Generalized hierarchical multivariate CAR models for areal data

In the fields of medicine and public health, a common application of areal data models is the study of geographical patterns of disease. When we have several measurements recorded at each spatial location (for example, information on p>/= 2 diseases from the same population groups or regions), we need to consider multivariate areal data models in order to handle the dependence among the multivariate components as well as the spatial dependence between sites. In this article, we propose a flexible new class of generalized multivariate conditionally autoregressive (GMCAR) models for areal data, and show how it enriches the MCAR class. Our approach differs from earlier ones in that it directly specifies the joint distribution for a multivariate Markov random field (MRF) through the specification of simpler conditional and marginal models. This in turn leads to a significant reduction in the computational burden in hierarchical spatial random effect modeling, where posterior summaries are computed using Markov chain Monte Carlo (MCMC). We compare our approach with existing MCAR models in the literature via simulation, using average mean square error (AMSE) and a convenient hierarchical model selection criterion, the deviance information criterion (DIC; Spiegelhalter et al., 2002, Journal of the Royal Statistical Society, Series B64, 583-639). Finally, we offer a real-data application of our proposed GMCAR approach that models lung and esophagus cancer death rates during 1991-1998 in Minnesota counties.     

Incorporating capture heterogeneity in the estimation of autoregressive coefficients of animal population dynamics using capture–recapture data

Population dynamic models combine density dependence and environmental effects. Ignoring sampling uncertainty might lead to biased estimation of the strength of density dependence. This is typically addressed using state‐space model approaches, which integrate sampling error and population process estimates. Such models seldom include an explicit link between the sampling procedures and the true abundance, which is common in capture–recapture settings. However, many of the models proposed to estimate abundance in the presence of capture heterogeneity lead to incomplete likelihood functions and cannot be straightforwardly included in state‐space models. We assessed the importance of estimating sampling error explicitly by taking an intermediate approach between ignoring uncertainty in abundance estimates and fully specified state‐space models for density‐dependence estimation based on autoregressive processes. First, we estimated individual capture probabilities based on a heterogeneity model for a closed population, using a conditional multinomial likelihood, followed by a Horvitz–Thompson estimate for abundance. Second, we estimated coefficients of autoregressive models for the log abundance. Inference was performed using the methodology of integrated nested Laplace approximation (INLA). We performed an extensive simulation study to compare our approach with estimates disregarding capture history information, and using R‐package VGAM, for different parameter specifications. The methods were then applied to a real data set of gray‐sided voles Myodes rufocanus from Northern Norway. We found that density‐dependence estimation was improved when explicitly modeling sampling error in scenarios with low process variances, in which differences in coverage reached up to 8% in estimating the coefficients of the autoregressive processes. In this case, the bias also increased assuming a Poisson distribution in the observational model. For high process variances, the differences between methods were small and it appeared less important to model heterogeneity.     

Semi-discrete host-parasitoid models

Arthropod host-parasitoid interactions constitute a very important class of consumer resource dynamics. Discrete-time models are a tradition for such interactions and are characterized by an updating function, which relates the population densities at a fixed date in one year to those at the same date in the previous year. Previous workers have investigated the effects of functional response and density dependence on the stability of the host-parasitoid interaction by heuristically incorporating them in the updating function. Such an approach ignores the effects of population changing continuously within a year due to different processes (for example intraspecific competition, mortality from parasitism) that may act simultaneously. Their cumulative effect on the updating function is not obvious and a more systematic methodology is needed. This paper uses a hybrid approach to formulate the updating function. This is done by modeling the dynamics of various within-year processes in continuous-time, and reproduction as a discrete event. Using this formalism we derive results connecting the stability of the host-parasitoid interaction with different forms of density dependence and the form of the functional response. The latter results contradict previous conclusions from heuristically formulated models, and illustrate the need for such a hybrid approach in discrete-time host-parasitoid theory.     

