Analysis of neighborhood dynamics of forest ecosystems using likelihood methods and modeling.

Advances in computing power in the past 20 years have led to a proliferation of spatially explicit, individual-based models of population and ecosystem dynamics. In forest ecosystems, the individual-based models encapsulate an emerging theory of "neighborhood" dynamics, in which fine-scale spatial interactions regulate the demography of component tree species. The spatial distribution of component species, in turn, regulates spatial variation in a whole host of community and ecosystem properties, with subsequent feedbacks on component species. The development of these models has been facilitated by development of new methods of analysis of field data, in which critical demographic rates and ecosystem processes are analyzed in terms of the spatial distributions of neighboring trees and physical environmental factors. The analyses are based on likelihood methods and information theory, and they allow a tight linkage between the models and explicit parameterization of the models from field data. Maximum likelihood methods have a long history of use for point and interval estimation in statistics. In contrast, likelihood principles have only more gradually emerged in ecology as the foundation for an alternative to traditional hypothesis testing. The alternative framework stresses the process of identifying and selecting among competing models, or in the simplest case, among competing point estimates of a parameter of a model. There are four general steps involved in a likelihood analysis: (1) model specification, (2) parameter estimation using maximum likelihood methods, (3) model comparison, and (4) model evaluation. Our goal in this paper is to review recent developments in the use of likelihood methods and modeling for the analysis of neighborhood processes in forest ecosystems. We will focus on a single class of processes, seed dispersal and seedling dispersion, because recent papers provide compelling evidence of the potential power of the approach, and illustrate some of the statistical challenges in applying the methods.     

On estimating the exponent of power-law frequency distributions

Power-law frequency distributions characterize a wide array of natural phenomena. In ecology, biology, and many physical and social sciences, the exponents of these power laws are estimated to draw inference about the processes underlying the phenomenon, to test theoretical models, and to scale up from local observations to global patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately, the binning-based methods traditionally used in ecology and other disciplines perform quite poorly. Here we discuss more sophisticated methods for fitting these exponents based on cumulative distribution functions and maximum likelihood estimation. We illustrate their superior performance at estimating known exponents and provide details on how and when ecologists should use them. Our results confirm that maximum likelihood estimation outperforms other methods in both accuracy and precision. Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.     

Detecting and Modeling Spatial and Temporal Dependence in Conservation Biology

Abstract: Due to the structuring forces and large-scale physical processes that shape our biosphere, we often find that environmental and ecological data are either spatially or temporally—or both spatially and temporally—dependent. When these data are analyzed, statistical techniques and models are frequently applied that were developed for independent data. We describe some of the detrimental consequences, such as inefficient parameter estimators, biased hypothesis test results, and inaccurate predictions, of ignoring spatial and temporal data dependencies, and we cite an example of adverse statistical results occurring when spatial dependencies were disregarded. We also discuss and recommend available techniques used to detect and model spatial and temporal dependence, including variograms, covariograms, autocorrelation and partial autocorrelation plots, geostatistical techniques, Gaussian autoregressive models, K functions, and ARIMA models, in environmental and ecological research to avoid the aforementioned difficulties.     

On Uncertain Sightings and Inference about Extinction

The extinction of many species can only be inferred from the record of sightings of individuals. Solow et al. (2012, Uncertain sightings and the extinction of the Ivory‐billed Woodpecker. Conservation Biology 26: 180–184) describe a Bayesian approach to such inference and apply it to a sighting record of the Ivory‐billed Woodpecker (Campephilus principalis). A feature of this sighting record is that all uncertain sightings occurred after the most recent certain sighting. However, this appears to be an artifact. We extended this earlier work in 2 ways. First, we allowed for overlap in time between certain and uncertain sightings. Second, we considered 2 plausible statistical models of a sighting record. In one of these models, certain and uncertain sightings that are valid arise from the same process whereas in the other they arise from independent processes. We applied both models to the case of the Ivory‐billed Woodpecker. The result from the first model did not favor extinction, whereas the result for the second model did. This underscores the importance, in applying tests for extinction, of understanding what could be called the natural history of the sighting record. Sobre Avistamientos Inciertos e Inferencia de la Extinción     

Robust benchmark dose determination based on profile score methods

We investigate several methods commonly used to obtain a benchmark dose and show that those based on full likelihood or profile likelihood methods might have severe shortcomings. We propose two new profile likelihood-based approaches which overcome these problems. Another contribution is the extension of the benchmark dose determination to non full likelihood models, such as quasi-likelihood, generalized estimating equations, which are widely used in settings such as developmental toxicity where clustered data are encountered. This widening of the scope of application is possible by the use of (robust) score statistics. Benchmark dose methods are applied to a data set from a developmental toxicity study.     

