Can the cause of aggregation be inferred from species distributions?

Species distributions often show an aggregated pattern, which can be due to a number of endo- and exogenous factors. While autologistic models have been used for modelling such data with statistical rigour, little emphasis has been put on disentangling potential causes of aggregation. In this paper we ask whether it is possible to infer sources of aggregation in species distributions from a single set of occurrence data by comparing the performance of various autologistic models. We create simulated data sets, which show similar occupancy patterns, but differ in the process that causes the aggregation. We model the distribution of these data with various autologistic models, and show how the relative performance of the models is sensitive to the factor causing aggregation in the data. This information can be used when modelling real species data, where causes of aggregation are typically unknown. To illustrate, we use our approach to assess the potential causes of aggregation in data of seven bird species with contrasting statistical patterns. Our findings have important implications for conservation, as understanding the mechanisms that drive population fluctuations in space and time is critical for the development of effective management actions for long-term conservation.     

A new method for dealing with residual spatial autocorrelation in species distribution models

Species distribution modelling (SDM) is a widely used tool and has many applications in ecology and conservation biology. Spatial autocorrelation (SAC), a pattern in which observations are related to one another by their geographic distance, is common in georeferenced ecological data. SAC in the residuals of SDMs violates the ‘independent errors’ assumption required to justify the use of statistical models in modelling species’ distributions. The autologistic modelling approach accounts for SAC by including an additional term (the autocovariate) representing the similarity between the value of the response variable at a location and neighbouring locations. However, autologistic models have been found to introduce bias in the estimation of parameters describing the influence of explanatory variables on habitat occupancy. To address this problem we developed an extension to the autologistic approach by calculating the autocovariate on SAC in residuals (the RAC approach). Performance of the new approach was tested on simulated data with a known spatial structure and on strongly autocorrelated mangrove species’ distribution data collected in northern Australia. The RAC approach was implemented as generalized linear models (GLMs) and boosted regression tree (BRT) models. We found that the BRT models with only environmental explanatory variables can account for some SAC, but applying the standard autologistic or RAC approaches further reduced SAC in model residuals and substantially improved model predictive performance. The RAC approach showed stronger inferential performance than the standard autologistic approach, as parameter estimates were more accurate and statistically significant variables were accurately identified. The new RAC approach presented here has the potential to account for spatial autocorrelation while maintaining strong predictive and inferential performance, and can be implemented across a range of modelling approaches.     

Predicting the distribution of the invasive alien Heracleum mantegazzianum at two different spatial scales

Modelling the distribution of invasive alien species is widely used for predicting future dispersal, response to climate change, and effects of management, but little information is available on the scale dependence of spatial models. This study is focused on Heracleum mantegazzianum , a problematic invasive plant in central and north-western Europe. The main objective was to model the current distribution of this species at national (43,000 km²) and regional scale (4900 km²) using autologistic regression with a Danish data set. Presence–absence data were used in a grid system with 5 × 5 km² or 2 × 2 km² as basic units. To avoid misleading presence–absence models and unreliable probability values due to unbalanced data, the prevalence was used as cut-off value, and a favourability function was applied to the model predictions. The national model showed a widespread distribution of H. mantegazzianum with highest habitat suitability in the eastern and northern parts of the country where human population density is high, winters more severe and/or loamy soils more common. At a regional scale the distribution of H. mantegazzianum is associated with alluvial sand cover, high human population density, spring precipitation, and presence of the species in neighbour grid units. The observed widespread national distribution is likely the result of anthropogenic spread of this ornamental plant, while the locally clumped distribution suggests that H. mantegazzianum naturally spreads mainly over short distances. The current distribution in Denmark resembles an intermediate invasion stage where long-distance dispersal is less important, while spread from suitable neighbour habitats is significant. The study demonstrates that the favourability function leads to improved mapping standards for invasive species.     

