Models for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums

Johnson and Wehrly (1978, Journal of the American Statistical Association 73, 602-606) and Wehrly and Johnson (1980, Biometrika 67, 255-256) show one way to construct the joint distribution of a circular and a linear random variable, or the joint distribution of a pair of circular random variables from their marginal distributions and the density of a circular random variable, which in this article is referred to as joining circular density. To construct flexible models, it is necessary that the joining circular density be able to present multimodality and/or skewness in order to model different dependence patterns. Fernández-Durán (2004, Biometrics 60, 499-503) constructed circular distributions based on nonnegative trigonometric sums that can present multimodality and/or skewness. Furthermore, they can be conveniently used as a model for circular-linear or circular-circular joint distributions. In the current work, joint distributions for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums are presented and applied to two data sets, one for circular-linear data related to the air pollution patterns in Mexico City and the other for circular-circular data related to the pair of dihedral angles between consecutive amino acids in a protein.     

A class of Bayesian shared gamma frailty models with multivariate failure time data

For multivariate failure time data, we propose a new class of shared gamma frailty models by imposing the Box-Cox transformation on the hazard function, and the product of the baseline hazard and the frailty. This novel class of models allows for a very broad range of shapes and relationships between the hazard and baseline hazard functions. It includes the well-known Cox gamma frailty model and a new additive gamma frailty model as two special cases. Due to the nonnegative hazard constraint, this shared gamma frailty model is computationally challenging in the Bayesian paradigm. The joint priors are constructed through a conditional-marginal specification, in which the conditional distribution is univariate, and it absorbs the nonlinear parameter constraints. The marginal part of the prior specification is free of constraints. The prior distributions allow us to easily compute the full conditionals needed for Gibbs sampling, while incorporating the constraints. This class of shared gamma frailty models is illustrated with a real dataset.     

Prioritizing global marine mammal habitats using density maps in place of range maps

Despite lessons from terrestrial systems, conservation efforts in marine systems continue to focus on identifying priority sites for protection based on high species richness inferred from range maps. Range maps oversimplify spatial variability in animal distributions by assuming uniform distribution within range and de facto giving equal weight to critical and marginal habitats. We used Marxan ver. 2.43 to compare species richness‐based systematic reserve network solutions using information about marine mammal range and relative abundance. At a global scale, reserve network solutions were strongly sensitive to model inputs and assumptions. Solutions based on different input data overlapped by a third at most, with agreement as low as 10% in some cases. At a regional scale, species richness was inversely related to density, such that species richness hotspots excluded highest‐density areas for all species. Based on these findings, we caution that species‐richness estimates derived from range maps and used as input in conservation planning exercises may inadvertently lead to protection of largely marginal habitat.     

Inclusion of soil data improves the performance of bioclimatic envelope models for insect species distributions in temperate Europe

Aim To analyse the effect of the inclusion of soil and land‐cover data on the performance of bioclimatic envelope models for the regional‐scale prediction of butterfly (Rhopalocera) and grasshopper (Orthoptera) distributions. Location Temperate Europe (Belgium). Methods Distributional data were extracted from butterfly and grasshopper atlases at a resolution of 5 km for the period 1991–2006 in Belgium. For each group separately, the well‐surveyed squares (n = 366 for butterflies and n = 322 for grasshoppers) were identified using an environmental stratification design and were randomly divided into calibration (70%) and evaluation (30%) datasets. Generalized additive models were applied to the calibration dataset to estimate occurrence probabilities for 63 butterfly and 33 grasshopper species, as a function of: (1) climate, (2) climate and land‐cover, (3) climate and soil, and (4) climate, land‐cover and soil variables. Models were evaluated as: (1) the amount of explained deviance in the calibration dataset, (2) Akaike’s information criterion, and (3) the number of omission and commission errors in the evaluation dataset. Results Information on broad land‐cover classes or predominant soil types led to similar improvements in the performance relative to the climate‐only models for both taxonomic groups. In addition, the joint inclusion of land‐cover and soil variables in the models provided predictions that fitted more closely to the species distributions than the predictions obtained from bioclimatic models incorporating only land‐cover or only soil variables. The combined models exhibited higher discrimination ability between the presence and absence of species in the evaluation dataset. Main conclusions These results draw attention to the importance of soil data for species distribution models at regional scales of analysis. The combined inclusion of land‐cover and soil data in the models makes it possible to identify areas with suitable climatic conditions but unsuitable combinations of vegetation and soil types. While contingent on the species, the results indicate the need to consider soil information in regional‐scale species–climate impact models, particularly when predicting future range shifts of species under climate change.     

