Simultaneous estimation of the temporal and spatial extent of animal migration using step lengths and turning angles.

BACKGROUND: Animals of many different species, trophic levels, and life history strategies migrate, and the improvement of animal tracking technology allows ecologists to collect increasing amounts of detailed data on these movements. Understanding when animals migrate is important for managing their populations, but is still difficult despite modelling advancements. METHODS: We designed a model that parametrically estimates the timing of migration from animal tracking data. Our model identifies the beginning and end of migratory movements as signaled by change-points in step length and turning angle distributions. To this end, we can also use the model to estimate how an animal's movement changes when it begins migrating. In addition to a thorough simulation analysis, we tested our model on three datasets: migratory ferruginous hawks (Buteo regalis) in the Great Plains, barren-ground caribou (Rangifer tarandus groenlandicus) in northern Canada, and non-migratory brown bears (Ursus arctos) from the Canadian Arctic. RESULTS: Our simulation analysis suggests that our model is most useful for datasets where an increase in movement speed or directional autocorrelation is clearly detectable. We estimated the beginning and end of migration in caribou and hawks to the nearest day, while confirming a lack of migratory behaviour in the brown bears. In addition to estimating when caribou and ferruginous hawks migrated, our model also identified differences in how they migrated; ferruginous hawks achieved efficient migrations by drastically increasing their movement rates while caribou migration was achieved through significant increases in directional persistence. CONCLUSIONS: Our approach is applicable to many animal movement studies and includes parameters that can facilitate comparison between different species or datasets. We hope that rigorous assessment of migration metrics will aid understanding of both how and why animals move.

Profile Likelihood for Hierarchical Models Using Data Doubling.

In scientific problems, an appropriate statistical model often involves a large number of canonical parameters. Often times, the quantities of scientific interest are real-valued functions of these canonical parameters. Statistical inference for a specified function of the canonical parameters can be carried out via the Bayesian approach by simply using the posterior distribution of the specified function of the parameter of interest. Frequentist inference is usually based on the profile likelihood for the parameter of interest. When the likelihood function is analytical, computing the profile likelihood is simply a constrained optimization problem with many numerical algorithms available. However, for hierarchical models, computing the likelihood function and hence the profile likelihood function is difficult because of the high-dimensional integration involved. We describe a simple computational method to compute profile likelihood for any specified function of the parameters of a general hierarchical model using data doubling. We provide a mathematical proof for the validity of the method under regularity conditions that assure that the distribution of the maximum likelihood estimator of the canonical parameters is non-singular, multivariate, and Gaussian.

A unifying theory for two-dimensional spatial redistribution kernels with applications in population spread modelling.

When building models to explain the dispersal patterns of organisms, ecologists often use an isotropic redistribution kernel to represent the distribution of movement distances based on phenomenological observations or biological considerations of the underlying physical movement mechanism. The Gaussian, two-dimensional (2D) Laplace and Bessel kernels are common choices for 2D space. All three are special (or limiting) cases of a kernel family, the Whittle-Matérn-Yasuda (WMY), first derived by Yasuda from an assumption of 2D Fickian diffusion with gamma-distributed settling times. We provide a novel derivation of this kernel family, using the simpler assumption of constant settling hazard, by means of a non-Fickian 2D diffusion equation representing movements through heterogeneous 2D media having a fractal structure. Our derivation reveals connections among a number of established redistribution kernels, unifying them under a single, flexible modelling framework. We demonstrate improvements in predictive performance in an established model for the spread of the mountain pine beetle upon replacing the Gaussian kernel by the Whittle-Matérn-Yasuda, and report similar results for a novel approximation, the product-Whittle-Matérn-Yasuda, that substantially speeds computations in applications to large datasets.

Estimating the intensity of use by interacting predators and prey using camera traps.

Understanding how organisms distribute themselves in response to interacting species, ecosystems, climate, human development and time is fundamental to ecological study and practice. A measure to quantify the relationship among organisms and their environments is intensity of use: the rate of use of a specific resource in a defined unit of time. Estimating the intensity of use differs from estimating probabilities of occupancy or selection, which can remain constant even when the intensity of use varies. We describe a method to evaluate the intensity of use across conditions that vary in both space and time. We demonstrate its application on a large mammal community where linear developments and human activity are conjectured to influence the interactions between white-tailed deer (Odocoileus virginianus) and wolves (Canis lupus) with possible consequences on threatened woodland caribou (Rangifer tarandus caribou). We collect and quantify intensity of use data for multiple, interacting species with the goal of assessing management efficacy, including a habitat restoration strategy for linear developments. We test whether blocking linear developments by spreading logs across a 200-m interval can be applied as an immediate mitigation to reduce the intensities of use by humans, predator and prey species in a boreal caribou range. We deployed camera traps on linear developments with and without restoration treatments in a landscape exposed to both timber and oil development. We collected a three-year dataset and employed spatial recurrent event models to analyse intensity of use by an interacting human and large mammal community across a range of environmental and climatic conditions. Spatial recurrent event models revealed that intensity of use by humans influenced the intensity of use by all five large mammal species evaluated, and the intensities of use by wolves and deer were inextricably linked in space and time. Conditions that resist travel on linear developments had a strong negative effect on the intensity of human and large mammal use. Mitigation strategies that resist, or redirect, animal travel on linear developments can reduce the effects of resource development on interacting human and predator-prey interactions. Our approach is easily applied to other continuous time point-based survey methodologies and shows that measuring the intensity of use within animal communities can help scientists monitor, mitigate and understand ecological states and processes.