The Role of Density Regulation in Extinction Processes and Population Viability Analysis

We review the role of density dependence in the stochastic extinction of populations and the role density dependence has played in population viability analysis (PVA) case studies. In total, 32 approaches have been used to model density regulation in theoretical or applied extinction models, 29 of them are mathematical functions of density dependence, and one approach uses empirical relationships between density and survival, reproduction, or growth rates. In addition, quasi-extinction levels are sometimes applied as a substitute for density dependence at low population size. Density dependence further has been modelled via explicit individual spacing behaviour and/or dispersal. We briefly summarise the features of density dependence available in standard PVA software, provide summary statistics about the use of density dependence in PVA case studies, and discuss the effects of density dependence on extinction probability. The introduction of an upper limit for population size has the effect that the probability of ultimate extinction becomes 1. Mean time to extinction increases with carrying capacity if populations start at high density, but carrying capacity often does not have any effect if populations start at low numbers. In contrast, the Allee effect is usually strong when populations start at low densities but has only a limited influence on persistence when populations start at high numbers. Contrary to previous opinions, other forms of density dependence may lead to increased or decreased persistence, depending on the type and strength of density dependence, the degree of environmental variability, and the growth rate. Furthermore, effects may be reversed for different quasi-extinction levels, making the use of arbitrary quasi-extinction levels problematic. Few systematic comparisons of the effects on persistence between different models of density dependence are available. These effects can be strikingly different among models. Our understanding of the effects of density dependence on extinction of metapopulations is rudimentary, but even opposite effects of density dependence can occur when metapopulations and single populations are contrasted. We argue that spatially explicit models hold particular promise for analysing the effects of density dependence on population viability provided a good knowledge of the biology of the species under consideration exists. Since the results of PVAs may critically depend on the way density dependence is modelled, combined efforts to advance statistical methods, field sampling, and modelling are urgently needed to elucidate the relationships between density, vital rates, and extinction probability.     

Density dependence and the control of helminth parasites

1. The transient dynamics and stability of a population are determined by the interplay between species density, its spatial distribution and the positive and negative density-dependent processes regulating population growth. 2. Using the human-helminth parasite system as an example, we propose that the life-stage upon which negative density dependence operates will influence the rate of host reinfection following anthelmintic chemotherapy, and the likely success of control programmes. 3. Simple deterministic models are developed which highlight how a parasite species whose population size is down-regulated by density-dependent establishment will reinfect a host population at a faster rate than a species with density-dependent parasite fecundity. 4. Different forms of density dependence can produce the same equilibrium behaviour but different transient dynamics. Under-representing the nature and magnitude of density-dependent mechanisms, and in particular those operating upon establishing life-stages, may cause the resilience of the parasite population to a control perturbation to be underestimated.     

Bayesian design for minimizing prediction uncertainty in bivariate spatial responses with applications to air quality monitoring

Model-based geostatistical design involves the selection of locations to collect data to minimize an expected loss function over a set of all possible locations. The loss function is specified to reflect the aim of data collection, which, for geostatistical studies, could be to minimize the prediction uncertainty at unobserved locations. In this paper, we propose a new approach to design such studies via a loss function derived through considering the entropy about the model predictions and the parameters of the model. The approach includes a multivariate extension to generalized linear spatial models, and thus can be used to design experiments with more than one response. Unfortunately, evaluating our proposed loss function is computationally expensive so we provide an approximation such that our approach can be adopted to design realistically sized geostatistical studies. This is demonstrated through a simulated study and through designing an air quality monitoring program in Queensland, Australia. The results show that our designs remain highly efficient in achieving each experimental objective individually, providing an ideal compromise between the two objectives. Accordingly, we advocate that our approach could be adopted more generally in model-based geostatistical design.     