Partitioning to uncover conditions for permutation tests to control multiple testing error rates

This article discusses specific assumptions necessary for permutation multiple tests to control the Familywise Error Rate (FWER). At issue is that, in comparing parameters of the marginal distributions of two sets of multivariate observations, validity of permutation testing is affected by all the parameters in the joint distributions of the observations. We show the surprising fact that, in the case of a linear model with i.i.d. errors such as in the analysis of Quantitative Trait Loci (QTL), this issue has no impact on control of FWER, if the test statistic is of a particular form. On the other hand, in the analysis of gene expression levels or multiple safety endpoints, unless some assumption connecting the marginal distributions of the observations to their joint distributions is made, permutation multiple tests may not control FWER.     

Foraminifer communities and environmental change in marginal marine sequences (Pliocene,Tuscany, Italy)

Fossil abundance data on foraminifer communities were collected in marginal marine sediments of the Pliocene Valdelsa succession, in Tuscany, Italy. This succession is organized in a hierarchy of elementary and composite depositional sequences. Multivariate techniques allowed to analyse the dataset and reconstruct gradients in species distributions. Species‐level data available on modern environmental distributions were used to reconstruct Pliocene environmental gradients and to infer absolute palaeodepths and palaeosalinities. Estimates were then compared with the sequence‐stratigraphic interpretation to check for consistency. The high‐resolution stratigraphic framework allowed us to test the stability of foraminifer communities against ecological variations related to high‐frequency glacio‐eustatic cycles. The results confirm that fossil distributions of foraminifer species can be used as a fine tool to detect environmental change and that multivariate techniques allow their interpretation in terms of absolute variations of controlling parameters. Salinity is the main contributor to the sum of depth‐related factors that regulate foraminifer distributions in coastal facies. In the same setting, nutrient levels and the presence of a sea grass cover are responsible for secondary changes in shallow‐water distributions. Below the wave base, however, depth‐related parameters other than salinity explain the largest variations. This study indicates that foraminifer communities are random associations of species that respond individualistically to environmental change.     

A bivariate quantitative genetic model for a threshold trait and a survival trait

Many of the functional traits considered in animal breeding can be analyzed as threshold traits or survival traits with examples including disease traits, conformation scores, calving difficulty and longevity. In this paper we derive and implement a bivariate quantitative genetic model for a threshold character and a survival trait that are genetically and environmentally correlated. For the survival trait, we considered the Weibull log-normal animal frailty model. A Bayesian approach using Gibbs sampling was adopted in which model parameters were augmented with unobserved liabilities associated with the threshold trait. The fully conditional posterior distributions associated with parameters of the threshold trait reduced to well known distributions. For the survival trait the two baseline Weibull parameters were updated jointly by a Metropolis-Hastings step. The remaining model parameters with non-normalized fully conditional distributions were updated univariately using adaptive rejection sampling. The Gibbs sampler was tested in a simulation study and illustrated in a joint analysis of calving difficulty and longevity of dairy cattle. The simulation study showed that the estimated marginal posterior distributions covered well and placed high density to the true values used in the simulation of data. The data analysis of calving difficulty and longevity showed that genetic variation exists for both traits. The additive genetic correlation was moderately favorable with marginal posterior mean equal to 0.37 and 95% central posterior credibility interval ranging between 0.11 and 0.61. Therefore, this study suggests that selection for improving one of the two traits will be beneficial for the other trait as well.     

Protein bioinformatics and mixtures of bivariate von Mises distributions for angular data

A fundamental problem in bioinformatics is to characterize the secondary structure of a protein, which has traditionally been carried out by examining a scatterplot (Ramachandran plot) of the conformational angles. We examine two natural bivariate von Mises distributions--referred to as Sine and Cosine models--which have five parameters and, for concentrated data, tend to a bivariate normal distribution. These are analyzed and their main properties derived. Conditions on the parameters are established which result in bimodal behavior for the joint density and the marginal distribution, and we note an interesting situation in which the joint density is bimodal but the marginal distributions are unimodal. We carry out comparisons of the two models, and it is seen that the Cosine model may be preferred. Mixture distributions of the Cosine model are fitted to two representative protein datasets using the expectation maximization algorithm, which results in an objective partition of the scatterplot into a number of components. Our results are consistent with empirical observations; new insights are discussed.     