Errors in Statistical Inference Under Model Misspecification: Evidence, Hypothesis Testing, and AIC.

The methods for making statistical inferences in scientific analysis have diversified even within the frequentist branch of statistics, but comparison has been elusive. We approximate analytically and numerically the performance of Neyman-Pearson hypothesis testing, Fisher significance testing, information criteria, and evidential statistics (Royall, 1997). This last approach is implemented in the form of evidence functions: statistics for comparing two models by estimating, based on data, their relative distance to the generating process (i.e., truth) (Lele, 2004). A consequence of this definition is the salient property that the probabilities of misleading or weak evidence, error probabilities analogous to Type 1 and Type 2 errors in hypothesis testing, all approach 0 as sample size increases. Our comparison of these approaches focuses primarily on the frequency with which errors are made, both when models are correctly specified, and when they are misspecified, but also considers ease of interpretation. The error rates in evidential analysis all decrease to 0 as sample size increases even under model misspecification. Neyman-Pearson testing on the other hand, exhibits great difficulties under misspecification. The real Type 1 and Type 2 error rates can be less, equal to, or greater than the nominal rates depending on the nature of model misspecification. Under some reasonable circumstances, the probability of Type 1 error is an increasing function of sample size that can even approach 1! In contrast, under model misspecification an evidential analysis retains the desirable properties of always having a greater probability of selecting the best model over an inferior one and of having the probability of selecting the best model increase monotonically with sample size. We show that the evidence function concept fulfills the seeming objectives of model selection in ecology, both in a statistical as well as scientific sense, and that evidence functions are intuitive and easily grasped. We find that consistent information criteria are evidence functions but the MSE minimizing (or efficient) information criteria (e.g., AIC, AICc, TIC) are not. The error properties of the MSE minimizing criteria switch between those of evidence functions and those of Neyman-Pearson tests depending on models being compared.

Relative Selection Strength: Quantifying effect size in habitat- and step-selection inference.

Habitat-selection analysis lacks an appropriate measure of the ecological significance of the statistical estimates-a practical interpretation of the magnitude of the selection coefficients. There is a need for a standard approach that allows relating the strength of selection to a change in habitat conditions across space, a quantification of the estimated effect size that can be compared both within and across studies. We offer a solution, based on the epidemiological risk ratio, which we term the relative selection strength (RSS). For a "used-available" design with an exponential selection function, the RSS provides an appropriate interpretation of the magnitude of the estimated selection coefficients, conditional on all other covariates being fixed. This is similar to the interpretation of the regression coefficients in any multivariable regression analysis. Although technically correct, the conditional interpretation may be inappropriate when attempting to predict habitat use across a given landscape. Hence, we also provide a simple graphical tool that communicates both the conditional and average effect of the change in one covariate. The average-effect plot answers the question: What is the average change in the space use probability as we change the covariate of interest, while averaging over possible values of other covariates? We illustrate an application of the average-effect plot for the average effect of distance to road on space use for elk (Cervus elaphus) during the hunting season. We provide a list of potentially useful RSS expressions and discuss the utility of the RSS in the context of common ecological applications.

Selection, use, choice and occupancy: clarifying concepts in resource selection studies.

1. During the last decade, there has been a proliferation of statistical methods for studying resource selection by animals. While statistical techniques are advancing at a fast pace, there is confusion in the conceptual understanding of the meaning of various quantities that these statistical techniques provide. 2. Terms such as selection, choice, use, occupancy and preference often are employed as if they are synonymous. Many practitioners are unclear about the distinctions between different concepts such as 'probability of selection,' 'probability of use,' 'choice probabilities' and 'probability of occupancy'. 3. Similarly, practitioners are not always clear about the differences between and relevance of 'relative probability of selection' vs. 'probability of selection' to effective management. 4. Practitioners also are unaware that they are using only a single statistical model for modelling resource selection, namely the exponential probability of selection, when other models might be more appropriate. Currently, such multimodel inference is lacking in the resource selection literature. 5. In this paper, we attempt to clarify the concepts and terminology used in animal resource studies by illustrating the relationships among these various concepts and providing their statistical underpinnings.

Predators choose prey over prey habitats: evidence from a lynx-hare system.