Distinguishing between Lévy walks and strong alternative models

Lévy walks are a widely used but contentious model of animal movement patterns. They are contentious because they have been wrongly ascribed to some animal species through use of incorrect statistical methods and because they have not been adequately compared against strong alternative models, such as composite correlated random walks. This lack of comparison has been partly because the strong alternative models do not have simple likelihood functions. Here I show that power-spectra and the distribution of the first significant digits (the leading non-zero digits) of the step lengths can distinguish between Lévy walks and composite correlated random walks. Using these diagnostic tools, I bolster previous claims that honey bees use a movement strategy that can be approximated by Lévy walks when searching for their hive or for a food source.     

Species distribution models and ecological theory: A critical assessment and some possible new approaches

Given the importance of knowledge of species distribution for conservation and climate change management, continuous and progressive evaluation of the statistical models predicting species distributions is necessary. Current models are evaluated in terms of ecological theory used, the data model accepted and the statistical methods applied. Focus is restricted to Generalised Linear Models (GLM) and Generalised Additive Models (GAM). Certain currently unused regression methods are reviewed for their possible application to species modelling.A review of recent papers suggests that ecological theory is rarely explicitly considered. Current theory and results support species responses to environmental variables to be unimodal and often skewed though process-based theory is often lacking. Many studies fail to test for unimodal or skewed responses and straight-line relationships are often fitted without justification.Data resolution (size of sampling unit) determines the nature of the environmental niche models that can be fitted. A synthesis of differing ecophysiological ideas and the use of biophysical processes models could improve the selection of predictor variables. A better conceptual framework is needed for selecting variables.Comparison of statistical methods is difficult. Predictive success is insufficient and a test of ecological realism is also needed. Evaluation of methods needs artificial data, as there is no knowledge about the true relationships between variables for field data. However, use of artificial data is limited by lack of comprehensive theory.Three potentially new methods are reviewed. Quantile regression (QR) has potential and a strong theoretical justification in Liebig's law of the minimum. Structural equation modelling (SEM) has an appealing conceptual framework for testing causality but has problems with curvilinear relationships. Geographically weighted regression (GWR) intended to examine spatial non-stationarity of ecological processes requires further evaluation before being used.Synthesis and applications: explicit theory needs to be incorporated into species response models used in conservation. For example, testing for unimodal skewed responses should be a routine procedure. Clear statements of the ecological theory used, the nature of the data model and sufficient details of the statistical method are needed for current models to be evaluated. New statistical methods need to be evaluated for compatibility with ecological theory before use in applied ecology. Some recent work with artificial data suggests the combination of ecological knowledge and statistical skill is more important than the precise statistical method used. The potential exists for a synthesis of current species modelling approaches based on their differing ecological insights not their methodology.     

Functional autoregressive forecasting of long-term seabed evolution

There is a need for decadal predictions of the seabed evolution, for example to inform resurvey strategies when maintaining navigation channels. The understanding of the physical processes involved in morphological evolution, and the viability of process models to accurately model evolution over these time scales, are currently limited. As a result, statistical approaches are used to supply long-term forecasts. In this paper, we introduce a novel statistical approach for this problem: the autoregressive Hilbertian model (ARH). This model naturally assesses the time evolution of spatially-distributed measurements. We apply the technique to a coastal area in the East Anglian coast over the period 1846 to 2002, and compare with two other statistical methods used recently for seabed prediction: the autoregressive model and the EOF model. We evaluate the performance of the three methods by comparing observations and predictions for 2002. The ARH model enables a reduction of 10% of the root mean squared errors. Finally, we compute the variability in the predictions related to time sampling using the jackknife, a method that uses subsamples to quantify uncertainties.     