A model for range expansion of an introduced species: the common waxbill Estrilda astrild in Portugal

Abstract. The common waxbill Estrilda astrild was first introduced to Portugal from Africa in 1964, and has spread across much of the country and into Spain. We modelled the expansion of the common waxbill on a 20 × 20 km UTM grid in 4‐year periods from 1964 to 1999. The time variation of the square root of the occupied area shows that this expansion process is stabilizing in Portugal, and reasons for this are discussed. Several methods used to model biological expansions are not appropriate for the present case, because little quantitative data are available on the species ecology and because this expansion has been spatially heterogeneous. Instead, colonization on a grid was modelled as a function of several biophysical and spatio‐temporal variables through the fitting of a multivariate autologistic equation. This approach allows examination of the underlying factors affecting the colonization process. In the case of the common waxbill it was associated positively with its occurrence in adjacent cells, and affected negatively by altitude and higher levels of solar radiation.     

The Mizon–Richard Encompassing Test for the Cox and Aalen Additive Hazards Models

Summary .   The Cox hazards model ( Cox, 1972 , Journal of the Royal Statistical Society, Series B 34, 187–220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model ( Aalen, 1980 , in Lecture Notes in Statistics-2 , 1–25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon–Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model ( Martinussen and Scheike, 2002 , Biometrika 89, 283–298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.     

Conditional generalized estimating equations for the analysis of clustered and longitudinal data

Summary .   A common and important problem in clustered sampling designs is that the effect of within-cluster exposures (i.e., exposures that vary within clusters) on outcome may be confounded by both measured and unmeasured cluster-level factors (i.e., measurements that do not vary within clusters). When some of these are ill/not accounted for, estimation of this effect through population-averaged models or random-effects models may introduce bias. We accommodate this by developing a general theory for the analysis of clustered data, which enables consistent and asymptotically normal estimation of the effects of within-cluster exposures in the presence of cluster-level confounders. Semiparametric efficient estimators are obtained by solving so-called conditional generalized estimating equations. We compare this approach with a popular proposal by Neuhaus and Kalbfleisch (1998, Biometrics 54, 638–645) who separate the exposure effect into a within- and a between-cluster component within a random intercept model. We find that the latter approach yields consistent and efficient estimators when the model is linear, but is less flexible in terms of model specification. Under nonlinear models, this approach may yield inconsistent and inefficient estimators, though with little bias in most practical settings.     

Distribution of selected macroinvertebrates in a mosaic of temporary and permanent freshwater ponds as explained by autologistic models

We investigated the aquatic macroinvertebrate fauna of 76 ponds and small pools in an urban fringe landscape, and related the presence of ten species to measures of water permanence, pond area and environmental conditions using logistic models. The incidence of all the species was strongly associated with variation in hydroperiod, but patterns were more variable with the other explanatory variables. To determine whether the presence of a species at neighbouring ponds increased its probability of occurance at a pond we constructed a series of autologistic models, that differed from the aspatial logistic model in that they included a spatial autocovariate in the predictor terms. The improvement of model fit on inclusion of this autocovariate, measured as the decline in deviance compared to the aspatial models, was determined across a range of lag distances. In seven of the ten species, the autologistic models explained the incidence of the species amongst the ponds better than the aspatial models. Spatial effects were typically over short distances (<200 m) before declining, but in two species appeared to reach an asymptote, and we propose that variation in dispersal ability is the most likely factor producing these spatial effects. We conclude that it is essential that some measure of spatial autocorrelation is considered when evaluating the distribution of aquatic macroinvertebrates at small or medium scales.     

Dynamic occupancy models for analyzing species' range dynamics across large geographic scales

Large‐scale biodiversity data are needed to predict species' responses to global change and to address basic questions in macroecology. While such data are increasingly becoming available, their analysis is challenging because of the typically large heterogeneity in spatial sampling intensity and the need to account for observation processes. Two further challenges are accounting for spatial effects that are not explained by covariates, and drawing inference on dynamics at these large spatial scales. We developed dynamic occupancy models to analyze large‐scale atlas data. In addition to occupancy, these models estimate local colonization and persistence probabilities. We accounted for spatial autocorrelation using conditional autoregressive models and autologistic models. We fitted the models to detection/nondetection data collected on a quarter‐degree grid across southern Africa during two atlas projects, using the hadeda ibis (Bostrychia hagedash) as an example. The model accurately reproduced the range expansion between the first (SABAP1: 1987–1992) and second (SABAP2: 2007–2012) Southern African Bird Atlas Project into the drier parts of interior South Africa. Grid cells occupied during SABAP1 generally remained occupied, but colonization of unoccupied grid cells was strongly dependent on the number of occupied grid cells in the neighborhood. The detection probability strongly varied across space due to variation in effort, observer identity, seasonality, and unexplained spatial effects. We present a flexible hierarchical approach for analyzing grid‐based atlas data using dynamical occupancy models. Our model is similar to a species' distribution model obtained using generalized additive models but has a number of advantages. Our model accounts for the heterogeneous sampling process, spatial correlation, and perhaps most importantly, allows us to examine dynamic aspects of species ranges.     

