Parameter redundancy in discrete state‐space and integrated models

Discrete state‐space models are used in ecology to describe the dynamics of wild animal populations, with parameters, such as the probability of survival, being of ecological interest. For a particular parametrization of a model it is not always clear which parameters can be estimated. This inability to estimate all parameters is known as parameter redundancy or a model is described as nonidentifiable. In this paper we develop methods that can be used to detect parameter redundancy in discrete state‐space models. An exhaustive summary is a combination of parameters that fully specify a model. To use general methods for detecting parameter redundancy a suitable exhaustive summary is required. This paper proposes two methods for the derivation of an exhaustive summary for discrete state‐space models using discrete analogues of methods for continuous state‐space models. We also demonstrate that combining multiple data sets, through the use of an integrated population model, may result in a model in which all parameters are estimable, even though models fitted to the separate data sets may be parameter redundant.     

Understanding species distribution in dynamic populations: a new approach using spatio‐temporal point process models

Understanding and predicting a species’ distribution across a landscape is of central importance in ecology, biogeography and conservation biology. However, it presents daunting challenges when populations are highly dynamic (i.e. increasing or decreasing their ranges), particularly for small populations where information about ecology and life history traits is lacking. Currently, many modelling approaches fail to distinguish whether a site is unoccupied because the available habitat is unsuitable or because a species expanding its range has not arrived at the site yet. As a result, habitat that is indeed suitable may appear unsuitable. To overcome some of these limitations, we use a statistical modelling approach based on spatio‐temporal log‐Gaussian Cox processes. These model the spatial distribution of the species across available habitat and how this distribution changes over time, relative to covariates. In addition, the model explicitly accounts for spatio‐temporal dynamics that are unaccounted for by covariates through a spatio‐temporal stochastic process. We illustrate the approach by predicting the distribution of a recently established population of Eurasian cranes Grus grus in England, UK, and estimate the effect of a reintroduction in the range expansion of the population. Our models show that wetland extent and perimeter‐to‐area ratio have a positive and negative effect, respectively, in crane colonisation probability. Moreover, we find that cranes are more likely to colonise areas near already occupied wetlands and that the colonisation process is progressing at a low rate. Finally, the reintroduction of cranes in SW England can be considered a human‐assisted long‐distance dispersal event that has increased the dispersal potential of the species along a longitudinal axis in S England. Spatio‐temporal log‐Gaussian Cox process models offer an excellent opportunity for the study of species where information on life history traits is lacking, since these are represented through the spatio‐temporal dynamics reflected in the model.     

Recommendations for photo‐identification methods used in capture‐recapture models with cetaceans

Capture‐recapture methods are frequently employed to estimate abundance of cetaceans using photographic techniques and a variety of statistical models. However, there are many unresolved issues regarding the selection and manipulation of images that can potentially impose bias on resulting estimates. To examine the potential impact of these issues we circulated a test data set of dorsal fin images from bottlenose dolphins to several independent research groups. Photo‐identification methods were generally similar, but the selection, scoring, and matching of images varied greatly amongst groups. Based on these results we make the following recommendations. Researchers should: (1) determine the degree of marking, or level of distinctiveness, and use images of sufficient quality to recognize animals of that level of distinctiveness; (2) ensure that markings are sufficiently distinct to eliminate the potential for “twins” to occur; (3) stratify data sets by distinctiveness and generate a series of abundance estimates to investigate the influence of including animals of varying degrees of markings; and (4) strive to examine and incorporate variability among analysts into capture‐recapture estimation. In this paper we summarize these potential sources of bias and provide recommendations for best practices for using natural markings in a capture‐recapture framework.     

A likelihood approach to estimating animal density from binary acoustic transects

We propose an approximate maximum likelihood method for estimating animal density and abundance from binary passive acoustic transects, when both the probability of detection and the range of detection are unknown. The transect survey is purposely designed so that successive data points are dependent, and this dependence is exploited to simultaneously estimate density, range of detection, and probability of detection. The data are assumed to follow a homogeneous Poisson process in space, and a second-order Markov approximation to the likelihood is used. Simulations show that this method has small bias under the assumptions used to derive the likelihood, although it performs better when the probability of detection is close to 1. The effects of violations of these assumptions are also investigated, and the approach is found to be sensitive to spatial trends in density and clustering. The method is illustrated using real acoustic data from a survey of sperm and humpback whales.     

