Fitting the truncated negative binomial distribution to count data

Modeling empirical distributions of repeated counts with parametric probability distributions is a frequent problem when studying species abundance. One must choose a family of distributions which is flexible enough to take into account very diverse patterns and possess parameters with clear biological/ecological interpretations. The negative binomial distribution fulfills these criteria and was selected for modeling counts of marine fish and invertebrates. This distribution depends on a vector \(\left( K,\mathfrak {P}\right) \) of parameters, and ranges from the Poisson distribution (when \(K\rightarrow +\infty \)) to Fisher’s log-series, when \(K\rightarrow 0\). Moreover, these parameters have biological/ecological interpretations which are detailed in the literature and in this study. We compared three estimators of K, \(\mathfrak {P}\) and the parameter \(\alpha \) of Fisher’s log-series, following the work of Rao CR (Statistical ecology. Pennsylvania State University Press, University Park, 1971) on a three-parameter unstandardized variant of the negative binomial distribution. We further investigated the coherence underlying parameter values resulting from the different estimators, using both real count data collected in the Mauritanian Exclusive Economic Zone (MEEZ) during the period 1987–2010 and realistic simulations of these data. In the case of the MEEZ, we first built homogeneous lists of counts (replicates), by gathering observations of each species with respect to “typical environments” obtained by clustering the sampled stations. The best estimation of \(\left( K,\mathfrak {P}\right) \) was generally obtained by penalized minimum Hellinger distance estimation. Interestingly, the parameters of most of the correctly sampled species seem compatible with the classical birth-and-dead model of population growth with immigration by Kendall (Biometrika 35:6–15, 1948).     

Using the expert model PERPEST to translate measured and predicted pesticide exposure data into ecological risks

An important topic in the registration of pesticides and the interpretation of monitoring data is the estimation of the consequences of a certain concentration of a pesticide for the ecology of aquatic ecosystems. Solving these problems requires predictions of the expected response of the ecosystem to chemical stress. Up until now, a dominant approach to come up with such a prediction is the use of simulation models or safety factors. The disadvantage of the use of safety factors is a crude method that does not provide any insight into the concentration–response relationships at the ecosystem level. On the other hand, simulation models also have serious drawbacks like that they are often very complex, lack transparency, their implementation is expensive and there may be a compilation of errors, due to uncertainties in parameters and processes. In this paper, we present the expert model prediction of the ecological risks of pesticides (PERPEST) that overcomes these problems. It predicts the effects of a given concentration of a pesticide based on the outcome of already performed experiments using experimental ecosystems. This has the great advantage that the outcome is more realistic. The paper especially discusses how this model can be used to translate measured and predicted concentrations of pesticides into ecological risks, by taking data on measured and predicted concentrations of atrazine as an example. It is argued that this model can be of great use to evaluate the outcome of chemical monitoring programmes (e.g. performed in the light of the Water Framework Directive) and can even be used to evaluate the effects of mixtures.     

Zero-inflated models with application to spatial count data

Count data arises in many contexts. Here our concern is with spatial count data which exhibit an excessive number of zeros. Using the class of zero-inflated count models provides a flexible way to address this problem. Available covariate information suggests formulation of such modeling within a regression framework. We employ zero-inflated Poisson regression models. Spatial association is introduced through suitable random effects yielding a hierarchical model. We propose fitting this model within a Bayesian framework considering issues of posterior propriety, informative prior specification and well-behaved simulation based model fitting. Finally, we illustrate the model fitting with a data set involving counts of isopod nest burrows for 1649 pixels over a portion of the Negev desert in Israel.     

Using data augmentation via the Gibbs Sampler to incorporate missing covariate structure in linear models for ecological assessments

Missing covariate values in linear regression models can be an important problem facing environmental researchers. Existing missing value treatment methods such as Multiple Imputation (MI), the EM algorithm and Data Augmentation (DA) have the assumption that both observed and unobserved data come from the same distribution, most commonly a multivariate normal or a conditionally multivariate normal family. These methods do try to incorporate the missing data mechanism and rely on the assumption of Missing At Random (MAR). We present a DA method which does not rely on the MAR assumption and can model missing data mechanisms and covariate structure. This method utilizes the Gibbs Sampler as a tool for incorporating these structures and mechanisms. We apply this method to an ecological data set that relates fish condition to environmental variables. Notice that the presented DA method detects relationships that are not detected when other missing data methods are employed.