Aberrant crypt foci and semiparametric modeling of correlated binary data

Summary .   Motivated by the spatial modeling of aberrant crypt foci (ACF) in colon carcinogenesis, we consider binary data with probabilities modeled as the sum of a nonparametric mean plus a latent Gaussian spatial process that accounts for short-range dependencies. The mean is modeled in a general way using regression splines. The mean function can be viewed as a fixed effect and is estimated with a penalty for regularization. With the latent process viewed as another random effect, the model becomes a generalized linear mixed model. In our motivating data set and other applications, the sample size is too large to easily accommodate maximum likelihood or restricted maximum likelihood estimation (REML), so pairwise likelihood, a special case of composite likelihood, is used instead. We develop an asymptotic theory for models that are sufficiently general to be used in a wide variety of applications, including, but not limited to, the problem that motivated this work. The splines have penalty parameters that must converge to zero asymptotically: we derive theory for this along with a data-driven method for selecting the penalty parameter, a method that is shown in simulations to improve greatly upon standard devices, such as likelihood crossvalidation. Finally, we apply the methods to the data from our experiment ACF. We discover an unexpected location for peak formation of ACF.     

Fitting conditional survival models to meta-analytic data by using a transformation toward mixed-effects models

Summary .   Frailty models are widely used to model clustered survival data. Classical ways to fit frailty models are likelihood-based. We propose an alternative approach in which the original problem of "fitting a frailty model" is reformulated into the problem of "fitting a linear mixed model" using model transformation. We show that the transformation idea also works for multivariate proportional odds models and for multivariate additive risks models. It therefore bridges segregated methodologies as it provides a general way to fit conditional models for multivariate survival data by using mixed models methodology. To study the specific features of the proposed method we focus on frailty models. Based on a simulation study, we show that the proposed method provides a good and simple alternative for fitting frailty models for data sets with a sufficiently large number of clusters and moderate to large sample sizes within covariate-level subgroups in the clusters. The proposed method is applied to data from 27 randomized trials in advanced colorectal cancer, which are available through the Meta-Analysis Group in Cancer.     

Insights into succession processes using temporally repeated habitat models: results from a long‐term study in a post‐mining landscape

Questions: The early phases of primary succession are governed by chance events and dispersal‐related processes in an environment that is largely free of competition. Thus, the predictability of vegetation patterns using environmental site factors can be expected to be low and spatial autocorrelation to be high. We asked whether the match between vegetation and environment becomes better in the course of succession, and whether vegetation types shift their realized niche with time. Location: Lignite mining region in Central Germany, the post‐mining landscape “Goitzsche” (Saxony‐Anhalt). Methods: Vegetation types were mapped in a 10‐m grid (total area 4.8 ha), starting in 1995, at 3‐year intervals until 2007. We used a temporal comparison of habitat models. We applied: GLS regression to partition the variation in coverage of vegetation types into environmental (soil pH) and spatial components; logistic regression to model the presence/absence of vegetation types along a soil acidity gradient; and autologistic regression allowing for soil acidity and neighbourhood effects. Results: For most vegetation types, the proportion of variation explained by space was high but declined during succession. The outcome of autologistic models suggests that soil acidity often plays a minor role compared to neighbourhood effects in the earlier phase of succession than 12 years later. For four vegetation types, the pH range in which the type was expected to be dominant clearly became smaller with time. These trends support the view that the role of processes related to chance and dispersal decrease with time, while those related to environmental filtering mediated by biotic interactions increase. Conclusions: We conclude that temporal comparisons of spatially explicit habitat models provide insights into changing biotic community processes and their effects on habitat specificity of species or their communities. Thus, this approach may be particularly important for analysis of ecological systems that are not in equilibrium with their environmental drivers.     

