Modeling continuous covariates with a “spike” at zero: Bivariate approaches

In epidemiology and clinical research, predictors often take value zero for a large amount of observations while the distribution of the remaining observations is continuous. These predictors are called variables with a spike at zero. Examples include smoking or alcohol consumption. Recently, an extension of the fractional polynomial (FP) procedure, a technique for modeling nonlinear relationships, was proposed to deal with such situations. To indicate whether or not a value is zero, a binary variable is added to the model. In a two stage procedure, called FP‐spike, the necessity of the binary variable and/or the continuous FP function for the positive part are assessed for a suitable fit. In univariate analyses, the FP‐spike procedure usually leads to functional relationships that are easy to interpret. This paper introduces four approaches for dealing with two variables with a spike at zero (SAZ). The methods depend on the bivariate distribution of zero and nonzero values. Bi‐Sep is the simplest of the four bivariate approaches. It uses the univariate FP‐spike procedure separately for the two SAZ variables. In Bi‐D3, Bi‐D1, and Bi‐Sub, proportions of zeros in both variables are considered simultaneously in the binary indicators. Therefore, these strategies can account for correlated variables. The methods can be used for arbitrary distributions of the covariates. For illustration and comparison of results, data from a case‐control study on laryngeal cancer, with smoking and alcohol intake as two SAZ variables, is considered. In addition, a possible extension to three or more SAZ variables is outlined. A combination of log‐linear models for the analysis of the correlation in combination with the bivariate approaches is proposed.     

Effect of historical land‐use and climate change on tree‐climate relationships in the upper Midwestern United States

Contemporary forest inventory data are widely used to understand environmental controls on tree species distributions and to construct models to project forest responses to climate change, but the stability and representativeness of contemporary tree‐climate relationships are poorly understood. We show that tree‐climate relationships for 15 tree genera in the upper Midwestern US have significantly altered over the last two centuries due to historical land‐use and climate change. Realised niches have shifted towards higher minimum temperatures and higher rainfall. A new attribution method implicates both historical climate change and land‐use in these shifts, with the relative importance varying among genera and climate variables. Most climate/land‐use interactions are compounding, in which historical land‐use reinforces shifts in species‐climate relationships toward wetter distributions, or confounding, in which land‐use complicates shifts towards warmer distributions. Compounding interactions imply that contemporary‐based models of species distributions may underestimate species resilience to climate change.     

Host range and distribution of small mammal fleas in South Africa,with a focus on species of medical and veterinary importance

The host range and distribution of flea species on rodents and insectivores across multiple vegetation types in South Africa were investigated. Habitat suitability for flea species considered as important vectors of disease in humans and domestic animals was modelled. Data originated from fleas that were recovered from small mammals captured at 29 localities during 2009–2013 and published literature searched for flea records. Climate‐based predictor variables, widely used in arthropod vector distribution, were selected and habitat suitability modelled for 10 flea vector species. A total of 2469 flea individuals representing 33 species and subspecies were collected from 1185 small mammals. Ten of each of the flea and rodent species are plague vectors and reservoirs, respectively. Multiple novel flea–host associations and locality records were noted. Three vector species were recorded from insectivores. Geographic distributions of flea species ranged from broad, across‐biome distributions to narrower distributions within one or two biomes. Habitat suitability models performed excellently for the majority of flea vectors and identified regions of summer and all‐year rainfall as representing suitable habitats for most vector species. Current knowledge of vector and disease ecology can benefit from similar sampling approaches that will be important not only for South Africa, but also for the sub‐region.     

Bayesian joint models with INLA exploring marine mobile predator–prey and competitor species habitat overlap

Understanding spatial physical habitat selection driven by competition and/or predator–prey interactions of mobile marine species is a fundamental goal of spatial ecology. However, spatial counts or density data for highly mobile animals often (1) include excess zeros, (2) have spatial correlation, and (3) have highly nonlinear relationships with physical habitat variables, which results in the need for complex joint spatial models. In this paper, we test the use of Bayesian hierarchical hurdle and zero‐inflated joint models with integrated nested Laplace approximation (INLA), to fit complex joint models to spatial patterns of eight mobile marine species (grey seal, harbor seal, harbor porpoise, common guillemot, black‐legged kittiwake, northern gannet, herring, and sandeels). For each joint model, we specified nonlinear smoothed effect of physical habitat covariates and selected either competing species or predator–prey interactions. Out of a range of six ecologically important physical and biologic variables that are predicted to change with climate change and large‐scale energy extraction, we identified the most important habitat variables for each species and present the relationships between these bio/physical variables and species distributions. In particular, we found that net primary production played a significant role in determining habitat preferences of all the selected mobile marine species. We have shown that the INLA method is well‐suited for modeling spatially correlated data with excessive zeros and is an efficient approach to fit complex joint spatial models with nonlinear effects of covariates. Our approach has demonstrated its ability to define joint habitat selection for both competing and prey–predator species that can be relevant to numerous issues in the management and conservation of mobile marine species.     

