rather stable shape and location, that might be controlled by behavioural processes. Our framework enables the development of robust, scale‐dependent hypotheses regarding the potential ecological processes that control species distributions.     

Selection on skewed characters and the paradox of stasis

Observed phenotypic responses to selection in the wild often differ from predictions based on measurements of selection and genetic variance. An overlooked hypothesis to explain this paradox of stasis is that a skewed phenotypic distribution affects natural selection and evolution. We show through mathematical modeling that, when a trait selected for an optimum phenotype has a skewed distribution, directional selection is detected even at evolutionary equilibrium, where it causes no change in the mean phenotype. When environmental effects are skewed, Lande and Arnold's (1983) directional gradient is in the direction opposite to the skew. In contrast, skewed breeding values can displace the mean phenotype from the optimum, causing directional selection in the direction of the skew. These effects can be partitioned out using alternative selection estimates based on average derivatives of individual relative fitness, or additive genetic covariances between relative fitness and trait (Robertson–Price identity). We assess the validity of these predictions using simulations of selection estimation under moderate sample sizes. Ecologically relevant traits may commonly have skewed distributions, as we here exemplify with avian laying date — repeatedly described as more evolutionarily stable than expected — so this skewness should be accounted for when investigating evolutionary dynamics in the wild.     

Physiology in ecological niche modeling: using zebra mussel's upper thermal tolerance to refine model predictions through Bayesian analysis

Climate change and human-mediated dispersal are increasingly influencing species’ geographic distributions. Ecological niche models (ENMs) are widely used in forecasting species’ distributions, but are weak in extrapolation to novel environments because they rely on available distributional data and do not incorporate mechanistic information, such as species’ physiological response to abiotic conditions. To improve accuracy of ENMs, we incorporated physiological knowledge through Bayesian analysis. In a case study of the zebra mussel Dreissena polymorpha, we used native and global occurrences to obtain native and global models representing narrower and broader understanding of zebra mussel’ response to temperature. We also obtained thermal limit and survival information for zebra mussel from peer-reviewed literature and used the two types of information separately and jointly to calibrate native models. We showed that, compared to global models, native models predicted lower relative probability of presence along zebra mussel's upper thermal limit, suggesting the shortcoming of native models in predicting zebra mussel's response to warm temperature. We also found that native models showed improved prediction of relative probability of presence when thermal limit was used alone, and best approximated global models when both thermal limit and survival data were used. Our result suggests that integration of physiological knowledge enhances extrapolation of ENM in novel environments. Our modeling framework can be generalized for other species or other physiological limits and may incorporate evolutionary information (e.g. evolved thermal tolerance), thus has the potential to improve predictions of species’ invasive potential and distributional response to climate change.     

Effects of censoring on parameter estimates and power in genetic modeling.

Genetic and environmental influences on variance in phenotypic traits may be estimated with normal theory Maximum Likelihood (ML). However, when the assumption of multivariate normality is not met, this method may result in biased parameter estimates and incorrect likelihood ratio tests. We simulated multivariate normal distributed twin data under the assumption of three different genetic models. Genetic model fitting was performed in six data sets: multivariate normal data, discrete uncensored data, censored data, square root transformed censored data, normal scores of censored data, and categorical data. Estimates were obtained with normal theory ML (data sets 1-5) and with categorical data analysis (data set 6). Statistical power was examined by fitting reduced models to the data. When fitting an ACE model to censored data, an unbiased estimate of the additive genetic effect was obtained. However, the common environmental effect was underestimated and the unique environmental effect was overestimated. Transformations did not remove this bias. When fitting an ADE model, the additive genetic effect was underestimated while the dominant and unique environmental effects were overestimated. In all models, the correct parameter estimates were recovered with categorical data analysis. However, with categorical data analysis, the statistical power decreased. The analysis of L-shaped distributed data with normal theory ML results in biased parameter estimates. Unbiased parameter estimates are obtained with categorical data analysis, but the power decreases.     