A general framework for modeling population abundance data

Time-series data resulting from surveying wild animals are often described using state-space population dynamics models, in particular with Gompertz, Beverton-Holt, or Moran-Ricker latent processes. We show how hidden Markov model methodology provides a flexible framework for fitting a wide range of models to such data. This general approach makes it possible to model abundance on the natural or log scale, include multiple observations at each sampling occasion and compare alternative models using information criteria. It also easily accommodates unequal sampling time intervals, should that possibility occur, and allows testing for density dependence using the bootstrap. The paper is illustrated by replicated time series of red kangaroo abundances, and a univariate time series of ibex counts which are an order of magnitude larger. In the analyses carried out, we fit different latent process and observation models using the hidden Markov framework. Results are robust with regard to the necessary discretization of the state variable. We find no effective difference between the three latent models of the paper in terms of maximized likelihood value for the two applications presented, and also others analyzed. Simulations suggest that ecological time series are not sufficiently informative to distinguish between alternative latent processes for modeling population survey data when data do not indicate strong density dependence.     

Spatially explicit analyses unveil density dependence

Density-dependent processes are fundamental in the understanding of species population dynamics. Whereas the benefits of considering the spatial dimension in population biology are widely acknowledged, the implications of doing so for the statistical detection of spatial density dependence have not been examined. The outcome of traditional tests may therefore differ from those that include ecologically relevant locational information on both the prey species and natural enemy. Here, we explicitly incorporate spatial information on individual counts when testing for density dependence between an insect herbivore and its parasitoids. The spatially explicit approach used identified significant density dependence more frequently and in different instances than traditional methods. The form of density dependence detected also differed between methods. These results demonstrate that the explicit consideration of patch location in density-dependence analyses is likely to significantly alter current understanding of the prevalence and form of spatial density dependence in natural populations.     

Joint models for multivariate longitudinal and multivariate survival data

Joint modeling of longitudinal and survival data is becoming increasingly essential in most cancer and AIDS clinical trials. We propose a likelihood approach to extend both longitudinal and survival components to be multidimensional. A multivariate mixed effects model is presented to explicitly capture two different sources of dependence among longitudinal measures over time as well as dependence between different variables. For the survival component of the joint model, we introduce a shared frailty, which is assumed to have a positive stable distribution, to induce correlation between failure times. The proposed marginal univariate survival model, which accommodates both zero and nonzero cure fractions for the time to event, is then applied to each marginal survival function. The proposed multivariate survival model has a proportional hazards structure for the population hazard, conditionally as well as marginally, when the baseline covariates are specified through a specific mechanism. In addition, the model is capable of dealing with survival functions with different cure rate structures. The methodology is specifically applied to the International Breast Cancer Study Group (IBCSG) trial to investigate the relationship between quality of life, disease-free survival, and overall survival.     

Spatial scales in the study of reef fishes: A theoretical perspective

Abstract Theoretical models imply that spatial scale derives its greatest importance through interactions between density-dependent processes and spatial variation in population densities and environmental variables. Such interactions cause population dynamics on large spatial scales to differ in important ways from predictions based on measurements of population dynamics at smaller scales, a phenomenon called the scale transition. These differences can account for large-scale population stability and species coexistence. The interactions between density dependence and spatial variation that lead to the scale transition can be understood by the process of non-linear averaging, which shows how variance originating on various spatial scales contributes to large-scale population dynamics. Variance originating below the scale of density dependence contributes less to the scale transition as the spatial scale of the variation declines, while variation originating on or above the scale of density dependence contributes independently of the spatial scale of the variation.     