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.     

Estimating Material and Energy Flows in Life Cycle Inventory with Statistical Models

Data (un)availability and uncertainty are recurring problems in life cycle assessment, and particularly inventory analysis. Advances in life cycle inventory have focused on the propagation and management of uncertainty, but this article addresses the question of how to account for unavailable data and corresponding uncertainty. Large and complicated systems often lack complete data due to confidential practices or the efforts required in the data collection process. Electricity production with multiple processes generating a single product is a classic example. Instead of the conventional process‐based models to estimate missing data, the approach developed in this article divides systems based on functionally equivalent objects. Each one of these objects is then described in terms of characteristic variables, such as power capacity. Kriging, a flexible statistical estimator, allows for the estimation of unknown material and energy flows based on the objects’ characteristic variables. Both univariate and multivariate kriging are tested and compared to regression analysis. It is found that kriging performs better than linear regression, according to the mean absolute error criterion. Multivariate kriging provides an even more accurate joint estimation method to bridge data gaps scattered across inventories and when observable values of material and energy flows differ from one object to the next. Parameters of the underlying models are interpreted in terms of data uncertainty.     

The role of El Niño–Southern Oscillation in the dynamics of a savanna large herbivore population

Our understanding of large‐scale climatic phenomena and dynamics of large herbivore populations comes principally from research in northern regions with temperate, seasonal climate and animal communities with relatively low species diversity. To assess the generality of that perspective, we investigated effects of El Niño–Southern Oscillation (ENSO) on population dynamics of African buffalo Syncerus caffer inhabiting a semi‐arid savanna with variable rainfall. We used linear and nonlinear‐threshold models to investigate relationships between population parameters and explanatory variables affecting forage conditions (seasonal rainfall, Southern Oscillation Index [SOI]). El Niño‐related droughts in 1982–1983 and 1991–1992 were associated with strongly negative population change, a pattern expected to coincide with a decrease in normally high and constant adult survival. Consistent with that nonlinear pattern, we detected threshold relationships between wet‐season rainfall and population change. Juvenile recruitment was described best by linear relationships with dry‐season. Because ENSO operates primarily through wet‐season rainfall, whereas population dynamics were also related to dry‐season rainfall, SOI did not have the predictive ability of individual weather components.     

Predicting species distributions: a critical comparison of the most common statistical models using artificial species

Aim To test statistical models used to predict species distributions under different shapes of occurrence–environment relationship. We addressed three questions: (1) Is there a statistical technique that has a consistently higher predictive ability than others for all kinds of relationships? (2) How does species prevalence influence the relative performance of models? (3) When an automated stepwise selection procedure is used, does it improve predictive modelling, and are the relevant variables being selected? Location We used environmental data from a real landscape, the state of California, and simulated species distributions within this landscape. Methods Eighteen artificial species were generated, which varied in their occurrence response to the environmental gradients considered (random, linear, Gaussian, threshold or mixed), in the interaction of those factors (no interaction vs. multiplicative), and on their prevalence (50% vs. 5%). The landscape was then randomly sampled with a large (n = 2000) or small (n = 150) sample size, and the predictive ability of each statistical approach was assessed by comparing the true and predicted distributions using five different indexes of performance (area under the receiver‐operator characteristic curve, Kappa, correlation between true and predictive probability of occurrence, sensitivity and specificity). We compared generalized additive models (GAM) with and without flexible degrees of freedom, logistic regressions (general linear models, GLM) with and without variable selection, classification trees, and the genetic algorithm for rule‐set production (GARP). Results Species with threshold and mixed responses, additive environmental effects, and high prevalence generated better predictions than did other species for all statistical models. In general, GAM outperforms all other strategies, although differences with GLM are usually not significant. The two variable‐selection strategies presented here did not discriminate successfully between truly causal factors and correlated environmental variables. Main conclusions Based on our analyses, we recommend the use of GAM or GLM over classification trees or GARP, and the specification of any suspected interaction terms between predictors. An expert‐based variable selection procedure was preferable to the automated procedures used here. Finally, for low‐prevalence species, variability in model performance is both very high and sample‐dependent. This suggests that distribution models for species with low prevalence can be improved through targeted sampling.     