Intraspecific body size frequency distributions of insects

Although interspecific body size frequency distributions are well documented for many taxa, including the insects, intraspecific body size frequency distributions (IaBSFDs) are more poorly known, and their variation among mass-based and linear estimates of size has not been widely explored. Here we provide IaBSFDs for 16 species of insects based on both mass and linear estimates and large sample sizes (n ≥ 100). In addition, we review the published IaBSFDs for insects, though doing so is complicated by their under-emphasis in the literature. The form of IaBSFDs can differ substantially between mass-based and linear measures. Nonetheless, in non-social insects they tend to be normally distributed (18 of 27 species) or in fewer instances positively skewed. Negatively skewed distributions are infrequently reported and log transformation readily removes the positive skew. Sexual size dimorphism does not generally cause bimodality in IaBSFDs. The available information on IaBSFDs in the social insects suggests that these distributions are usually positively skewed or bimodal (24 of 30 species). However, only c. 15% of ant genera are polymorphic, suggesting that normal distributions are probably more common, but less frequently investigated. Although only 57 species, representing seven of the 29 orders of insects, have been considered here, it appears that whilst IaBSFDs are usually normal, other distribution shapes can be found in several species, though most notably among the social insects. By contrast, the interspecific body size frequency distribution is typically right-skewed in insects and in most other taxa.     

Stepping inside the niche: microclimate data are critical for accurate assessment of species' vulnerability to climate change

To assess a species'' vulnerability to climate change, we commonly use mapped environmental data that are coarsely resolved in time and space. Coarsely resolved temperature data are typically inaccurate at predicting temperatures in microhabitats used by an organism and may also exhibit spatial bias in topographically complex areas. One consequence of these inaccuracies is that coarsely resolved layers may predict thermal regimes at a site that exceed species'' known thermal limits. In this study, we use statistical downscaling to account for environmental factors and develop high-resolution estimates of daily maximum temperatures for a 36 000 km² study area over a 38-year period. We then demonstrate that this statistical downscaling provides temperature estimates that consistently place focal species within their fundamental thermal niche, whereas coarsely resolved layers do not. Our results highlight the need for incorporation of fine-scale weather data into species'' vulnerability analyses and demonstrate that a statistical downscaling approach can yield biologically relevant estimates of thermal regimes.     

Robust estimation of microbial diversity in theory and in practice

Quantifying diversity is of central importance for the study of structure, function and evolution of microbial communities. The estimation of microbial diversity has received renewed attention with the advent of large-scale metagenomic studies. Here, we consider what the diversity observed in a sample tells us about the diversity of the community being sampled. First, we argue that one cannot reliably estimate the absolute and relative number of microbial species present in a community without making unsupported assumptions about species abundance distributions. The reason for this is that sample data do not contain information about the number of rare species in the tail of species abundance distributions. We illustrate the difficulty in comparing species richness estimates by applying Chao''s estimator of species richness to a set of in silico communities: they are ranked incorrectly in the presence of large numbers of rare species. Next, we extend our analysis to a general family of diversity metrics (‘Hill diversities''), and construct lower and upper estimates of diversity values consistent with the sample data. The theory generalizes Chao''s estimator, which we retrieve as the lower estimate of species richness. We show that Shannon and Simpson diversity can be robustly estimated for the in silico communities. We analyze nine metagenomic data sets from a wide range of environments, and show that our findings are relevant for empirically-sampled communities. Hence, we recommend the use of Shannon and Simpson diversity rather than species richness in efforts to quantify and compare microbial diversity.     

Does Optimal Packing of Flowers in Syconia Shape Style Length Variation in Monoecious Figs?1

Style length variation in female flowers of monoecious figs has been shown to play an important role in regulating the proportion of flowers that develop into seeds and those that become infested by the pollinator wasp. In this study, we tested the suggestion that style length variation in figs is a consequence of optimal packing of the flowers. We show that optimal packing of flowers in fig syconia will result in a highly skewed distribution of style lengths and a positive skewness of pedicel lengths. These predictions were qualitatively tested in eight species of figs and the results indicate that observed style length distributions did not conform to those expected. We argue that while the pedicel lengths are likely to be a spinoff of optimal packing of the flowers, style lengths are probably shaped by independent selective fotces. The pedicel lengths also are subjected to compensatory growth so as to place the stigmas at a common height in the central cavity for effective pollination of the flowers. This was substantiated by the response (dependence) of pedicels to (on) the length of the style, neck, and other floral features. In all species the coefficient of variation of styles was very consistent (ca 30 %). This is in accordance with expectations if style lengths are shaped to regulate the proportion of flowers for wasp and seed production.