Estimating species occurrence, abundance, and detection probability using zero-inflated distributions

Researchers have developed methods to account for imperfect detection of species with either occupancy (presence absence) or count data using replicated sampling. We show how these approaches can be combined to simultaneously estimate occurrence, abundance, and detection probability by specifying a zero-inflated distribution for abundance. This approach may be particularly appropriate when patterns of occurrence and abundance arise from distinct processes operating at differing spatial or temporal scales. We apply the model to two data sets: (1) previously published data for a species of duck, Anas platyrhynchos, and (2) data for a stream fish species, Etheostoma scotti. We show that in these cases, an incomplete-detection zero-inflated modeling approach yields a superior fit to the data than other models. We propose that zero-inflated abundance models accounting for incomplete detection be considered when replicate count data are available.     

Comparing species abundance models

Five regression models (Poisson, negative binomial, quasi-Poisson, the hurdle model and the zero-inflated Poisson) were used to assess the relationship between the abundance of a vulnerable plant species, Leionema ralstonii, and the environment. The methods differed in their capacity to deal with common properties of ecological data. They were assessed theoretically, and their predictive performance was evaluated with correlation, calibration and error statistics calculated within a bootstrap evaluation procedure that simulated performance for independent data.     

Traditional occupancy–abundance models are inadequate for zero-inflated ecological count data

Traditional occupancy–abundance and abundance–variance–occupancy models do not take into account zero-inflation, which occurs when sampling rare species or in correlated counts arising from repeated measures. In this paper we propose a novel approach extending occupancy–abundance relationships to zero-inflated count data. This approach involves three steps: (1) selecting distributional assumptions and parsimonious models for the count data, (2) estimating abundance, occupancy and variance parameters as functions of site- and/or time-specific covariates, and (3) modelling the occupancy–abundance relationship using the parameters estimated in step 2. Five count datasets were used for comparing standard Poisson and negative binomial distribution (NBD) occupancy–abundance models. Zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) occupancy–abundance models were introduced for the first time, and these were compared with the Poisson, NBD, He and Gaston's and Wilson and Room's abundance–variance–occupancy models. The percentage of zero counts ranged from 45 to 80% in the datasets analysed. For most of the datasets, the ZINB occupancy–abundance model performed better than the traditional Poisson, NBD and Wilson and Room's model. He and Gaston's model performed better than the ZINB in two out of the five datasets. However, the occupancy predicted by all models increased faster than the observed as density increased resulting in significant mismatch at the highest densities. Limitations of the various models are discussed, and the need for careful choice of count distributions and predictors in estimating abundance and occupancy parameter are indicated.     

Quasi-Poisson vs. negative binomial regression: how should we model overdispersed count data?

Quasi-Poisson and negative binomial regression models have equal numbers of parameters, and either could be used for overdispersed count data. While they often give similar results, there can be striking differences in estimating the effects of covariates. We explain when and why such differences occur. The variance of a quasi-Poisson model is a linear function of the mean while the variance of a negative binomial model is a quadratic function of the mean. These variance relationships affect the weights in the iteratively weighted least-squares algorithm of fitting models to data. Because the variance is a function of the mean, large and small counts get weighted differently in quasi-Poisson and negative binomial regression. We provide an example using harbor seal counts from aerial surveys. These counts are affected by date, time of day, and time relative to low tide. We present results on a data set that showed a dramatic difference on estimating abundance of harbor seals when using quasi-Poisson vs. negative binomial regression. This difference is described and explained in light of the different weighting used in each regression method. A general understanding of weighting can help ecologists choose between these two methods.     

Modeling and inference of forest coverage ratio using zero-one inflated distributions with spatial dependence

This paper explores statistical modeling of forest area with two covariates. The forest coverage ratio of grid-cell data was modeled by taking human population density and relief energy into account. The likelihood of the forest ratios was decomposed into the product of two likelihoods. The first likelihood was due to trinomial logistic distributions on three categories: the forest ratios take zero, or one, or values between zero and one. The second one was due to a logistic-normal regression model for the ratios between zero and one. This model was applied to real grid-cell data and it fit better than zero-inflated beta regression models.     