Nonparametric estimation of the survival function for ordered multivariate failure time data: A comparative study

In longitudinal studies of disease, patients may experience several events through a follow‐up period. In these studies, the sequentially ordered events are often of interest and lead to problems that have received much attention recently. Issues of interest include the estimation of bivariate survival, marginal distributions, and the conditional distribution of gap times. In this work, we consider the estimation of the survival function conditional to a previous event. Different nonparametric approaches will be considered for estimating these quantities, all based on the Kaplan–Meier estimator of the survival function. We explore the finite sample behavior of the estimators through simulations. The different methods proposed in this article are applied to a dataset from a German Breast Cancer Study. The methods are used to obtain predictors for the conditional survival probabilities as well as to study the influence of recurrence in overall survival.     

Predictability of stream insect distributions is dependent on niche position,but not on biological traits or taxonomic relatedness of species

Species distributions can be analysed under two perspectives: the niche‐based approach, which focuses on species–environment relationships; and the dispersal‐based approach, which focuses on metapopulation dynamics. The degree to which each of these two components affect species distributions may depend on habitat fragmentation, species traits and phylogenetic constraints. We analysed the distributions of 36 stream insect species across 60 stream sites in three drainage basins at high latitudes in Finland. We used binomial generalised linear models (GLMs) in which the predictor variables were environmental factors (E models), within‐basin spatial variables as defined by Moran's eigenvector maps (M models), among‐basin variability (B models), or a combination of the three (E + M + B models) sets of variables. Based on a comparative analysis, model performance was evaluated across all the species using Gaussian GLMs whereby the deviance accounted for by binomial GLMs was fitted on selected explanatory variables: niche position, niche breadth, site occupancy, biological traits and taxonomic relatedness. For each type of model, a reduced Gaussian GLM was eventually obtained after variable selection (Bayesian information criterion). We found that niche position was the only variable selected in all reduced models, implying that marginal species were better predicted than non‐marginal species. The influence of niche position was strongest in models based on environmental variables (E models) or a combination of all types of variables (E + M + B models), and weakest in spatial autocorrelation models (M models). This suggests that species–environment relationships prevail over dispersal processes in determining stream insect distributions at a regional scale. Our findings have clear implications for biodiversity conservation strategies, and they also emphasise the benefits of considering both the niche‐based and dispersal‐based approaches in species distribution modelling studies.     

Relative importance of water, energy, and heterogeneity in determining regional pteridophyte and seed plant richness in China

Environmental variables, such as ambient energy, water availability, and environmental heterogeneity have been frequently proposed to account for species diversity gradients. How taxon-specific functional traits define large-scale richness gradients is a fundamental issue in understanding spatial patterns of species diversity, but has not been well documented. Using a large dataset on the regional flora from China, we examine the contrast spatial patterns and environmental determinants between pteridophytes and seed plants which differ in dispersal capacity and environmental requirements. Pteridophyte richness shows more pronounced spatial variation and stronger environmental associations than seed plant richness. Water availability generally accounts for more spatial variance in species richness of pteridophytes and seed plants than energy and heterogeneity do, especially for pteridophytes which have high dependence on moist and shady environments. Thus, pteridophyte richness is disproportionally affected by water-related variables; this in turn results in a higher proportion of pteridophytes in regional vascular plant floras (pteridophyte proportion) in wet regions. Most of the variance in seed plant richness, pteridophyte richness, and pteridophyte proportion explained by energy is included in variation that water and heterogeneity account for, indicating the redundancy of energy in the study extent. However, heterogeneity is more important for determining seed plant distributions. Pteridophyte and seed plant richness is strongly correlated, even after the environmental effects have been removed, implying functional linkages between them. Our study highlights the importance of incorporating biological traits of different taxonomic groups into the studies of macroecology and global change biology.     

Intra-generic competition among Nothofagus in New Zealand's primary indigenous forests

Competitive interactions between New Zealand's four Nothofagus or southern beech species were analysed using an extensive dataset describing the composition of natural forests, supplemented by environmental estimates describing both climate and landform. Using multiple regression models of progressively increasing complexity, the analysis first accounted for variation in tree abundance attributable to both environment and regional-scale distributional disjunctions of likely historic origin. Intra-generic competition, expressed as variation in tree abundance dependent on the presence or absence of each congener, was then assessed by adding (1) simple terms to assess the magnitude of gross changes in abundance, and (2) interaction terms to assess variation in abundance along the dominant temperature gradient given different competitive contexts. Results indicate the presence of substantial intra-generic interactions, with simple interaction terms giving marginal increases in explained deviance equal to that explained by initial regressions using environment alone. Addition of interaction terms brought about smaller improvements in model fit, but confirm that variation in abundance along the dominant annual temperature gradient is strongly influenced by the competitive context provided by the remaining congeners. Such results are consistent with current understanding of the niche concept, and underline the difficulty inherent in using current species limits to predict likely changes in species distributions consequent on global warming.     