Model selection and information theory in geographical ecology

Aim   Although parameter estimates are not as affected by spatial autocorrelation as Type I errors, the change from classical null hypothesis significance testing to model selection under an information theoretic approach does not completely avoid problems caused by spatial autocorrelation. Here we briefly review the model selection approach based on the Akaike information criterion (AIC) and present a new routine for Spatial Analysis in Macroecology (SAM) software that helps establishing minimum adequate models in the presence of spatial autocorrelation.
Innovation    We illustrate how a model selection approach based on the AIC can be used in geographical data by modelling patterns of mammal species in South America represented in a grid system ( n  = 383) with 2° of resolution, as a function of five environmental explanatory variables, performing an exhaustive search of minimum adequate models considering three regression methods: non-spatial ordinary least squares (OLS), spatial eigenvector mapping and the autoregressive (lagged-response) model. The models selected by spatial methods included a smaller number of explanatory variables than the one selected by OLS, and minimum adequate models contain different explanatory variables, although model averaging revealed a similar rank of explanatory variables.
Main conclusions    We stress that the AIC is sensitive to the presence of spatial autocorrelation, generating unstable and overfitted minimum adequate models to describe macroecological data based on non-spatial OLS regression. Alternative regression techniques provided different minimum adequate models and have different uncertainty levels. Despite this, the averaged model based on Akaike weights generates consistent and robust results across different methods and may be the best approach for understanding of macroecological patterns.     

Allometric models for non-destructive leaf area estimation in coffee (Coffea arabica and Coffea canephora)

We aimed to evaluate the currently used allometric models, as well as to propose a reliable and accurate model using non-destructive measurements of leaf length (L) and/or width (W), for estimating the area of leaves of eight field-grown coffee cultivars. For model construction, a total of 1563 leaves were randomly selected from different levels of the tree canopies and encompassed the full spectrum of measurable leaf sizes (0.3–263 cm²) for each genotype. Power models better fit coffee leaf area (LA) than linear models. To validate the model, an independent data set of 388 leaves was used. We demonstrated that the currently used allometric models are biased, underestimating the area of a coffee leaf. We developed a single power model     based on two leaf dimensions [LA = 0.6626 (LW)^1.0116; standard errors: β₀ = 0.0064, β₁ = 0.0019; R² = 0.996] with high precision and accuracy, random dispersion pattern of residuals and also unbiased, irrespective of cultivar and leaf size and shape. Even when the L (but not width) alone was used as the single leaf dimension, the power model developed still predicted with good accuracy the LA but at the expense of some loss of precision, as particularly found for 8% of the leaves sampled with length-to-width ratios below 2.0 or above 3.0.     

Autologistic network model on binary data for disease progression study

This paper focuses on analysis of spatiotemporal binary data with absorbing states. The research was motivated by a clinical study on amyotrophic lateral sclerosis (ALS), a neurological disease marked by gradual loss of muscle strength over time in multiple body regions. We propose an autologistic regression model to capture complex spatial and temporal dependencies in muscle strength among different muscles. As it is not clear how the disease spreads from one muscle to another, it may not be reasonable to define a neighborhood structure based on spatial proximity. Relaxing the requirement for prespecification of spatial neighborhoods as in existing models, our method identifies an underlying network structure empirically to describe the pattern of spreading disease. The model also allows the network autoregressive effects to vary depending on the muscles’ previous status. Based on the joint distribution derived from this autologistic model, the joint transition probabilities of responses among locations can be estimated and the disease status can be predicted in the next time interval. Model parameters are estimated through maximization of penalized pseudo‐likelihood. Postmodel selection inference was conducted via a bias‐correction method, for which the asymptotic distributions were derived. Simulation studies were conducted to evaluate the performance of the proposed method. The method was applied to the analysis of muscle strength loss from the ALS clinical study.     

Gamma generalized linear models for pharmacokinetic data

Summary .   This article considers the modeling of single-dose pharmacokinetic data. Traditionally, so-called compartmental models have been used to analyze such data. Unfortunately, the mean function of such models are sums of exponentials for which inference and computation may not be straightforward. We present an alternative to these models based on generalized linear models, for which desirable statistical properties exist, with a logarithmic link and gamma distribution. The latter has a constant coefficient of variation, which is often appropriate for pharmacokinetic data. Inference is convenient from either a likelihood or a Bayesian perspective. We consider models for both single and multiple individuals, the latter via generalized linear mixed models. For single individuals, Bayesian computation may be carried out with recourse to simulation. We describe a rejection algorithm that, unlike Markov chain Monte Carlo, produces independent samples from the posterior and allows straightforward calculation of Bayes factors for model comparison. We also illustrate how prior distributions may be specified in terms of model-free pharmacokinetic parameters of interest. The methods are applied to data from 12 individuals following administration of the antiasthmatic agent theophylline.     

Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation

Summary We estimate the parameters of a stochastic process model for a macroparasite population within a host using approximate Bayesian computation (ABC). The immunity of the host is an unobserved model variable and only mature macroparasites at sacrifice of the host are counted. With very limited data, process rates are inferred reasonably precisely. Modeling involves a three variable Markov process for which the observed data likelihood is computationally intractable. ABC methods are particularly useful when the likelihood is analytically or computationally intractable. The ABC algorithm we present is based on sequential Monte Carlo, is adaptive in nature, and overcomes some drawbacks of previous approaches to ABC. The algorithm is validated on a test example involving simulated data from an autologistic model before being used to infer parameters of the Markov process model for experimental data. The fitted model explains the observed extra‐binomial variation in terms of a zero‐one immunity variable, which has a short‐lived presence in the host.     

Uncovering a Latent Multinomial: Analysis of Mark–Recapture Data with Misidentification

Summary .  Natural tags based on DNA fingerprints or natural features of animals are now becoming very widely used in wildlife population biology. However, classic capture–recapture models do not allow for misidentification of animals which is a potentially very serious problem with natural tags. Statistical analysis of misidentification processes is extremely difficult using traditional likelihood methods but is easily handled using Bayesian methods. We present a general framework for Bayesian analysis of categorical data arising from a latent multinomial distribution. Although our work is motivated by a specific model for misidentification in closed population capture–recapture analyses, with crucial assumptions which may not always be appropriate, the methods we develop extend naturally to a variety of other models with similar structure. Suppose that observed frequencies  f  are a known linear transformation    f = A ' x    of a latent multinomial variable  x  with cell probability vector    π = π ( θ )  . Given that full conditional distributions   [ θ  |  x ]   can be sampled, implementation of Gibbs sampling requires only that we can sample from the full conditional distribution   [ x  |  f , θ ]  , which is made possible by knowledge of the null space of A ' . We illustrate the approach using two data sets with individual misidentification, one simulated, the other summarizing recapture data for salamanders based on natural marks.     

Using a Bayesian latent growth curve model to identify trajectories of positive affect and negative events following myocardial infarction

Positive and negative affect data are often collected over time in psychiatric care settings, yet no generally accepted means are available to relate these data to useful diagnoses or treatments. Latent class analysis attempts data reduction by classifying subjects into one of K unobserved classes based on observed data. Latent class models have recently been extended to accommodate longitudinally observed data. We extend these approaches in a Bayesian framework to accommodate trajectories of both continuous and discrete data. We consider whether latent class models might be used to distinguish patients on the basis of trajectories of observed affect scores, reported events, and presence or absence of clinical depression.     

An autologistic model for the genetic analysis of familial binary data.

Regressive logistic models specify the probability distribution of familial binary traits by conditioning each individual's phenotype on those of preceding relatives; therefore, the expression of the joint probability of the familial data necessitates ordering the observations. In the present paper, we propose an autologistic model of this familial dependence structure, which does not require specification of a particular ordering of the phenotypic observations. Genetic effects are introduced into the model in order to perform segregation analysis that is aimed at detecting the role of a major locus in the expression of familial phenotypes. In this model, the conditional probabilities have a logistic form, and large patterns of dependence between relatives can be considered with a simple interpretation of the parameters measuring the relationship between two phenotypes. The model is compared with the regressive logistic approach in terms of odds ratios and by using a simulation study.     

Latent-Model Robustness in Joint Models for a Primary Endpoint and a Longitudinal Process

Summary .  Joint modeling of a primary response and a longitudinal process via shared random effects is widely used in many areas of application. Likelihood-based inference on joint models requires model specification of the random effects. Inappropriate model specification of random effects can compromise inference. We present methods to diagnose random effect model misspecification of the type that leads to biased inference on joint models. The methods are illustrated via application to simulated data, and by application to data from a study of bone mineral density in perimenopausal women and data from an HIV clinical trial.