Sampling variability and estimates of density dependence: a composite-likelihood approach

It is well known that sampling variability, if not properly taken into account, affects various ecologically important analyses. Statistical inference for stochastic population dynamics models is difficult when, in addition to the process error, there is also sampling error. The standard maximum-likelihood approach suffers from large computational burden. In this paper, I discuss an application of the composite-likelihood method for estimation of the parameters of the Gompertz model in the presence of sampling variability. The main advantage of the method of composite likelihood is that it reduces the computational burden substantially with little loss of statistical efficiency. Missing observations are a common problem with many ecological time series. The method of composite likelihood can accommodate missing observations in a straightforward fashion. Environmental conditions also affect the parameters of stochastic population dynamics models. This method is shown to handle such nonstationary population dynamics processes as well. Many ecological time series are short, and statistical inferences based on such short time series tend to be less precise. However, spatial replications of short time series provide an opportunity to increase the effective sample size. Application of likelihood-based methods for spatial time-series data for population dynamics models is computationally prohibitive. The method of composite likelihood is shown to have significantly less computational burden, making it possible to analyze large spatial time-series data. After discussing the methodology in general terms, I illustrate its use by analyzing a time series of counts of American Redstart (Setophaga ruticilla) from the Breeding Bird Survey data, San Joaquin kit fox (Vulpes macrotis mutica) population abundance data, and spatial time series of Bull trout (Salvelinus confluentus) redds count data.     

Alternatives to statistical hypothesis testing in ecology: a guide to self teaching.

Statistical methods emphasizing formal hypothesis testing have dominated the analyses used by ecologists to gain insight from data. Here, we review alternatives to hypothesis testing including techniques for parameter estimation and model selection using likelihood and Bayesian techniques. These methods emphasize evaluation of weight of evidence for multiple hypotheses, multimodel inference, and use of prior information in analysis. We provide a tutorial for maximum likelihood estimation of model parameters and model selection using information theoretics, including a brief treatment of procedures for model comparison, model averaging, and use of data from multiple sources. We discuss the advantages of likelihood estimation, Bayesian analysis, and meta-analysis as ways to accumulate understanding across multiple studies. These statistical methods hold promise for new insight in ecology by encouraging thoughtful model building as part of inquiry, providing a unified framework for the empirical analysis of theoretical models, and by facilitating the formal accumulation of evidence bearing on fundamental questions.     

Continuous dose-response modeling and risk analysis with the gamma and reciprocal gamma distributions

Kodell and West (1993) describe two methods for calculating pointwise upper confidence limits on the risk function with normally distributed responses and using a certain definition of adverse quantitative effect. But Banga et al. (2000) have shown that these normal theory methods break down when applied to skew data. We accordingly develop a risk analysis model and associated likelihood-based methodology when the response follows either a gamma or reciprocal gamma distribution. The model supposes that the shape (index) parameter k of the response distribution is held fixed while the logarithm of the scale parameter is a linear model in terms of the dose level. Existence and uniqueness of the maximum likelihood estimates is established. Asymptotic likelihood-based upper and lower confidence limits on the risk are solutions of the Lagrange equations associated with a constrained optimization problem. Starting values for an iterative solution are obtained by replacing the Lagrange equations by the lowest order terms in their asymptotic expansions. Three methods are then compared for calculating confidence limits on the risk: (i) the aforementioned starting values (LRAL method), (ii) full iterative solution of the Lagrange equations (LREL method), and (iii) bounds obtained using approximate normality of the maximum likelihood estimates with standard errors derived from the information matrix (MLE method). Simulation is used to assess coverage probabilities for the resulting upper confidence limits when the log of the scale parameter is quadratic in the dose level. Results indicate that coverage for the MLE method can be off by as much as 15% points and converges very slowly to nominal coverage levels as the sample size increases. Coverage for the LRAL and LREL methods, on the other hand, is close to nominal levels unless (a) the sample size is small, say N < 25, (b) the index parameter is small, say k 1, and (c) the direction of adversity is to the left for the gamma distribution or to the right for the reciprocal gamma distribution.     