Hierarchical modeling of count data with application to nuclear fall-out

We propose a method for a Bayesian hierarchical analysis of count data that are observed at irregular locations in a bounded domain of R². We model the data as having been observed on a fine regular lattice, where we do not have observations at all the sites. The counts are assumed to be independent Poisson random variables whose means are given by a log Gaussian process. In this article, the Gaussian process is assumed to be either a Markov random field (MRF) or a geostatistical model, and we compare the two models on an environmental data set. To make the comparison, we calibrate priors for the parameters in the geostatistical model to priors for the parameters in the MRF. The calibration is obtained empirically. The main goal is to predict the hidden Poisson-mean process at all sites on the lattice, given the spatially irregular count data; to do this we use an efficient MCMC. The spatial Bayesian methods are illustrated on radioactivity counts analyzed by Diggle et al. (1998).     

Comparison of some homogeneity tests in analysis of over-dispersed binomial data

This paper compares the procedures based on the extended quasi-likelihood, pseudo-likelihood and quasi-likelihood approaches for testing homogeneity of several proportions for over-dispersed binomial data. The type I error of the Wald tests using the model-based and robust variance estimates, the score test, and the extended quasi-likelihood ratio test (deviance reduction test) were examined by simulation. The extended quasi-likelihood method performs less well when mean responses are close to 1 or 0. The model-based Wald test based on the quasi-likelihood performs the best in maintaining the nominal level. The score test performs less well when the intracluster correlations are large or heterogeneous. In summary: (i) both the quasilikelihood and pseudo-likelihood methods appear to be acceptable but care must be taken when overfitting a variance function with small sample sizes; (ii) the extended quasi-likelihood approach is the least favourable method because its nominal level is much too high; and (iii) the robust variance estimator performs poorly, particularly when the sample size is small.     

Analyzing spatial count data,with an application to weed counts

Count data on a lattice may arise in observational studies of ecological phenomena. In this paper a hierarchical spatial model is used to analyze weed counts. Anisotropy is introduced, and a bivariate extension of the model is presented.     

Dealing with collinearity in behavioural and ecological data: model averaging and the problems of measurement error

There has been a great deal of recent discussion of the practice of regression analysis (or more generally, linear modelling) in behaviour and ecology. In this paper, I wish to highlight two factors that have been under-considered, collinearity and measurement error in predictors, as well as to consider what happens when both exist at the same time. I examine what the consequences are for conventional regression analysis (ordinary least squares, OLS) as well as model averaging methods, typified by information theoretic approaches based around Akaike’s information criterion. Collinearity causes variance inflation of estimated slopes in OLS analysis, as is well known. In the presence of collinearity, model averaging reduces this variance for predictors with weak effects, but also can lead to parameter bias. When collinearity is strong or when all predictors have strong effects, model averaging relies heavily on the full model including all predictors and hence the results from this and OLS are essentially the same. I highlight that it is not safe to simply eliminate collinear variables without due consideration of their likely independent effects as this can lead to biases. Measurement error is also considered and I show that when collinearity exists, this can lead to extreme biases when predictors are collinear, have strong effects but differ in their degree of measurement error. I highlight techniques for dealing with and diagnosing these problems. These results reinforce that automated model selection techniques should not be relied on in the analysis of complex multivariable datasets.     

Simplicity and complexity in ecological data analysis

I argue that ecological data analyses are often needlessly complicated, and I present two examples of published analyses for which simpler alternatives are available. Unnecessary complexity is often introduced when analysts focus on subunits of the key experimental or observational units in a study, or use a very general framework to present an analysis that is a simple special case. Simpler analyses are easier to explain and understand; they clarify what the key units in a study are; they reduce the chances for computational mistakes; and they are more likely to lead to the same conclusions when applied by different analysts to the same data.     

