Identifying occupancy model inadequacies: can residuals separately assess detection and presence?

Occupancy models are widely applied to estimate species distributions, but few methods exist for model checking. Thorough model assessments can uncover inadequacies and allow for deeper ecological insight by exploring structure in the observed data not accounted for by a model. We introduce occupancy model residual definitions that utilize the posterior distribution of the partially latent occupancy states. Residual-based assessments are valuable because they can target specific assumptions and identify ways to improve a model, such as adding spatial correlation or meaningful covariates. Our approach defines separate residuals for occupancy and detection, and we use simulation to examine whether missing structure for modeling detection probabilities can be distinguished from that for occupancy probabilities. In many scenarios, our residual diagnostics were able to separate inadequacies at the different model levels successfully, but we describe other situations when this may not be the case. Applying Moran's I residual diagnostics to assess models for silver-haired (Lasionycteris noctivagans) and little brown (Myotis lucifugus) bats only provided evidence of residual spatial correlation among detections. Targeting specific model assumptions using carefully chosen residual diagnostics is valuable for any analysis, and we remove previous barriers for occupancy analyses-lack of examples and practical advice.

Considerations for assessing model averaging of regression coefficients.

Model choice is usually an inevitable source of uncertainty in model-based statistical analyses. While the focus of model choice was traditionally on methods for choosing a single model, methods to formally account for multiple models within a single analysis are now accessible to many researchers. The specific technique of model averaging was developed to improve predictive ability by combining predictions from a set of models. However, it is now often used to average regression coefficients across multiple models with the ultimate goal of capturing a variable's overall effect. This use of model averaging implicitly assumes the same parameter exists across models so that averaging is sensible. While this assumption may initially seem tenable, regression coefficients associated with particular explanatory variables may not hold equivalent interpretations across all of the models in which they appear, making explanatory inference about covariates challenging. Accessibility to easily implementable software, concerns about being criticized for ignoring model uncertainty, and the chance to avoid having to justify choice of a final model have all led to the increasing popularity of model averaging in practice. We see a gap between the theoretical development of model averaging and its current use in practice, potentially leaving well-intentioned researchers with unclear inferences or difficulties justifying reasons for using (or not using) model averaging. We attempt to narrow this gap by revisiting some relevant foundations of regression modeling, suggesting more explicit notation and graphical tools, and discussing how individual model results are combined to obtain a model averaged result. Our goal is to help researchers make informed decisions about model averaging and to encourage question-focused modeling over method-focused modeling.

Use of posterior predictive checks as an inferential tool for investigating individual heterogeneity in animal population vital rates.

The investigation of individual heterogeneity in vital rates has recently received growing attention among population ecologists. Individual heterogeneity in wild animal populations has been accounted for and quantified by including individually varying effects in models for mark-recapture data, but the real need for underlying individual effects to account for observed levels of individual variation has recently been questioned by the work of Tuljapurkar et al. (Ecology Letters, 12, 93, 2009) on dynamic heterogeneity. Model-selection approaches based on information criteria or Bayes factors have been used to address this question. Here, we suggest that, in addition to model-selection, model-checking methods can provide additional important insights to tackle this issue, as they allow one to evaluate a model's misfit in terms of ecologically meaningful measures. Specifically, we propose the use of posterior predictive checks to explicitly assess discrepancies between a model and the data, and we explain how to incorporate model checking into the inferential process used to assess the practical implications of ignoring individual heterogeneity. Posterior predictive checking is a straightforward and flexible approach for performing model checks in a Bayesian framework that is based on comparisons of observed data to model-generated replications of the data, where parameter uncertainty is incorporated through use of the posterior distribution. If discrepancy measures are chosen carefully and are relevant to the scientific context, posterior predictive checks can provide important information allowing for more efficient model refinement. We illustrate this approach using analyses of vital rates with long-term mark-recapture data for Weddell seals and emphasize its utility for identifying shortfalls or successes of a model at representing a biological process or pattern of interest. We show how posterior predictive checks can be used to strengthen inferences in ecological studies. We demonstrate the application of this method on analyses dealing with the question of individual reproductive heterogeneity in a population of Antarctic pinnipeds.

Individual heterogeneity in reproductive rates and cost of reproduction in a long-lived vertebrate.

Individual variation in reproductive success is a key feature of evolution, but also has important implications for predicting population responses to variable environments. Although such individual variation in reproductive outcomes has been reported in numerous studies, most analyses to date have not considered whether these realized differences were due to latent individual heterogeneity in reproduction or merely random chance causing different outcomes among like individuals. Furthermore, latent heterogeneity in fitness components might be expressed differently in contrasted environmental conditions, an issue that has only rarely been investigated. Here, we assessed (i) the potential existence of latent individual heterogeneity and (ii) the nature of its expression (fixed vs. variable) in a population of female Weddell seals (Leptonychotes weddellii), using a hierarchical modeling approach on a 30-year mark-recapture data set consisting of 954 individual encounter histories. We found strong support for the existence of latent individual heterogeneity in the population, with "robust" individuals expected to produce twice as many pups as "frail" individuals. Moreover, the expression of individual heterogeneity appeared consistent, with only mild evidence that it might be amplified when environmental conditions are severe. Finally, the explicit modeling of individual heterogeneity allowed us to detect a substantial cost of reproduction that was not evidenced when the heterogeneity was ignored.

Discretized and aggregated: modeling dive depth of harbor seals from ordered categorical data with temporal autocorrelation.

Ordered categorical data are pervasive in environmental and ecological data, and often arise from constraints that require discretizing a continuous variable into ordered categories. A great deal of data have been collected toward the study of marine mammal dive behavior using satellite depth recorders (SDRs), which often discretize a continuous variable such as depth. Additionally, data storage or transmission constraints may also necessitate the aggregation of data over time intervals of a specified length. The categorization and aggregation create a time series of ordered multicategory counts for each animal, which present challenges in terms of statistical modeling and practical interpretation. We describe an intuitive strategy for modeling such aggregated, ordered categorical data allowing for inference regarding the category probabilities and a measure of central tendency on the original scale of the data (e.g., meters), along with incorporation of temporal correlation and overdispersion. The strategy extends covariate-specific cutpoint models for ordinal data. We demonstrate the method in an analysis of SDR dive-depth data collected on harbor seals in Alaska. The primary goal of the analysis is to assess the relationship of covariates, such as time of day, with number of dives and maximum depth of dives. We also predict missing values and introduce novel graphical summaries of the data and results.