Characterizing source-sink dynamics with genetic parentage assignments

Source-sink dynamics have been suggested to characterize the population structure of many species, but the prevalence of source-sink systems in nature is uncertain because of inherent challenges in estimating migration rates among populations. Migration rates are often difficult to estimate directly with demographic methods, and indirect genetic methods are subject to a variety of assumptions that are difficult to meet or to apply to evolutionary timescales. Furthermore, such methods cannot be rigorously applied to high-gene-flow species. Here, we employ genetic parentage assignments in conjunction with demographic simulations to infer the level of immigration into a putative sink population. We use individual-based demographic models to estimate expected distributions of parent-offspring dyads under competing sink and closed-population models. By comparing the actual number of parent-offspring dyads (identified from multilocus genetic profiles) in a random sample of individuals taken from a population to expectations under these two contrasting demographic models, it is possible to estimate the rate of immigration and test hypotheses related to the role of immigration on population processes on an ecological timescale. The difference in the expected number of parent-offspring dyads between the two population models was greatest when immigration into the sink population was high, indicating that unlike traditional population genetic inference models, the highest degree of statistical power is achieved for the approach presented here when migration rates are high. We used the proposed genetic parentage approach to demonstrate that a threatened population of Marbled Murrelets (Braclhyrarmphus marmotus) appears to be supplemented by a low level of immigration (approximately 2-6% annually) from other populations.     

Efficient Markov chain Monte Carlo sampling for hierarchical hidden Markov models

Traditional Markov chain Monte Carlo (MCMC) sampling of hidden Markov models (HMMs) involves latent states underlying an imperfect observation process, and generates posterior samples for top-level parameters concurrently with nuisance latent variables. When potentially many HMMs are embedded within a hierarchical model, this can result in prohibitively long MCMC runtimes. We study combinations of existing methods, which are shown to vastly improve computational efficiency for these hierarchical models while maintaining the modeling flexibility provided by embedded HMMs. The methods include discrete filtering of the HMM likelihood to remove latent states, reduced data representations, and a novel procedure for dynamic block sampling of posterior dimensions. The first two methods have been used in isolation in existing application-specific software, but are not generally available for incorporation in arbitrary model structures. Using the NIMBLE package for R, we develop and test combined computational approaches using three examples from ecological capture–recapture, although our methods are generally applicable to any embedded discrete HMMs. These combinations provide several orders of magnitude improvement in MCMC sampling efficiency, defined as the rate of generating effectively independent posterior samples. In addition to being computationally significant for this class of hierarchical models, this result underscores the potential for vast improvements to MCMC sampling efficiency which can result from combinations of known algorithms.     

GIS and geostatistics: Essential partners for spatial analysis

Initially, geographical information systems (GIS) concentrated on two issues: automated map making, and facilitating the comparison of data on thematic maps. The first required high quality graphics, vector data models and powerful data bases, the second is based on grid cells that can be manipulated by suites of mathematical operators collectively termed map algebra. Both kinds of GIS are widely available and are taught in many universities and technical colleges. After more than 20 years of development, most standard GIS provide both kinds of functionality and good quality graphic display, but until recently they have not included the methods of statistics and geostatistics as tools for spatial analysis. Recently, standard statistical packages have been linked to GIS for both exploratory data analysis and statistical analysis and hypothesis testing. Standard statistical packages include methods for the analysis of random samples of cases or objects that are not necessarily co-located in space—if the results of statistical analysis display a spatial pattern then that is because the underlying data also share that pattern. Geostatistics addresses the need to make predictions of sampled attributes (i.e., maps) at unsampled locations from sparse, often expensive data. To make up for lack of hard data geostatistics has concentrated on the development of powerful methods based on stochastic theory. Though there have been recent moves to incorporate ancillary data in geostatistical analyses, insufficient attention has been paid to using modern methods of data display for the visualization of results. GIS can serve geostatistics by aiding geo-registration of data, facilitating spatial exploratory data analysis, providing a spatial context for interpolation and conditional simulation, as well as providing easy-to-use and effective tools for data display and visualization. The value of geostatistics for GIS lies in the provision of reliable interpolation methods with known errors, methods of upscaling and generalization, and for supplying multiple realizations of spatial patterns that can be used in environmental modeling. These stochastic methods are improving understanding of how errors in models of spatial processes accrue from errors in data or incompleteness in the structure of the models. New developments in GIS, based on ideas taken from map algebra, cellular automata and image analysis are providing high level programming languages for modeling dynamic processes such as erosion or the development of alluvial fans and deltas. Research has demonstrated that these models need stochastic inputs to yield realistic results. Non-stochastic tools such as fuzzy subsets have been shown to be useful for spatial analysis when probabilistic approaches are inappropriate or impossible. The conclusion is that in spite of differences in history and approach, the linkage of GIS, statistics and geostatistics provides a powerful, and complementary suite of tools for spatial analysis in the agricultural, earth and environmental sciences.     