Copula‐based semiparametric models for spatiotemporal data

The joint analysis of spatial and temporal processes poses computational challenges due to the data's high dimensionality. Furthermore, such data are commonly non‐Gaussian. In this paper, we introduce a copula‐based spatiotemporal model for analyzing spatiotemporal data and propose a semiparametric estimator. The proposed algorithm is computationally simple, since it models the marginal distribution and the spatiotemporal dependence separately. Instead of assuming a parametric distribution, the proposed method models the marginal distributions nonparametrically and thus offers more flexibility. The method also provides a convenient way to construct both point and interval predictions at new times and locations, based on the estimated conditional quantiles. Through a simulation study and an analysis of wind speeds observed along the border between Oregon and Washington, we show that our method produces more accurate point and interval predictions for skewed data than those based on normality assumptions.     

The exponential‐Poisson model for recurrent event data: An application to a set of data on malaria in Brazil

In this paper, we introduce a new model for recurrent event data characterized by a baseline rate function fully parametric, which is based on the exponential‐Poisson distribution. The model arises from a latent competing risk scenario, in the sense that there is no information about which cause was responsible for the event occurrence. Then, the time of each recurrence is given by the minimum lifetime value among all latent causes. The new model has a particular case, which is the classical homogeneous Poisson process. The properties of the proposed model are discussed, including its hazard rate function, survival function, and ordinary moments. The inferential procedure is based on the maximum likelihood approach. We consider an important issue of model selection between the proposed model and its particular case by the likelihood ratio test and score test. Goodness of fit of the recurrent event models is assessed using Cox‐Snell residuals. A simulation study evaluates the performance of the estimation procedure in the presence of a small and moderate sample sizes. Applications on two real data sets are provided to illustrate the proposed methodology. One of them, first analyzed by our team of researchers, considers the data concerning the recurrence of malaria, which is an infectious disease caused by a protozoan parasite that infects red blood cells.     

Semiparametric frailty models for zero‐inflated event count data in the presence of informative dropout

Recurrent events data are commonly encountered in medical studies. In many applications, only the number of events during the follow‐up period rather than the recurrent event times is available. Two important challenges arise in such studies: (a) a substantial portion of subjects may not experience the event, and (b) we may not observe the event count for the entire study period due to informative dropout. To address the first challenge, we assume that underlying population consists of two subpopulations: a subpopulation nonsusceptible to the event of interest and a subpopulation susceptible to the event of interest. In the susceptible subpopulation, the event count is assumed to follow a Poisson distribution given the follow‐up time and the subject‐specific characteristics. We then introduce a frailty to account for informative dropout. The proposed semiparametric frailty models consist of three submodels: (a) a logistic regression model for the probability such that a subject belongs to the nonsusceptible subpopulation; (b) a nonhomogeneous Poisson process model with an unspecified baseline rate function; and (c) a Cox model for the informative dropout time. We develop likelihood‐based estimation and inference procedures. The maximum likelihood estimators are shown to be consistent. Additionally, the proposed estimators of the finite‐dimensional parameters are asymptotically normal and the covariance matrix attains the semiparametric efficiency bound. Simulation studies demonstrate that the proposed methodologies perform well in practical situations. We apply the proposed methods to a clinical trial on patients with myelodysplastic syndromes.     

Open Capture–Recapture Models with Heterogeneity: II. Jolly–Seber Model

Summary Estimation of abundance is important in both open and closed population capture–recapture analysis, but unmodeled heterogeneity of capture probability leads to negative bias in abundance estimates. This article defines and develops a suite of open population capture–recapture models using finite mixtures to model heterogeneity of capture and survival probabilities. Model comparisons and parameter estimation use likelihood‐based methods. A real example is analyzed, and simulations are used to check the main features of the heterogeneous models, especially the quality of estimation of abundance, survival, recruitment, and turnover. The two major advances in this article are the provision of realistic abundance estimates that take account of heterogenetiy of capture, and an appraisal of the amount of overestimation of survival arising from conditioning on the first capture when heterogeneity of survival is present.     