Is more data always better? A simulation study of benefits and limitations of integrated distribution models

Species distribution models are popular and widely applied ecological tools. Recent increases in data availability have led to opportunities and challenges for species distribution modelling. Each data source has different qualities, determined by how it was collected. As several data sources can inform on a single species, ecologists have often analysed just one of the data sources, but this loses information, as some data sources are discarded. Integrated distribution models (IDMs) were developed to enable inclusion of multiple datasets in a single model, whilst accounting for different data collection protocols. This is advantageous because it allows efficient use of all data available, can improve estimation and account for biases in data collection. What is not yet known is when integrating different data sources does not bring advantages. Here, for the first time, we explore the potential limits of IDMs using a simulation study integrating a spatially biased, opportunistic, presence-only dataset with a structured, presence–absence dataset. We explore four scenarios based on real ecological problems; small sample sizes, low levels of detection probability, correlations between covariates and a lack of knowledge of the drivers of bias in data collection. For each scenario we ask; do we see improvements in parameter estimation or the accuracy of spatial pattern prediction in the IDM versus modelling either data source alone? We found integration alone was unable to correct for spatial bias in presence-only data. Including a covariate to explain bias or adding a flexible spatial term improved IDM performance beyond single dataset models, with the models including a flexible spatial term producing the most accurate and robust estimates. Increasing the sample size of presence–absence data and having no correlated covariates also improved estimation. These results demonstrate under which conditions integrated models provide benefits over modelling single data sources.     

The biogeography of invasive alien plants in California: an application of GIS and spatial regression analysis

The spatial distribution of invasive alien plants has been poorly documented in California. However, with the increased availability of GIS software and spatially explicit data, the distribution of invasive alien plants can be explored. Using bioregions as defined in Hickman (1993 ), I compared the distribution of invasive alien plants (n = 78) and noninvasive alien plants (n = 1097). The distribution of both categories of alien plants was similar with the exception of a higher concentration of invasive alien plants in the North Coast bioregion. Spatial autocorrelation analysis using Moran's I indicated significant spatial dependence for both invasive and noninvasive alien plant species. I used both ordinary least squares (OLS) and spatial autoregressive (SAR) models to assess the relationship between alien plant species distribution and native plant species richness, road density, population density, elevation, area of sample unit, and precipitation. The OLS model for invasive alien plants included two significant effects; native plant species richness and elevation. The SAR model for invasive alien plants included three significant effects; elevation, road density, and native plant species richness. The SAR model for noninvasive alien plants resulted in the same significant effects as invasive alien plants. Both invasive and noninvasive alien plants are found in regions with low elevation, high road density, and high native‐plant species richness. This is in congruity with previous spatial pattern studies of alien plant species. However, the similarity in effects for both categories of alien plants alludes to the importance of autecological attributes, such as pollination system, dispersal system and differing responses to disturbance in the distribution of invasive plant species. In addition, this study emphasizes the critical importance of testing for spatial autocorrelation in spatial pattern studies and using SAR models when appropriate.     

Spatial variation in the magnitude and functional form of density‐dependent processes on the large skipper butterfly Ochlodes sylvanus

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FISHER'S GEOMETRIC MODEL WITH A MOVING OPTIMUM

Fisher's geometric model has been widely used to study the effects of pleiotropy and organismic complexity on phenotypic adaptation. Here, we study a version of Fisher's model in which a population adapts to a gradually moving optimum. Key parameters are the rate of environmental change, the dimensionality of phenotype space, and the patterns of mutational and selectional correlations. We focus on the distribution of adaptive substitutions, that is, the multivariate distribution of the phenotypic effects of fixed beneficial mutations. Our main results are based on an “adaptive‐walk approximation,” which is checked against individual‐based simulations. We find that (1) the distribution of adaptive substitutions is strongly affected by the ecological dynamics and largely depends on a single composite parameter γ, which scales the rate of environmental change by the “adaptive potential” of the population; (2) the distribution of adaptive substitution reflects the shape of the fitness landscape if the environment changes slowly, whereas it mirrors the distribution of new mutations if the environment changes fast; (3) in contrast to classical models of adaptation assuming a constant optimum, with a moving optimum, more complex organisms evolve via larger adaptive steps.     