Wrong,but useful: regional species distribution models may not be improved by range‐wide data under biased sampling

Species distribution modeling (SDM) is an essential method in ecology and conservation. SDMs are often calibrated within one country's borders, typically along a limited environmental gradient with biased and incomplete data, making the quality of these models questionable. In this study, we evaluated how adequate are national presence‐only data for calibrating regional SDMs. We trained SDMs for Egyptian bat species at two different scales: only within Egypt and at a species‐specific global extent. We used two modeling algorithms: Maxent and elastic net, both under the point‐process modeling framework. For each modeling algorithm, we measured the congruence of the predictions of global and regional models for Egypt, assuming that the lower the congruence, the lower the appropriateness of the Egyptian dataset to describe the species' niche. We inspected the effect of incorporating predictions from global models as additional predictor (“prior”) to regional models, and quantified the improvement in terms of AUC and the congruence between regional models run with and without priors. Moreover, we analyzed predictive performance improvements after correction for sampling bias at both scales. On average, predictions from global and regional models in Egypt only weakly concur. Collectively, the use of priors did not lead to much improvement: similar AUC and high congruence between regional models calibrated with and without priors. Correction for sampling bias led to higher model performance, whatever prior used, making the use of priors less pronounced. Under biased and incomplete sampling, the use of global bats data did not improve regional model performance. Without enough bias‐free regional data, we cannot objectively identify the actual improvement of regional models after incorporating information from the global niche. However, we still believe in great potential for global model predictions to guide future surveys and improve regional sampling in data‐poor regions.     

A sib-pair strategy for the use of restriction fragment length polymorphisms to study the mode of transmission of type II diabetes.

Three models for explaining the joint distribution of DNA insertional elements and type II diabetes in sibships are given as potential candidates for resolving the mode of transmission for this common non-Mendelian disorder. While the distributions predicted from all models are subject to the same rigid marginal constraints, only two models can be considered genetic (a combined major locus multifactorial background model and a two-locus model). A sequential probability ratio test is proposed to distinguish between these two models. Once the probands have been ascertained and using realistic parameter estimates, it is shown that, on the average, fewer than 100 affected sib-pairs are required to reach a decision at the 1% significance level with 99% power.     

Integrating correlation between traits improves spatial predictions of plant functional composition

Functional trait composition is increasingly recognized as key to better understand and predict community responses to environmental gradients. Predictive approaches traditionally model the weighted mean trait values of communities (CWMs) as a function of environmental gradients. However, most approaches treat traits as independent regardless of known tradeoffs between them, which could lead to spurious predictions. To address this issue, we suggest jointly modeling a suit of functional traits along environmental gradients while accounting for relationships between traits. We use generalized additive mixed effect models to predict the functional composition of alpine grasslands in the Guisane Valley (France). We demonstrate that, compared to traditional approaches, joint trait models explain considerable amounts of variation in CWMs, yield less uncertainty in trait CWM predictions and provide more realistic spatial projections when extrapolating to novel environmental conditions. Modeling traits and their co‐variation jointly is an alternative and superior approach to predicting traits independently. Additionally, compared to a ‘predict first, assemble later’ approach that estimates trait CWMs post hoc based on stacked species distribution models, our ‘assemble first, predict later’ approach directly models trait‐responses along environmental gradients, and does not require data and models on species’ distributions, but only mean functional trait values per community plot. This highlights the great potential of joint trait modeling approaches in large‐scale mapping applications, such as spatial projections of the functional composition of vegetation and associated ecosystem services as a response to contemporary global change.     

Flexible copula model for integrating correlated multi-omics data from single-cell experiments

With recent advances in technologies to profile multi-omics data at the single-cell level, integrative multi-omics data analysis has been increasingly popular. It is increasingly common that information such as methylation changes, chromatin accessibility, and gene expression are jointly collected in a single-cell experiment. In biomedical studies, it is often of interest to study the associations between various data types and to examine how these associations might change according to other factors such as cell types and gene regulatory components. However, since each data type usually has a distinct marginal distribution, joint analysis of these changes of associations using multi-omics data is statistically challenging. In this paper, we propose a flexible copula-based framework to model covariate-dependent correlation structures independent of their marginals. In addition, the proposed approach could jointly combine a wide variety of univariate marginal distributions, either discrete or continuous, including the class of zero-inflated distributions. The performance of the proposed framework is demonstrated through a series of simulation studies. Finally, it is applied to a set of experimental data to investigate the dynamic relationship between single-cell RNA sequencing, chromatin accessibility, and DNA methylation at different germ layers during mouse gastrulation.     