Modelling the impacts of flow regulation on fish distributions in naturally intermittent lowland streams: an approach for predicting restoration responses

1. Understanding the relationships between flow regime and the distribution of biota is critical for managing flows in regulated rivers. In northern Victoria, Australia, efforts are presently underway to restore a natural, intermittent flow regime to several streams which, for over 100 years, have received perennial diversions as part of a stock, irrigation and domestic water supply. 2. Bayesian, model‐averaged, binomial regression was used to predict probabilities of occurrence for 13 fish species, including five non‐native species, based on hydrologic variables characterising both the current and modelled future flow regimes at 10 sites representing a range of hydrologic regimes (categorised here as heavily regulated, moderately regulated and unregulated). 3. Regression models accurately predicted present probabilities of occurrence for most species across all sites. The models predicted a reduced likelihood of large, native, flow‐dependent species occurring at regulated sites following flow restoration. Predictions regarding the future distribution of widespread species including two small‐bodied native and four exotic species were less certain as current distributions of these widespread species were unrelated to hydrologic variables we examined and thus unlikely to be significantly affected by flow restoration. The distributions of two small native species currently restricted to unregulated sites are predicted to increase throughout the system. 4. This study illustrates the effects of artificially induced perennial flow on lowland fish distributions. Furthermore, the combination of pre‐restoration data together with predictive modelling provides valuable insights into the likely outcomes of flow regime shifts. 5. This study clearly demonstrates the value of combining empirical research and modelling in guiding environmental flow and ecosystem restoration decisions. Knowledge from the study is now helping guide management decisions and the development of mitigation strategies to protect highly valued species in the system from potential future threats.     

Bayesian Dynamical Systems Modelling in the Social Sciences

Data arising from social systems is often highly complex, involving non-linear relationships between the macro-level variables that characterize these systems. We present a method for analyzing this type of longitudinal or panel data using differential equations. We identify the best non-linear functions that capture interactions between variables, employing Bayes factor to decide how many interaction terms should be included in the model. This method punishes overly complicated models and identifies models with the most explanatory power. We illustrate our approach on the classic example of relating democracy and economic growth, identifying non-linear relationships between these two variables. We show how multiple variables and variable lags can be accounted for and provide a toolbox in R to implement our approach.     

Age‐specific fecundity of mammalian populations: A test of three mathematical models

Empirical age‐specific fecundity distributions are often based on small samples and hence include high levels of sampling error, particularly at the older ages. One solution to this problem is to smooth the distributions using appropriate models. The aim of this article is to compare the utility of three models for smoothing and/or graduating these distributions. The three models examined are 1) the Gamma distribution, 2) the Hadwiger function, and 3) the Brass polynomial. Test data sets consist of four types of primates (including humans), Asian elephants, and Przewalski’s horse (an extinct species). The results indicate that all three models work well with a variety of mammalian data. The simplest of these models, the Brass polynomial, cannot be rejected based on available data and appears to be the optimum choice. Zoo Biol 20:487–499, 2001. © 2002 Wiley‐Liss, Inc.     

The role of fiber-matrix interactions in a nonlinear fiber-reinforced strain energy model of tendon

The objective of this study was to develop a nonlinear and anisotropic three-dimensional mathematical model of tendon behavior in which the structural components of fibers, matrix, and fiber-matrix interactions are explicitly incorporated and to use this model to infer the contributions of these structures to tendon mechanical behavior. We hypothesized that this model would show that: (i) tendon mechanical behavior is not solely governed by the isotropic matrix and fiber stretch, but is also influenced by fiber-matrix interactions; and (ii) shear fiber-matrix interaction terms will better describe tendon mechanical behavior than bulk fiber-matrix interaction terms. Model versions that did and did not include fiber-matrix interaction terms were applied to experimental tendon stress-strain data in longitudinal and transverse orientations, and the R2 goodness-of-fit was evaluated. This study showed that models that included fiber-matrix interaction terms improved the fit to longitudinal data (R2(toe) = 0.88, R2(Lin) = 0.94) over models that only included isotropic matrix and fiber stretch terms (R2(Toe) = 0.36, R2(Lin) = 0.84). Shear fiber-matrix interaction terms proved to be responsible for the best fit to data and to contribute to stress-strain nonlinearity. The mathematical model of tendon behavior developed in this study showed that fiber-matrix interactions are an important contributor to tendon behavior The more complete characterization of mechanical behavior afforded by this mathematical model can lead to an improved understanding of structure-function relationships in soft tissues and, ultimately, to the development of tissue-engineered therapies for injury or degeneration.     