A Bayesian approach to retransformation bias in transformed regression

Ecological data analysis often involves fitting linear or nonlinear equations to data after transforming either the response variable, the right side of the equation, or both, so that the standard suite of regression assumptions are more closely met. However, inference is usually done in the natural metric and it is well known that retransforming back to the original metric provides a biased estimator for the mean of the response variable. For the normal linear model, fit under a log-transformation, correction factors are available to reduce this bias, but these factors may not be generally applicable to all model forms or other transformations. We demonstrate that this problem is handled in a straightforward manner using a Bayesian approach, which is general for linear and nonlinear models and other transformations and model error structures. The Bayesian framework provides a predictive distribution for the response variable so that inference can be made at the mean, or over the entire distribution to incorporate the predictive uncertainty.     

A spatial zero-inflated poisson regression model for oak regeneration

Ecological counts data are often characterized by an excess of zeros and spatial dependence. Excess zeros can occur in regions outside the range of the distribution of a given species. A zero-inflated Poisson regression model is developed, under which the species range is determined by a spatial probit model, including physical variables as covariates. Within that range, species counts are independently drawn from a Poisson distribution whose mean depends on biotic variables. Bayesian inference for this model is illustrated using data on oak seedling counts. Received: May 2004 / Revised: December 2004     

The application of generalized linear mixed models to multi-level sampling for insect population monitoring

Iwao's quadratic regression or Taylor's Power Law (TPL) are commonly used to model the variance as a function of the mean for sample counts of insect populations which exhibit spatial aggregation. The modeled variance and distribution of the mean are typically used in pest management programs to decide if the population is above the action threshold in any management unit (MU) (e.g., orchard, forest compartment). For nested or multi-level sampling the usual two-stage modeling procedure first obtains the sample variance for each MU and sampling level using ANOVA and then fits a regression of variance on the mean for each level using either Iwao or TPL variance models. Here this approach is compared to the single-stage procedure of fitting a generalized linear mixed model (GLMM) directly to the count data with both approaches demonstrated using 2-level sampling. GLMMs and additive GLMMs (AGLMMs) with conditional Poisson variance function as well as the extension to the negative binomial are described. Generalization to more than two sampling levels is outlined. Formulae for calculating optimal relative sample sizes (ORSS) and the operating characteristic curve for the control decision are given for each model. The ORSS are independent of the mean in the case of the AGLMMs. The application described is estimation of the variance of the mean number of leaves per shoot occupied by immature stages of a defoliator of eucalypts, the Tasmanian Eucalyptus leaf beetle, based on a sample of trees within plots from each forest compartment. Historical population monitoring data were fitted using the above approaches.     

Modelling spatial zero-inflated continuous data with an exponentially compound Poisson process

A parsimonious model is presented as an alternative to delta approaches to modelling zero-inflated continuous data. The data model relies on an exponentially compound Poisson process, also called the law of leaks (LOL). It represents the process of sampling resources that are spatially distributed as Poisson distributed patches, each containing a certain quantity of biomass drawn from an exponential distribution. In an application of the LOL, two latent structures are proposed to account for spatial dependencies between zero values at different scales within a hierarchical Bayesian framework. The LOL is compared to the delta-gamma (ΔΓ) distribution using bottom-trawl survey data. Results of this case study emphasize that the LOL provides slightly better fits to learning samples with a very high proportion of zero values and small strictly positive abundance data. Additionally, it offers better predictions of validation samples.     

Using Age Structure to Detect Impacts on Threatened Populations: a Case Study with Steller Sea Lions

