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On the Reliability of N-Mixture Models for Count Data
N-mixture models describe count data replicated in time and across sites in terms of abundance N and detectability p. They are popular because they allow inference about N while controlling for factors that influence p without the need for marking animals. Using a capture-recapture perspective, we show that the loss of information that results from not marking animals is critical, making reliable statistical modeling of N and p problematic using just count data. One cannot reliably fit a model in which the detection probabilities are distinct among repeat visits as this model is overspecified. This makes uncontrolled variation in p problematic. By counter example, we show that even if p is constant after adjusting for covariate effects (the \"constant p\" assumption) scientifically plausible alternative models in which N (or its expectation) is non-identifiable or does not even exist as a parameter, lead to data that are practically indistinguishable from data generated under an N-mixture model. This is particularly the case for sparse data as is commonly seen in applications. We conclude that under the constant p assumption reliable inference is only possible for relative abundance in the absence of questionable and/or untestable assumptions or with better quality data than seen in typical applications. Relative abundance models for counts can be readily fitted using Poisson regression in standard software such as R and are sufficiently flexible to allow controlling for p through the use covariates while simultaneously modeling variation in relative abundance. If users require estimates of absolute abundance, they should collect auxiliary data that help with estimation of p.
Journal Article
On the robustness of N-mixture models
N-mixture models provide an appealing alternative to mark–recapture models, in that they allow for estimation of detection probability and population size from count data, without requiring that individual animals be identified. There is, however, a cost to using the N-mixture models: inference is very sensitive to the model’s assumptions. We consider the effects of three violations of assumptions that might reasonably be expected in practice: double counting, unmodeled variation in population size over time, and unmodeled variation in detection probability over time. These three examples show that small violations of assumptions can lead to large biases in estimation. The violations of assumptions we consider are not only small qualitatively, but are also small in the sense that they are unlikely to be detected using goodness-of-fit tests. In cases where reliable estimates of population size are needed, we encourage investigators to allocate resources to acquiring additional data, such as recaptures of marked individuals, for estimation of detection probabilities.
Journal Article
Model Weights and the Foundations of Multimodel Inference
Statistical thinking in wildlife biology and ecology has been profoundly influenced by the introduction of AIC (Akaike's information criterion) as a tool for model selection and as a basis for model averaging. In this paper, we advocate the Bayesian paradigm as a broader framework for multimodel inference, one in which model averaging and model selection are naturally linked, and in which the performance of AIC-based tools is naturally evaluated. Prior model weights implicitly associated with the use of AIC are seen to highly favor complex models: in some cases, all but the most highly parameterized models in the model set are virtually ignored a priori. We suggest the usefulness of the weighted BIC (Bayesian information criterion) as a computationally simple alternative to AIC, based on explicit selection of prior model probabilities rather than acceptance of default priors associated with AIC. We note, however, that both procedures are only approximate to the use of exact Bayes factors. We discuss and illustrate technical difficulties associated with Bayes factors, and suggest approaches to avoiding these difficulties in the context of model selection for a logistic regression. Our example highlights the predisposition of AIC weighting to favor complex models and suggests a need for caution in using the BIC for computing approximate posterior model weights.
Journal Article
Identifying Pareto‐efficient eradication strategies for invasive populations
Invasive species are a major cause of biodiversity loss and are notoriously expensive and challenging to manage. We developed a decision‐analytic framework for evaluating invasive species removal strategies, given objectives of maximizing eradication probability and minimizing costs. The framework uses an existing estimation model for spatially referenced removal data—one of the most accessible types of invasive species data—to obtain estimates of population growth rate, movement probability, and detection probability. We use these estimates in simulations to identify Pareto‐efficient strategies—strategies where increases in eradication probability cannot be obtained without increases in cost—from a set of proposed strategies. We applied the framework post hoc to a successful eradication of veiled chameleons (Chamaeleo calyptratus) and identified the potential for substantial improvements in efficiency. Our approach provides managers and policymakers with tools to identify cost‐effective strategies for a range of invasive species using only prior knowledge or data from initial physical removals.
Journal Article
Tigers and Their Prey: Predicting Carnivore Densities from Prey Abundance
The goal of ecology is to understand interactions that determine the distribution and abundance of organisms. In principle, ecologists should be able to identify a small number of limiting resources for a species of interest, estimate densities of these resources at different locations across the landscape, and then use these estimates to predict the density of the focal species at these locations. In practice, however, development of functional relationships between abundances of species and their resources has proven extremely difficult, and examples of such predictive ability are very rare. Ecological studies of prey requirements of tigers Panthera tigris led us to develop a simple mechanistic model for predicting tiger density as a function of prey density. We tested our model using data from a landscape-scale long-term (1995-2003) field study that estimated tiger and prey densities in 11 ecologically diverse sites across India. We used field techniques and analytical methods that specifically addressed sampling and detectability, two issues that frequently present problems in macroecological studies of animal populations. Estimated densities of ungulate prey ranged between 5.3 and 63.8 animals per km2. Estimated tiger densities (3.2-16.8 tigers per 100 km2) were reasonably consistent with model predictions. The results provide evidence of a functional relationship between abundances of large carnivores and their prey under a wide range of ecological conditions. In addition to generating important insights into carnivore ecology and conservation, the study provides a potentially useful model for the rigorous conduct of macroecological science.
Journal Article
Individual Covariation in Life‐History Traits: Seeing the Trees Despite the Forest
We investigated the influence of age on survival and breeding rates in a long‐lived speciesRissa tridactylausing models with individual random effects permitting variation and covariation in fitness components among individuals. Differences in survival or breeding probabilities among individuals are substantial, and there was positive covariation between survival and breeding probability; birds that were more likely to survive were also more likely to breed, given that they survived. The pattern of age‐related variation in these rates detected at the individual level differed from that observed at the population level. Our results provided confirmation of what has been suggested by other investigators: within‐cohort phenotypic selection can mask senescence. Although this phenomenon has been extensively studied in humans and captive animals, conclusive evidence of the discrepancy between population‐level and individual‐level patterns of age‐related variation in life‐history traits is extremely rare in wild animal populations. Evolutionary studies of the influence of age on life‐history traits should use approaches differentiating population level from the genuine influence of age: only the latter is relevant to theories of life‐history evolution. The development of models permitting access to individual variation in fitness is a promising advance for the study of senescence and evolutionary processes.
Journal Article
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Analysis of the North American Breeding Bird Survey Using Hierarchical Models
We analyzed population change for 420 bird species from the North American Breeding Bird Survey (BBS) using a hierarchical log-linear model and compared the results with those obtained through route-regression analysis. Survey-wide trend estimates based on the hierarchical model were generally more precise than estimates from the earlier analysis. No consistent pattern of differences existed in the magnitude of trends between the analysis methods. Survey-wide trend estimates changed substantially for 15 species between route-regression and hierarchical-model analyses. We compared regional estimates for states, provinces, and Bird Conservation Regions; differences observed in these regional analyses are likely a consequence of the route-regression procedure's inadequate accommodation of temporal differences in survey effort. We used species-specific hierarchical-model results to estimate composite change for groups of birds associated with major habitats and migration types. Grassland, aridland, and eastern-forest-obligate bird species declined, whereas urban—suburban species increased over the interval 1968–2008. No migration status group experienced significant changes, although Nearctic—Neotropical migrant species showed intervals of decline and permanent resident species increased almost 20% during the interval. Hierarchical-model results better portrayed patterns of population change over time than route-regression results. We recommend use of hierarchical models for BBS analyses.
Journal Article