Modeling fecundity in birds: Conceptual overview, current models, and considerations for future developments

Fecundity is fundamental to the fitness, population dynamics, conservation, and management of birds. For all the efforts made to measure fecundity or its surrogates over the past century of avian research, it is still mismeasured, misrepresented, and misunderstood. Fundamentally, these problems arise because of partial observability of underlying processes such as renesting, multiple brooding, and temporary emigration. Over the last several decades, various analytical approaches have been developed to estimate fecundity from incomplete and biased data. These, include scalar arithmetic formulae, partial differential equations, individual-based simulations, and Markov chain methodology. In this paper, we: (1) identify component processes of avian reproduction; (2) review existing methods for modeling fecundity; (3) place these diverse models under a common conceptual framework; (4) describe the parameterization, validation, and limitations of such models; and (5) point out future considerations and challenges in the application of fecundity models. We hope this synthesis of existing literature will help direct researchers toward the most appropriate methods to assess avian reproductive success for answering questions in evolutionary ecology, natural history, population dynamics, reproductive toxicology, and management.     

Hierarchical Bayesian space-time models

Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.     

Hierarchical Bayesian Spatial Models for Multispecies Conservation Planning and Monitoring

Abstract: Biologists who develop and apply habitat models are often familiar with the statistical challenges posed by their data's spatial structure but are unsure of whether the use of complex spatial models will increase the utility of model results in planning. We compared the relative performance of nonspatial and hierarchical Bayesian spatial models for three vertebrate and invertebrate taxa of conservation concern (Church's sideband snails [Monadenia churchi], red tree voles [Arborimus longicaudus], and Pacific fishers [Martes pennanti pacifica]) that provide examples of a range of distributional extents and dispersal abilities. We used presence–absence data derived from regional monitoring programs to develop models with both landscape and site‐level environmental covariates. We used Markov chain Monte Carlo algorithms and a conditional autoregressive or intrinsic conditional autoregressive model framework to fit spatial models. The fit of Bayesian spatial models was between 35 and 55% better than the fit of nonspatial analogue models. Bayesian spatial models outperformed analogous models developed with maximum entropy (Maxent) methods. Although the best spatial and nonspatial models included similar environmental variables, spatial models provided estimates of residual spatial effects that suggested how ecological processes might structure distribution patterns. Spatial models built from presence–absence data improved fit most for localized endemic species with ranges constrained by poorly known biogeographic factors and for widely distributed species suspected to be strongly affected by unmeasured environmental variables or population processes. By treating spatial effects as a variable of interest rather than a nuisance, hierarchical Bayesian spatial models, especially when they are based on a common broad‐scale spatial lattice (here the national Forest Inventory and Analysis grid of 24 km² hexagons), can increase the relevance of habitat models to multispecies conservation planning.     

A method for assigning species into groups based on generalized Mahalanobis distance between habitat model coefficients

Habitat association models are commonly developed for individual animal species using generalized linear modeling methods such as logistic regression. We considered the issue of grouping species based on their habitat use so that management decisions can be based on sets of species rather than individual species. This research was motivated by a study of western landbirds in northern Idaho forests. The method we examined was to separately fit models to each species and to use a generalized Mahalanobis distance between coefficient vectors to create a distance matrix among species. Clustering methods were used to group species from the distance matrix, and multidimensional scaling methods were used to visualize the relations among species groups. Methods were also discussed for evaluating the sensitivity of the conclusions because of outliers or influential data points. We illustrate these methods with data from the landbird study conducted in northern Idaho. Simulation results are presented to compare the success of this method to alternative methods using Euclidean distance between coefficient vectors and to methods that do not use habitat association models. These simulations demonstrate that our Mahalanobis-distance-based method was nearly always better than Euclidean-distance-based methods or methods not based on habitat association models. The methods used to develop candidate species groups are easily explained to other scientists and resource managers since they mainly rely on classical multivariate statistical methods.