Abstract:   A delayed response to change is often a characteristic of long-lived species and presents a major challenge to monitoring their status. However, rapid shifts in age structure can occur even while population size remains relatively static. We used time-varying matrix models to study age-structure information as a tool for improving detection of survivorship and fecundity change and status. We applied the methods to Steller sea lions (  Eumetopias jubatus ), a long-lived endangered marine mammal found throughout the North Pacific Rim. Population and newborn counts were supplemented with information on the fraction of the population that was juvenile, obtained by measuring animals in aerial photographs taken during range-wide censuses. By fitting the model to 1976–1998 data, we obtained maximum-likelihood estimates and 95% confidence intervals for juvenile survivorship, adult survivorship, and adult fecundity in the mid-1980s, late 1980s, and 1990s. We used a series of nested models to test whether the data were best fit by a model with one, two, or three temporal changes in demographic rates, and we fit the models to different lengths of data to test the number of years of data needed to detect a demographic change. The declines in the early 1980s were associated with severely low juvenile survivorship, whereas declines in the 1990s were associated with disproportionately low fecundity. We repeated these analyses, fitting only to the count data without the juvenile-fraction information, to determine whether the age-structure information changed the conclusions and/or changed the certainty and speed with which demographic-rate changes could be detected. The juvenile-fraction data substantially improved the degree to which estimates from the model were consistent with field data and significantly improved the speed and certainty with which changes in demographic rates were detected.     

A spatial scan statistic for zero-inflated Poisson process

The scan statistic is widely used in spatial cluster detection applications of inhomogeneous Poisson processes. However, real data may present substantial departure from the underlying Poisson process. One of the possible departures has to do with zero excess. Some studies point out that when applied to data with excess zeros, the spatial scan statistic may produce biased inferences. In this work, we develop a closed-form scan statistic for cluster detection of spatial zero-inflated count data. We apply our methodology to simulated and real data. Our simulations revealed that the Scan-Poisson statistic steadily deteriorates as the number of zeros increases, producing biased inferences. On the other hand, our proposed Scan-ZIP and Scan-ZIP+EM statistics are, most of the time, either superior or comparable to the Scan-Poisson statistic.     

Modeling and inference of animal movement using artificial neural networks

Movement of animals in relation to objects in their environment is important in many areas of ecology and wildlife conservation. Tools for analysis of movement data, however, still remain rather limited. In previous work, we developed nonlinear regression models for movement in relation to a single landscape feature. Here we greatly expand these previous models by using artificial neural networks. The new models add substantial flexibility and capabilities, including the ability to incorporate multiple factors and covariates. We devise a likelihood-based model fitting procedure that utilizes genetic algorithms and demonstrate the approach with movement data for red diamond rattlesnakes. The proposed methodology can be useful for global positioning system tracking data that are becoming more common in studies of animal movement behavior.     

基于GWR的中国地级城市SO2年均质量浓度模拟

中国城市空气污染问题已经引起广泛关注。目前相关研究很多,但是以空间位置为拟合参数,对空气质量进行回归模拟的研究较少。以2010年中国地级以上城市SO2年均质量浓度为因变量,分别应用普通线性回归和地理加权回归（GWR）模型模拟SO2年均质量浓度,其中地理加权回归方法考虑了空间位置的影响并以此作为回归参数。回归的自变量指标体系包括气象要素（多年平均温度、光照、降水）、植被覆盖（NDVI）、地形要素（坡度、坡向、起伏度）、人为因素（GDP、能源消费）几个方面。由于各指标之间存在较强的相关性,用主成分分析方法计算得到温度、日照、降水、NDVI表征的气象植被综合指标,高程、坡度、起伏度表征的地形综合指标,和GDP、能源消费表征的人为因素综合指标。用3个综合指标值作为自变量进行回归模拟。普通回归结果较差,其r^2为0.11,矫正的r^2为0.10;GWR模型模拟结果相对较好,其拟合优度显著提高,r^2为0.66,矫正的r^2为0.47。因此,地理加权回归适合进行此类拟合,普通线性回归不适合。通过对比地理加权回归模拟的各个城市的拟合优度,发现年均质量浓度数值较高的地区拟合效果较差,这些地区主要集中在中国华北和南部部分地区。与基于机理的模型相比,GWR 模型和其各具优缺点,GWR 的优势主要表现在数据及其格式化要求低,计算机软硬件条件要求低,运算速度快等。     

Efficient statistical mapping of avian count data

We develop a spatial modeling framework for count data that is efficient to implement in high-dimensional prediction problems. We consider spectral parameterizations for the spatially varying mean of a Poisson model. The spectral parameterization of the spatial process is very computationally efficient, enabling effective estimation and prediction in large problems using Markov chain Monte Carlo techniques. We apply this model to creating avian relative abundance maps from North American Breeding Bird Survey (BBS) data. Variation in the ability of observers to count birds is modeled as spatially independent noise, resulting in over-dispersion relative to the Poisson assumption. This approach represents an improvement over existing approaches used for spatial modeling of BBS data which are either inefficient for continental scale modeling and prediction or fail to accommodate important distributional features of count data thus leading to inaccurate accounting of prediction uncertainty.     