The use of polynomial regression analysis with indicator variables for interpretation of mercury in fish data

Mercury levels in fish in reservoirs and natural lakes have been monitored on a regular basis since 1978 at the La Grande hydroelectric complex located in the James Bay region of Québec, Canada. The main analytical tools historically used were analysis of covariance (ANCOVA), linear regression of the mercury-to-length relationship and Student-Newman-Keuls (SNK) multiple comparisons of mean mercury levels. Inadequacy of linear regression (mercury-to-length relationships are often curvilinear) and difficulties in comparing mean mercury levels when regressions differ lead us to use polynomial regression with indicator variables.For comparisons between years, polynomial regression models relate mercury levels to length (L), length squared (L²), binary (dummy) indicator variables (B_n), each representing a sampled year, and the products of each of these explanatory variables (L × B₁, L² × B₁, L × B₂, etc.). Optimal transformations of the mercury levels (for normality and homogeneity) were found by the Box-Cox procedure. The models so obtained formed a partially nested series corresponding to four situations: (a) all years are well represented by a single polynomial model; (b) the year-models are of the same shape, but the means may differ; (c) the means are the same, but the year-models differ in shape; (d) both the means and shapes may differ among years. Since year-specific models came from the general one, rigorous statistical comparisons are possible between models.Polynomial regression with indicator variables allows rigorous statistical comparisons of mercury-to-length relationships among years, even when the shape of the relationships differ. It is simple to obtain accurate estimates of mercury levels at standardized length, and multiple comparisons of these estimations are simple to perform. The method can also be applied to spatial analysis (comparison of sampling stations), or to the comparison of different biological forms of the same species (dwarf and normal lake whitefish).     

Multivariate Association and Dimension Reduction: A Generalization of Canonical Correlation Analysis

Summary In this article, we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an L ₂ distance between each projected vector and an unknown function of the other. The unknown functions are estimated using the Nadaraya–Watson smoother. Extensions to multiple sets and groups of multiple sets are also discussed, and a bootstrap procedure is developed to detect the number of significant relationships. All the proposed methods are assessed through extensive simulations and real data analyses. In particular, for environmental data from Los Angeles County, we apply our multiple‐set methodology to study relationships between mortality, weather, and pollutants vectors. Here, we detect existence of both linear and nonlinear relationships between the dimension‐reduced vectors, which are then used to build nonlinear time‐series regression models for the dimension‐reduced mortality vector. These findings also illustrate potential use of our method in many other applications. A comprehensive assessment of our methodologies along with their theoretical properties are given in a Web Appendix.     

Investigating the relationships between the distribution of British ground beetle species (Coleoptera, Carabidae) and temperature, precipitation and altitude

Aims We examine the relationships between the distribution of British ground beetle species and climatic and altitude variables with a view to developing models for evaluating the impact of climate change. Location Data from 1684 10‐km squares in Britain were used to model species–climate/altitude relationships. A validation data set was composed of data from 326 British 10‐km squares not used in the model data set. Methods The relationships between incidence and climate and altitude variables for 137 ground beetle species were investigated using logistic regression. The models produced were subjected to a validation exercise using the Kappa statistic with a second data set of 30 species. Distribution patterns for four species were predicted for Britain using the regression equations generated. Results As many as 136 ground beetle species showed significant relationships with one or more of the altitude and climatic variables but the amount of variation explained by the models was generally poor. Models explaining 20% or more of the variation in species incidence were generated for only 10 species. Mean summer temperature and mean annual temperature were the best predictors for eight and six of these 10 species respectively. Few models based on altitude, annual precipitation and mean winter temperature were good predictors of ground beetle species distribution. The results of the validation exercise were mixed, with models for four species showing good or moderate fits whilst the remainder were poor. Main conclusions Whilst there were many significant relationships between British ground beetle species distributions and altitude and climatic variables, these variables did not appear to be good predictors of ground beetle species distribution. The poor model performance appears to be related to the coarse nature of the response and predictor data sets and the absence of key predictors from the models.     