Modelling directional spatial processes in ecological data

Distributions of species, animals or plants, terrestrial or aquatic, are influenced by numerous factors such as physical and biogeographical gradients. Dominant wind and current directions cause the appearance of gradients in physical conditions whereas biogeographical gradients can be the result of historical events (e.g. glaciations). No spatial modelling technique has been developed to this day that considers the direction of an asymmetric process controlling species distributions along a gradient or network. This paper presents a new method that can model species spatial distributions generated by a hypothesized asymmetric, directional physical process. This method is an eigenfunction-based spatial filtering technique that offers as much flexibility as the Moran's eigenvector maps (MEM) framework; it is called asymmetric eigenvector maps (AEM) modelling. Information needed to construct eigenfunctions through the AEM framework are the spatial coordinates of the sampling or experimental sites, a connexion diagram linking the sites to one another, prior information about the direction of the hypothesized asymmetric process influencing the response variable(s), and optionally, weights attached to the edges (links). To illustrate how this new method works, AEM is compared to MEM analysis through simulations and in the analysis of an ecological example where a known asymmetric forcing is present. The ecological example reanalyses the dietary habits of brook trout (Salvelinus fontinalis) sampled in 42 lakes of the Mastigouche Reserve, Québec.     

Estimating trend precision and power to detect trends across grouped count data

Ecologists commonly use grouped or clustered count data to estimate temporal trends in counts, abundance indices, or abundance. For example, the U.S. Breeding Bird Survey data represent multiple counts of birds from within each of multiple, spatially defined routes. Despite a reliance on grouped counts, analytical methods for prospectively estimating precision of trend estimates or statistical power to detect trends that explicitly acknowledge the characteristics of grouped count data are undescribed. These characteristics include the fact that the sampling variance is an increasing function of the mean, and that sampling and group-level variance estimates are generally estimated on different scales (the sampling and log scales, respectively). We address these issues for repeated sampling of a single population using an analytical approach that has the flavor of a generalized linear mixed model, specifically that of a negative binomial-distributed count variable with random group effects. The count mean, including grand intercept, trend, and random group effects, is modeled linearly on the log scale, while sampling variance of the mean is estimated on the log scale via the delta method. Results compared favorably with those derived using Monte Carlo simulations. For example, at trend = 5% per temporal unit, differences in standard errors and in power were modest relative to those estimated by simulation (< or = /11/% and < or = /16/%, respectively), with relative differences among power estimates decreasing to < or = /7/% when power estimated by simulations was > or = 0.50. Similar findings were obtained using data from nine surveys of fingernail clams in the Mississippi River. The proposed method is suggested (1) where simulations are not practical and relative precision or power is desired, or (2) when multiple precision or power calculations are required and where the accuracy of a fraction of those calculations will be confirmed using simulations.     

Using structural equation modeling to investigate relationships among ecological variables

Structural equation modeling is an advanced multivariate statistical process with which a researcher can construct theoretical concepts, test their measurement reliability, hypothesize and test a theory about their relationships, take into account measurement errors, and consider both direct and indirect effects of variables on one another. Latent variables are theoretical concepts that unite phenomena under a single term, e.g., ecosystem health, environmental condition, and pollution (Bollen, 1989). Latent variables are not measured directly but can be expressed in terms of one or more directly measurable variables called indicators. For some researchers, defining, constructing, and examining the validity of latent variables may be the end task of itself. For others, testing hypothesized relationships of latent variables may be of interest. We analyzed the correlation matrix of eleven environmental variables from the U.S. Environmental Protection Agency's (USEPA) Environmental Monitoring and Assessment Program for Estuaries (EMAP-E) using methods of structural equation modeling. We hypothesized and tested a conceptual model to characterize the interdependencies between four latent variables-sediment contamination, natural variability, biodiversity, and growth potential. In particular, we were interested in measuring the direct, indirect, and total effects of sediment contamination and natural variability on biodiversity and growth potential. The model fit the data well and accounted for 81% of the variability in biodiversity and 69% of the variability in growth potential. It revealed a positive total effect of natural variability on growth potential that otherwise would have been judged negative had we not considered indirect effects. That is, natural variability had a negative direct effect on growth potential of magnitude –0.3251 and a positive indirect effect mediated through biodiversity of magnitude 0.4509, yielding a net positive total effect of 0.1258. Natural variability had a positive direct effect on biodiversity of magnitude 0.5347 and a negative indirect effect mediated through growth potential of magnitude –0.1105 yielding a positive total effects of magnitude 0.4242. Sediment contamination had a negative direct effect on biodiversity of magnitude –0.1956 and a negative indirect effect on growth potential via biodiversity of magnitude –0.067. Biodiversity had a positive effect on growth potential of magnitude 0.8432, and growth potential had a positive effect on biodiversity of magnitude 0.3398. The correlation between biodiversity and growth potential was estimated at 0.7658 and that between sediment contamination and natural variability at –0.3769.     