Spatial heterogeneity and population risk analysis of cotton bollworm, Helicoverpa armigera, in China

In this study, semivariance was used to quantitatively measure the spatial heterogeneity for the egg population of cotton bollworm during a growing season. The typical characteristic parameters of theoretical semivariance models against lag distances were applied to measure components of spatial heterogeneity： trend, range, spatial dependence, and the strength of spatial dependence. Then, kriging interpolation was used to evaluate the population risk of cotton bollworm exceeding economic thresholds. From early June through early September, the population densities were sampled 10 times in the study field. Results showed that the spatial patterns were related to population density. For its low-density population, the spatially heterogeneous trends were usually of spherical shapes; but for highdensity ones, the trends shifted to Gaussian shapes. The spatial dependence appeared at varied distances ranging from 52 meters to 936 meters, and the spatial dependence was in the range of 0.39-288.60, which changed with population densities. While having high heterogeneity, the strength of spatial dependence became much stronger. Results of population risk analysis showed that there was a high risk during its early stages, especially in mid-June. In August, population risk was so low that it did not need to be controlled.     

Coregionalized single- and multiresolution spatially varying growth curve modeling with application to weed growth

Modeling of longitudinal data from agricultural experiments using growth curves helps understand conditions conducive or unconducive to crop growth. Recent advances in Geographical Information Systems (GIS) now allow geocoding of agricultural data that help understand spatial patterns. A particularly common problem is capturing spatial variation in growth patterns over the entire experimental domain. Statistical modeling in these settings can be challenging because agricultural designs are often spatially replicated, with arrays of subplots, and interest lies in capturing spatial variation at possibly different resolutions. In this article, we develop a framework for modeling spatially varying growth curves as Gaussian processes that capture associations at single and multiple resolutions. We provide Bayesian hierarchical models for this setting, where flexible parameterization enables spatial estimation and prediction of growth curves. We illustrate using data from weed growth experiments conducted in Waseca, Minnesota, that recorded growth of the weed Setaria spp. in a spatially replicated design.     

Model Diagnostic Plots for Repeated Measures Data

In the analysis of repeated measurements, multivariate regression models that account for the correlations among the observations from the same subject are widely used. Like the usual univariate regression models, these multivariate regression models also need some model diagnostic procedures. Though these models have been widely used, not many studies have been performed in model diagnostic areas. In this paper, we propose simple residual plots to investigate the goodness of model fit for repeated measures data. Here, we mainly focus on the mean model diagnostics. The proposed residual plots are based on the quantile‐quantile(Q–Q) plots of a χ² distribution and a normal distribution. In particular, the proposed model is useful in comparing several models simultaneously. The proposed method is illustrated using two examples. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal strategies of spatial distribution: the Olli effect

The selection of strategies of spatial distribution of individuals has been studied. In case of non-monotonous dependence of reproduction coefficient on the mean population density, a cluster formation is possible. At low mean densities, parity strategies of spatial distribution are realized, and at high densities, non-parity ones. A generalized notion of parity strategy of spatial distribution has been proposed. It includes such expenditures as expenditure for the movement of an individual, defense of the territory etc. A problem of evolutionary stability of different strategies of spatial distribution has been discussed.     

Effects of climate on population fluctuations of ibex

Predicting the effects of the expected changes in climate on the dynamics of populations require that critical periods for climate‐induced changes in population size are identified. Based on time series analyses of 26 Swiss ibex (Capra ibex) populations, we show that variation in winter climate affected the annual changes in population size of most of the populations after accounting for the effects of density dependence and demographic stochasticity. In addition, precipitation during early summer also influenced the population fluctuations. This suggests that the major influences of climate on ibex population dynamics operated either through loss of individuals during winter or early summer, or through an effect on fecundity. However, spatial covariation in these climate variables was not able to synchronize the population fluctuations of ibex over larger distances, probably due to large spatial heterogeneity in the effects of single climate variables on different populations. Such spatial variation in the influence of the same climate variable on the local population dynamics suggests that predictions of influences of climate change need to account for local differences in population dynamical responses to climatic conditions.