Ethnicity,sibling, and family correlates of young adults’ status attainment: A follow‐up study

Abstract

Longitudinal data were used to examine relationships among sibling variables, perceptions of family environments, and measures of educational attainment, occupational status, and occupational aspirations. The analyses involved 21‐year‐old Australians from Anglo‐Australian, Greek, and Southern Italian families. Regression surfaces were plotted from models that included terms to test for possible linear, interaction, and curvilinear associations among the variables. The study indicated that sibsize and birth order continued to have many significant associations with young adults’ status attainment even after taking into account the mediating influence of perceived family environments. Also, the investigation suggested that there are ethnic‐group differences in relations among sibling variables, perceived parents’ support for learning, and young adults’ status attainment.     

Incorporating ecological principles into statistical models for the prediction of species’ distribution and abundance

Understanding the determinants of species’ distributions and abundances is a central theme in ecology. The development of statistical models to achieve this has a long history and the notion that the model should closely reflect underlying scientific understanding has encouraged ecologists to adopt complex statistical methods as they arise. In this paper we describe a Bayesian hierarchical model that reflects a conceptual ecological model of multi‐scaled environmental determinants of riverine fish species’ distributions and abundances. We illustrate this with distribution and abundance data of a small‐bodied fish species, the Empire gudgeon Hypseleotris galii, in the Mary and Albert Rivers, Queensland, Australia. Specifically, the model sought to address; 1) the extent that landscape‐scale abiotic variables can explain the species’ distribution compared to local‐scale variables, 2) how local‐scale abiotic variables can explain species’ abundances, and 3) how are these local‐scale relationships mediated by landscape‐scale variables. Overall, the model accounted for around 60% of variation in the distribution and abundance of H. galii. The findings show that the landscape‐scale variables explain much of the distribution of the species; however, there was considerable improvement in estimating the species’ distribution with the addition of local‐scale variables. There were many strong relationships between abundance and local‐scale abiotic variables; however, several of these relationships were mediated by some of the landscape‐scale variables. The extent of spatial autocorrelation in the data was relatively low compared to the distances among sampling reaches. Our findings exemplify that Bayesian statistical modelling provides a robust framework for statistical modelling that reflects our ecological understanding. This allows ecologists to address a range of ecological questions with a single unified probability model rather than a series of disconnected analyses.     

Sibling and environmental correlates of adolescents’ perceptions of family environments: Ethnic group differences

Abstract

Hierarchical regression analyses and response surfaces were used in an investigation of relationships between sibling variables and adolescents’ perceptions of their family environments, after the impact on the perceptions of earlier family measures was taken into account. The analyses involved Australian adolescents and included 260 Anglo‐Australian, 120 Greek, and 90 Southern Italian families. Regression models included terms to test for possible linear, interaction, and curvilinear associations between the sibling and family environment variables. The analysis suggested the propositions that: (a) parents from different ethnic groups create different learning environments for their children and that there are ethnic‐group variations in how children perceive their family environments, (b) there are ethnic‐group variations in relationships between sibling variables and adolescents’ perceptions of their parents’ support for learning, and (c) within ethnic groups the relationships between sibling variables and perceived parental support for learning differ, depending on whether the support of fathers or mothers is being considered.     

Cost effective prediction of the eutrophication status of lakes and reservoirs

1. Eutrophication is a serious threat in many parts of the world, and identifying the environmental factors that determine the spatial distribution of eutrophicated waterbodies as well as the development of management tools is a challenge. 2. In this study, data from the Ile‐de‐France region were analysed to determine if catchment scale environmental variables could predict concentrations of chlorophyll a (used as a proxy for eutrophication status) of artificial lakes and reservoirs. 3. General additive models (GAM) and random forest models (RF) displayed greater predictive power than generalised linear models, indicating the importance of non‐monotonic relationships. Using RF modelling, very high predictive accuracy was achieved for both continuous and binomial (eutrophic or not) response variables (continuous: R² = 0.715; binomial: kappa = 0.764, 89% of waterbodies were accurately predicted). The better predictive power and robustness of RF versus GAM was attributed to the formers ability to better handle complex interactions between predictors and to account for threshold effects. 4. Our results confirmed the close link between the water quality of lakes and reservoirs and the characteristics of their catchments. Moreover, we also showed that (i) simple (e.g. linear and/or monotonic) relationships between catchment land use and water quality were only found for sub‐regional datasets, and (ii) land use needs to be considered in association with complementary environmental variables (hydromorphological variables) to best assess its impact on water quality.