Using and comparing two nonparametric methods (CART and MARS) to model the potential distribution of gullies

Gully erosion represents an important soil degradation process in rangelands. In order to take preventive or control measures and to reduce its environmental damages and economical costs it is useful to localize the points in the landscape where gullying takes place and to determine the importance of the different factors involved. The study is carried out in Extremadura, southwest Spain. The main objectives of this work are: (a) comparing two nonparametric schemes to model the potential distribution of gullies, (b) evaluating the importance of the different factors involved in gullying processes, (c) analyzing the role of prevalence in the success of the model and finally, (d) implementing and mapping the results with the help of a Geographical Information System (GIS). Two methods were used to model the response of a dependent variable (gullying) from a set of independent variables: Classification And Regression Trees (CART) and Multivariate Adaptive Regression Splines (MARS). Three different datasets were used; the first one for constructing the model (training dataset) and the others for validating the model (external datasets). These datasets are formed by a target variable (presence or absence of gullies) and a set of independent variables. The dependent variable was obtained by mapping the locations of gullies with the help of a GPS and high resolution aerial ortophotographs. A set of 32 independent variables reflecting topography, lithology, soil type, climate, land use and vegetation cover of each area were used. The performance of the models was evaluated using a non-dependent threshold method: the Receiver Operating Characteristic (ROC) curve. The results showed a better performance of MARS for predicting gullying with areas under the ROC curve of 0.98 and 0.97 for the validation datasets, while CART presented values of 0.96 and 0.66.     

A model for ecological assessment to pesticide pollution management

Applying the relational analysis in the Grey System Theory and Method, the comprehensive evaluation on five pesticide pollution controlling techniques in the vegetable production has been made and a comprehensive profit (cp–comprehensive cost (cc) evaluation system (composed of 15 comprehensive cost indices and 14 comprehensive profit indices) has been established, with a index optimization matrix of comprehensive cost indices and comprehensive profit indices obtained and a ratio model of comprehensive cost to comprehensive profit (R_cc/cp) built. Results show that the R_cc/cp value of vegetables intercropping soybeans in insect-proof thin film greenhouses is the smallest and the R_cc/cp value of vegetables intercropping taros in insect-proof net greenhouses, pheromones in insect-proof thin film greenhouses, pheromones in insect-proof thin film greenhouses and ground planting (only using chemical pesticide for insect-proof without covering materials and synthetic sex pheromone) other four techniques are 0.6268, 0.6393, 0.6407, 0.9809 respectively. In accordance with the R_cc/cp value, vegetables intercropping soybeans in insect-proof thin film greenhouses can be the most optimized pesticide pollution controlling technique in the vegetable growing.     

Potential impact model to assess agricultural pressure to landscape ecological functions

Conservation and protection of soil and water resources and visual aspects of landscape, as well as the promotion of biodiversity, are some of the central tasks of environmental policy development and social politics in the future. One of the main questions is: ‘which agricultural systems are able to guarantee sustained resource-conserving land use?’ Based on the ecological risk concepts of the 1970s and 1980s, a potential impact model was developed using a universal assessment algorithm derived from fuzzy logic. The model estimated the potential impact of agricultural land use on ecosystem function using a few resource indicators. Intervention intensities of agricultural land-use are set in relation to site conditions and aggregated for each of several defined potential impact categories. The interpretation with respect to risk and the calculation of potential impact values are explained.     

Urban expansion and its contribution to the regional carbon emissions: Using the model based on the population density distribution

The method is used for calculating regional urban area dynamics and the resulting carbon emissions (from the land-conversion) for the period of 1980 till 2050 for the eight world regions. This approach is based on the fact that the spatial distribution of population density is close to the two-parametric Γ-distribution [Kendall, M.G., Stuart, A., 1958. The Advanced Theory of Statistics, vol. 1.2. Academic Press, New York; Vaughn, R., 1987. Urban Spatial Traffic Patterns, Pion, London]. The developed model provides us with the scenario of urbanisation, based on which the regional and world dynamics of carbon emissions and export from cities, and the annual total urban carbon balance are estimated. According to our estimations, world annual emissions of carbon as a result of urbanisation increase up to 1.25 GtC in 2005 and begin to decrease afterwards. If we compare the emission maximum with the annual emission caused by deforestation, 1.36 GtC per year, then we can say that the role of urbanised territories (UT) in the global carbon balance is of a comparable magnitude. Regarding the world annual export of carbon from UT, we observe its monotonous growth by three times, reaching 505 MtC. The latter, is comparable to the amount of carbon transported by rivers into the ocean (196–537 MtC). The current model shows that urbanisation is inhibited in the interval 2020–2030, and by 2050 the growth of urbanised areas would almost stop. Hence, the total balance, being almost constant until 2000, then starts to decrease at an almost constant rate. By the end of the XXI century, the total carbon balance will be equal to zero, with the exchange flows fully balanced, and may even be negative, with the system beginning to take up carbon from the atmosphere, i.e., becomes a “sink”. The regional dynamics is somewhat more complex, i.e., some regions, like China, Asia and Pacific are being active sources of Carbon through the studied period, while others are changing from source to sink or continue to be neutral in respect the GCC.     