Updating coarse-scale species distribution models using small fine-scale samples

Large, fine-grained samples are ideal for predictive species distribution models used for management purposes, but such datasets are not available for most species and conducting such surveys is costly. We attempted to overcome this obstacle by updating previously available coarse-grained logistic regression models with small fine-grained samples using a recalibration approach. Recalibration involves re-estimation of the intercept or slope of the linear predictor and may improve calibration (level of agreement between predicted and actual probabilities). If reliable estimates of occurrence likelihood are required (e.g., for species selection in ecological restoration) calibration should be preferred to other model performance measures. This updating approach is not expected to improve discrimination (the ability of the model to rank sites according to species suitability), because the rank order of predictions is not altered. We tested different updating methods and sample sizes with tree distribution data from Spain. Updated models were compared to models fitted using only fine-grained data (refitted models). Updated models performed reasonably well at fine scales and outperformed refitted models with small samples (10-100 occurrences). If a coarse-grained model is available (or could be easily developed) and fine-grained predictions are to be generated from a limited sample size, updating previous models may be a more accurate option than fitting a new model. Our results encourage further studies on model updating in other situations where species distribution models are used under different conditions from their training (e.g., different time periods, different regions).     

Tropical deforestation in Madagascar: analysis using hierarchical,spatially explicit,Bayesian regression models

Establishing cause–effect relationships for deforestation at various scales has proven difficult even when rates of deforestation appear well documented. There is a need for better explanatory models, which also provide insight into the process of deforestation. We propose a novel hierarchical modeling specification incorporating spatial association. The hierarchical aspect allows us to accommodate misalignment between the land-use (response) data layer and explanatory data layers. Spatial structure seems appropriate due to the inherently spatial nature of land use and data layers explaining land use. Typically, there will be missing values or holes in the response data. To accommodate this we propose an imputation strategy. We apply our modeling approach to develop a novel deforestation model for the eastern wet forested zone of Madagascar, a global rain forest “hot spot”. Using five data layers created for this region, we fit a suitable spatial hierarchical model. Though fitting such models is computationally much more demanding than fitting more standard models, we show that the resulting interpretation is much richer. Also, we employ a model choice criterion to argue that our fully Bayesian model performs better than simpler ones. To the best of our knowledge, this is the first work that applies hierarchical Bayesian modeling techniques to study deforestation processes. We conclude with a discussion of our findings and an indication of the broader ecological applicability of our modeling style.     

Using the negative binomial distribution to model overdispersion in ecological count data

A Poisson process is a commonly used starting point for modeling stochastic variation of ecological count data around a theoretical expectation. However, data typically show more variation than implied by the Poisson distribution. Such overdispersion is often accounted for by using models with different assumptions about how the variance changes with the expectation. The choice of these assumptions can naturally have apparent consequences for statistical inference. We propose a parameterization of the negative binomial distribution, where two overdispersion parameters are introduced to allow for various quadratic mean-variance relationships, including the ones assumed in the most commonly used approaches. Using bird migration as an example, we present hypothetical scenarios on how overdispersion can arise due to sampling, flocking behavior or aggregation, environmental variability, or combinations of these factors. For all considered scenarios, mean-variance relationships can be appropriately described by the negative binomial distribution with two overdispersion parameters. To illustrate, we apply the model to empirical migration data with a high level of overdispersion, gaining clearly different model fits with different assumptions about mean-variance relationships. The proposed framework can be a useful approximation for modeling marginal distributions of independent count data in likelihood-based analyses.     