How do disturbances and environmental heterogeneity affect the pace of forest distribution shifts under climate change?

Although it is widely predicted that the geographic distributions of tree species and forest types will undergo substantial shifts in future, modelling approaches used to date are largely unable to project the pace at which forest distributions will respond to environmental change. The expansion and contraction of forest distributions act against considerable demographic inertia in the present composition and size‐structure of forest stands as climate‐induced changes in growth, mortality, and recruitment alter population dynamics through time. We aimed to better understand how shifts in forest distributions reflect long‐term changes in tree demographic rates and population dynamics, and how such shifts are influenced by 1) disturbance from forest harvesting and 2) local environmental heterogeneity. Using a simple, data‐constrained gap model, we simulated regional forest dynamics in the eastern United States over the next 500 yr. We then compared the geographic distributions of five different forest types through time under present and altered climatic conditions, in scenarios that variously included and excluded forest harvesting and environmental heterogeneity. Although we held climate fixed after 100 yr, it took another 160 yr after this for these forest types to collectively experience 90% of their eventual climate‐related distribution gains and losses. Competition strongly affected the nature of responses to climate change. Harvesting accelerated and amplified gains by an early‐successional forest type at the expense of a late‐successional one, but these gains did not occur faster than those for other forest types. Environmental heterogeneity had little effect on distribution gains or losses through time. These findings indicate that forest distributions should respond quite slowly to climate change, with the leading and trailing edges of different forest types shifting over a span of centuries. Disturbances can expedite some transitions, but are unlikely to lead to wholesale changes in forest types in the coming decades.     

Loglinear Symmetry and Quasi-Symmetry Models for the Analysis of Change

Loglinear symmetry and quasi-symmetry models are proposed as tools for investigating various hypotheses about change. First, a survey of model representations is provided, including model specification in terms of hierarchical loglinear models and in design matrix notation. Secondly, the range of symmetry and quasi-symmetry models is extended to the joint analysis of several groups. Parameter constraints are discussed which allow one to test specific hypotheses about group differences in symmetric frequency distributions. Finally, symmetry and quasi-symmetry models are considered for multiway contigency tables. In this context, loglinear total score models are proposed for the analysis of symmetry in several marginal distributions. The proposed models reflect cross-sectional as well as longitudinal facets of development.     

The effect of 50 years of landscape change on species richness and community composition

We analysed a 50-year dataset of avian species observations to determine how richness and community composition varied over a period of landscape-scale environmental change. Our study area, northern lower Michigan, has experienced substantial land-use and land-cover change over time. Like much of the northern Midwest, it has shifted from a largely unpopulated, post-logging shrubland to a moderately populated closed-canopy forest. Such changes are generally expected to influence overall richness and community composition. We found that regional richness per year remained virtually unchanged over the study period. Year-to-year variation in species number was surprisingly low. Richness totals included vastly different species groups as the composition of the regional bird community changed substantially over time. Changes in the types of species present appear to reflect deterministic changes in habitat. The number of grassland and open-habitat species decreased, for example, while species associated with older forests and urban habitats increased. Our results suggest that habitat changes at the landscape scale do not necessarily lead to changes in the number of species a region can support. Such changes, however, do appear to influence the types of species that will occupy a region, and can lead to substantial changes in community composition.     

Stability and distribution of predator–prey systems: local and regional mechanisms and patterns

Explaining the coexistence and distribution of species in time and space remains a fundamental challenge. While species coexistence depends on both local and regional mechanisms, it is sometimes unclear which role each mechanism takes in a given ecosystem. Consequently, it is very hard to predict the response of the ecosystem to environmental changes. Here, we develop a model to study spatial patterns of coexistence, focusing on predator–prey and host–parasite populations. We show, both theoretically and empirically, that these systems may exhibit both local and regional patterns and mechanisms of coexistence. Changes in environmental parameters, such as spatial connectivity, may lead to a transition from regional to local coexistence or it may lead directly to extinction, depending on demographic parameters. This demonstrates the importance of simultaneously analysing interacting mechanisms that act at different spatial scales to understand the response of ecosystems to environmental changes.