Misspecification of within-area exposure distribution in ecological Poisson models

Ecological studies enable investigation of geographic variations in exposure to environmental variables, across groups, in relation to health outcomes measured on a geographic scale. Such studies are subject to ecological biases, including pure specification bias which arises when a nonlinear individual exposure-risk model is assumed to apply at the area level. Introduction of the within-area variance of exposure should induce a marked reduction in this source of ecological bias. Assuming several measurements per area of exposure and no confounding risk factors, we study the model including the within-area exposure variability when Gaussian within-area exposure distribution is assumed. The robustness is assessed when the within-area exposure distribution is misspecified. Two underlying exposure distributions are studied: the Gamma distribution and an unimodal mixture of two Gaussian distributions. In case of strong ecological association, this model can reduce the bias and improve the precision of the individual parameter estimates when the within-area exposure means and variances are correlated. These different models are applied to analyze the ecological association between radon concentration and childhood acute leukemia in France.

Using a social–ecological framework to inform the implementation of conservation plans

One of the key determinants of success in biodiversity conservation is how well conservation planning decisions account for the social system in which actions are to be implemented. Understanding elements of how the social and ecological systems interact can help identify opportunities for implementation. Utilizing data from a large‐scale conservation initiative in southwestern of Australia, we explored how a social–ecological system framework can be applied to identify how social and ecological factors interact to influence the opportunities for conservation. Using data from semistructured interviews, an online survey, and publicly available data, we developed a conceptual model of the social–ecological system associated with the conservation of the Fitz‐Stirling region. We used this model to identify the relevant variables (remnants of vegetation, stakeholder presence, collaboration between stakeholders, and their scale of management) that affect the implementation of conservation actions in the region. We combined measures for these variables to ascertain how areas associated with different levels of ecological importance coincided with areas associated with different levels of stakeholder presence, stakeholder collaboration, and scales of management. We identified areas that could benefit from different implementation strategies, from those suitable for immediate conservation action to areas requiring implementation over the long term to increase on‐the‐ground capacity and identify mechanisms to incentivize implementation. The application of a social–ecological framework can help conservation planners and practitioners facilitate the integration of ecological and social data to inform the translation of priorities for action into implementation strategies that account for the complexities of conservation problems in a focused way.     

Inference about the ratio of means from Negative Binomial paired count data

We derive some statistical properties of the distribution of two Negative Binomial random variables conditional on their total. This type of model can be appropriate for paired count data with Poisson over-dispersion such that the variance is a quadratic function of the mean. This statistical model is appropriate in many ecological applications including comparative fishing studies of two vessels and or gears. The parameter of interest is the ratio of pair means. We show that the conditional means and variances are different from the more commonly used Binomial model with variance adjusted for over-dispersion, or the Beta-Binomial model. The conditional Negative Binomial model is complicated because it does not eliminate nuisance parameters like in the Poisson case. Maximum likelihood estimation with the unconditional Negative Binomial model can result in biased estimates of the over-dispersion parameter and poor confidence intervals for the ratio of means when there are many nuisance parameters. We propose three approaches to deal with nuisance parameters in the conditional Negative Binomial model. We also study a random effects Binomial model for this type of data, and we develop an adjustment to the full-sample Negative Binomial profile likelihood to reduce the bias caused by nuisance parameters. We use simulations with these methods to examine bias, precision, and accuracy of estimators and confidence intervals. We conclude that the maximum likelihood method based on the full-sample Negative Binomial adjusted profile likelihood produces the best statistical inferences for the ratio of means when paired counts have Negative Binomial distributions. However, when there is uncertainty about the type of Poisson over-dispersion then a Binomial random effects model is a good choice.