Improving inferences about private land conservation by accounting for incomplete reporting

Private lands provide key habitat for imperiled species and are core components of function protectected area networks; yet, their incorporation into national and regional conservation planning has been challenging. Identifying locations where private landowners are likely to participate in conservation initiatives can help avoid conflict and clarify trade-offs between ecological benefits and sociopolitical costs. Empirical, spatially explicit assessment of the factors associated with conservation on private land is an emerging tool for identifying future conservation opportunities. However, most data on private land conservation are voluntarily reported and incomplete, which complicates these assessments. We used a novel application of occupancy models to analyze the occurrence of conservation easements on private land. We compared multiple formulations of occupancy models with a logistic regression model to predict the locations of conservation easements based on a spatially explicit social–ecological systems framework. We combined a simulation experiment with a case study of easement data in Idaho and Montana (United States) to illustrate the utility of the occupancy framework for modeling conservation on private land. Occupancy models that explicitly accounted for variation in reporting produced estimates of predictors that were substantially less biased than estimates produced by logistic regression under all simulated conditions. Occupancy models produced estimates for the 6 predictors we evaluated in our case study that were larger in magnitude, but less certain than those produced by logistic regression. These results suggest that occupancy models result in qualitatively different inferences regarding the effects of predictors on conservation easement occurrence than logistic regression and highlight the importance of integrating variable and incomplete reporting of participation in empirical analysis of conservation initiatives. Failure to do so can lead to emphasizing the wrong social, institutional, and environmental factors that enable conservation and underestimating conservation opportunities in landscapes where social norms or institutional constraints inhibit reporting.     

Making more out of sparse data: hierarchical modeling of species communities

Community ecologists and conservation biologists often work with data that are too sparse for achieving reliable inference with species-specific approaches. Here we explore the idea of combining species-specific models into a single hierarchical model. The community component of the model seeks for shared patterns in how the species respond to environmental covariates. We illustrate the modeling framework in the context of logistic regression and presence-absence data, but a similar hierarchical structure could also be used in many other types of applications. We first use simulated data to illustrate that the community component can improve parameterization of species-specific models especially for rare species, for which the data would be too sparse to be informative alone. We then apply the community model to real data on 500 diatom species to show that it has much greater predictive power than a collection of independent species-specific models. We use the modeling approach to show that roughly one-third of distance decay in community similarity can be explained by two variables characterizing water quality, rare species typically preferring nutrient-poor waters with high pH, and common species showing a more general pattern of resource use.     

Bayesian Joint Estimation of Binary Outcome and Time-to-event Data: Effects of Leaf Quality on Pupal Survival and Time-to-Emergence in the Winter Moth

Plant–herbivore interactions are complex and affect herbivore fitness components and life history traits in many different ways. In this paper, we present results from an experiment studying the effects of leaf quality on pupal survival and duration of pupation (as measured by time-to-emergence) in the winter moth. Because only surviving pupae are at risk of emerging, analysis of time-to-emergence should exclude the dead pupae. However, due to right censoring, the survival status could not be determined for each individual. This failure to determine the group of moths at risk of emerging a priori motivated the development of a joint model of both survival probability and time-to-emergence. We formulate the model in a Bayesian framework and apply Monte Carlo Markov Chain (MCMC) to obtain posterior distributions. Time-to-emergence is modeled by a Cox Proportional Hazards (CPH) model where only the surviving pupae are at risk of emergence. Probability of pupal survival was modeled by a Generalized Linear Mixed Model (GLMM). The censored individuals were included in the analysis as a missing value in the GLMM. The GLMM then generated prior distributions of survival probabilties—and thus of the probability of being at risk of emergence—for these 19 individuals, conditional on the model parameters. The CPH model was formulated as a count process and the binary frailty was incorporated as a zero-inflated Poisson model. Zeros in this model represent the non-survivors. Leaf quality did not appear to influence time-to-emergence. Pupal survival was affected in a complex and unexpected way showing opposite effects in males and females. We also explored the robustness of our model against increased levels of censoring. While the degree of censoring was low in our study (< 1%), we artificially increased it to 67%. Although further study is required to study the generality of these results in a theoretical framework, our explorations suggest that the newly proposed technique may be widely applicable in a variety of situations where the identification of the at risk population cannot be done in a straightforward way. Received: January 2005 / Revised: June 2005