A spatially explicit capture–recapture estimator for single‐catch traps

Single‐catch traps are frequently used in live‐trapping studies of small mammals. Thus far, a likelihood for single‐catch traps has proven elusive and usually the likelihood for multicatch traps is used for spatially explicit capture–recapture (SECR) analyses of such data. Previous work found the multicatch likelihood to provide a robust estimator of average density. We build on a recently developed continuous‐time model for SECR to derive a likelihood for single‐catch traps. We use this to develop an estimator based on observed capture times and compare its performance by simulation to that of the multicatch estimator for various scenarios with nonconstant density surfaces. While the multicatch estimator is found to be a surprisingly robust estimator of average density, its performance deteriorates with high trap saturation and increasing density gradients. Moreover, it is found to be a poor estimator of the height of the detection function. By contrast, the single‐catch estimators of density, distribution, and detection function parameters are found to be unbiased or nearly unbiased in all scenarios considered. This gain comes at the cost of higher variance. If there is no interest in interpreting the detection function parameters themselves, and if density is expected to be fairly constant over the survey region, then the multicatch estimator performs well with single‐catch traps. However if accurate estimation of the detection function is of interest, or if density is expected to vary substantially in space, then there is merit in using the single‐catch estimator when trap saturation is above about 60%. The estimator's performance is improved if care is taken to place traps so as to span the range of variables that affect animal distribution. As a single‐catch likelihood with unknown capture times remains intractable for now, researchers using single‐catch traps should aim to incorporate timing devices with their traps.     

Upper bound estimators of the population size based on ordinal models for capture‐recapture experiments

Capture‐recapture studies have attracted a lot of attention over the past few decades, especially in applied disciplines where a direct estimate for the size of a population of interest is not available. Epidemiology, ecology, public health, and biodiversity are just a few examples. The estimation of the number of unseen units has been a challenge for theoretical statisticians, and considerable progress has been made in providing lower bound estimators for the population size. In fact, it is well known that consistent estimators for this cannot be provided in the very general case. Considering a case where capture‐recapture studies are summarized by a frequency of frequencies distribution, we derive a simple upper bound of the population size based on the cumulative distribution function. We introduce two estimators of this bound, without any specific parametric assumption on the distribution of the observed frequency counts. The behavior of the proposed estimators is investigated using several benchmark datasets and a large‐scale simulation experiment based on the scheme discussed by Pledger.     

Wrong,but useful: regional species distribution models may not be improved by range‐wide data under biased sampling

Species distribution modeling (SDM) is an essential method in ecology and conservation. SDMs are often calibrated within one country's borders, typically along a limited environmental gradient with biased and incomplete data, making the quality of these models questionable. In this study, we evaluated how adequate are national presence‐only data for calibrating regional SDMs. We trained SDMs for Egyptian bat species at two different scales: only within Egypt and at a species‐specific global extent. We used two modeling algorithms: Maxent and elastic net, both under the point‐process modeling framework. For each modeling algorithm, we measured the congruence of the predictions of global and regional models for Egypt, assuming that the lower the congruence, the lower the appropriateness of the Egyptian dataset to describe the species' niche. We inspected the effect of incorporating predictions from global models as additional predictor (“prior”) to regional models, and quantified the improvement in terms of AUC and the congruence between regional models run with and without priors. Moreover, we analyzed predictive performance improvements after correction for sampling bias at both scales. On average, predictions from global and regional models in Egypt only weakly concur. Collectively, the use of priors did not lead to much improvement: similar AUC and high congruence between regional models calibrated with and without priors. Correction for sampling bias led to higher model performance, whatever prior used, making the use of priors less pronounced. Under biased and incomplete sampling, the use of global bats data did not improve regional model performance. Without enough bias‐free regional data, we cannot objectively identify the actual improvement of regional models after incorporating information from the global niche. However, we still believe in great potential for global model predictions to guide future surveys and improve regional sampling in data‐poor regions.