Resource selection is grounded in the understanding that animals select resources based on fitness requirements. Despite uncertainty in how mechanisms relate to the landscape, resource selection studies often assume, but rarely demonstrate, a relationship between modeled variables and fitness mechanisms. Using Canada lynx (Lynx canadensis) and snowshoe hare (Lepus americanus) as a model system, we assess whether prey habitat is a viable surrogate for encounters between predators and prey. We simultaneously collected winter track data for lynx and hare in two study areas. We used information criteria to determine whether selection by lynx is best characterized by a hare resource selection probability function (RSPF) or by the amount of hare resource use. Results show that lynx selection is better explained by the amount of hare use (SIC = -21.9; Schwarz's Information Criterion) than by hare RSPF (SIC = -16.71), and that hare RSPF cannot be assumed to reveal the amount of resource use, a primary mechanism of predator selection. Our study reveals an obvious but important distinction between selection and use that is applicable to all resource selection studies. We recommend that resource selection studies be coupled with mechanistic data (e.g., metrics of diet, forage, fitness, or abundance) when investigating mechanisms of resource selection.

Improved estimation of site occupancy using penalized likelihood.

When detection or occupancy probability is small or when the number of sites and number of visits per site is small, maximum likelihood estimators (MLE) of site occupancy parameters have large biases, are numerically unstable, and the corresponding confidence intervals have smaller than nominal coverage. We propose an alternative method of estimation, based on penalized likelihood. This method is numerically stable, the estimators have smaller mean square error than the MLE, and associated confidence intervals have close to nominal coverage.

Hierarchical models in ecology: confidence intervals, hypothesis testing, and model selection using data cloning.

Hierarchical statistical models are increasingly being used to describe complex ecological processes. The data cloning (DC) method is a new general technique that uses Markov chain Monte Carlo (MCMC) algorithms to compute maximum likelihood (ML) estimates along with their asymptotic variance estimates for hierarchical models. Despite its generality, the method has two inferential limitations. First, it only provides Wald-type confidence intervals, known to be inaccurate in small samples. Second, it only yields ML parameter estimates, but not the maximized likelihood values used for profile likelihood intervals, likelihood ratio hypothesis tests, and information-theoretic model selection. Here we describe how to overcome these inferential limitations with a computationally efficient method for calculating likelihood ratios via data cloning. The ability to calculate likelihood ratios allows one to do hypothesis tests, construct accurate confidence intervals and undertake information-based model selection with hierarchical models in a frequentist context. To demonstrate the use of these tools with complex ecological models, we reanalyze part of Gause's classic Paramecium data with state-space population models containing both environmental noise and sampling error. The analysis results include improved confidence intervals for parameters, a hypothesis test of laboratory replication, and a comparison of the Beverton-Holt and the Ricker growth forms based on a model selection index.

Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods.

We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise.

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.

Weighted distributions and estimation of resource selection probability functions.

Understanding how organisms selectively use resources is essential for designing wildlife management strategies. The probability that an individual uses a given resource, as characterized by environmental factors, can be quantified in terms of the resource selection probability function (RSPF). The present literature on the topic has claimed that, except when both used and unused sites are known, the RSPF is non-estimable and that only a function proportional to RSPF, namely, the resource selection function (RSF) can be estimated. This paper describes a close connection between the estimation of the RSPF and the estimation of the weight function in the theory of weighted distributions. This connection can be used to obtain fully efficient, maximum likelihood estimators of the resource selection probability function under commonly used survey designs in wildlife management. The method is illustrated using GPS collar data for mountain goats (Oreamnos americanus de Blainville 1816) in northwest British Columbia, Canada.

The promise of geometric morphometrics.

Nontraditional or geometric morphometric methods have found wide application in the biological sciences, especially in anthropology, a field with a strong history of measurement of biological form. Controversy has arisen over which method is the "best" for quantifying the morphological difference between forms and for making proper statistical statements about the detected differences. This paper explains that many of these arguments are superfluous to the real issues that need to be understood by those wishing to apply morphometric methods to biological data. Validity, the ability of a method to find the correct answer, is rarely discussed and often ignored. We explain why demonstration of validity is a necessary step in the evaluation of methods used in morphometrics. Focusing specifically on landmark data, we discuss the concepts of size and shape, and reiterate that since no unique definition of size exists, shape can only be recognized with reference to a chosen surrogate for size. We explain why only a limited class of information related to the morphology of an object can be known when landmark data are used. This observation has genuine consequences, as certain morphometric methods are based on models that require specific assumptions, some of which exceed what can be known from landmark data. We show that orientation of an object with reference to other objects in a sample can never be known, because this information is not included in landmark data. Consequently, a descriptor of form difference that contains information on orientation is flawed because that information does not arise from evidence within the data, but instead is a product of a chosen orientation scheme. To illustrate these points, we apply superimposition, deformation, and linear distance-based morphometric methods to the analysis of a simulated data set for which the true differences are known. This analysis demonstrates the relative efficacy of various methods to reveal the true difference between forms. Our discussion is intended to be fair, but it will be obvious to the reader that we favor a particular approach. Our bias comes from the realization that morphometric methods should operate with a definition of form and form difference consistent with the limited class of information that can be known from landmark data. Answers based on information that can be known from the data are of more use to biological inquiry than those based on unjustifiable assumptions.