Hierarchical Bayesian Analysis of Correlated Zero‐inflated Count Data

This article presents two‐component hierarchical Bayesian models which incorporate both overdispersion and excess zeros. The components may be resultants of some intervention (treatment) that changes the rare event generating process. The models are also expanded to take into account any heterogeneity that may exist in the data. Details of the model fitting, checking and selecting alternative models from a Bayesian perspective are also presented. The proposed methods are applied to count data on the assessment of an efficacy of pesticides in controlling the reproduction of whitefly. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hierarchical multivariate mixture generalized linear models for the analysis of spatial data: An application to disease mapping

Disease mapping of a single disease has been widely studied in the public health setup. Simultaneous modeling of related diseases can also be a valuable tool both from the epidemiological and from the statistical point of view. In particular, when we have several measurements recorded at each spatial location, we need to consider multivariate models in order to handle the dependence among the multivariate components as well as the spatial dependence between locations. It is then customary to use multivariate spatial models assuming the same distribution through the entire population density. However, in many circumstances, it is a very strong assumption to have the same distribution for all the areas of population density. To overcome this issue, we propose a hierarchical multivariate mixture generalized linear model to simultaneously analyze spatial Normal and non‐Normal outcomes. As an application of our proposed approach, esophageal and lung cancer deaths in Minnesota are used to show the outperformance of assuming different distributions for different counties of Minnesota rather than assuming a single distribution for the population density. Performance of the proposed approach is also evaluated through a simulation study.     

Testing Homogeneity of Two Zero‐inflated Poisson Populations

The problem of testing treatment difference in the occurrence of a safety parameter in a randomized parallel‐group comparative clinical trial under the assumption that the number of occurrence follows a zero‐inflated Poisson (ZIP) distribution is considered. Likelihood ratio tests (LRT) for homogeneity of two ZIP populations are derived under the hypotheses that (i) there is no difference in inflation parameters, (ii) there is no difference in non‐zero means; and (iii) there is no difference in both inflation parameters and non‐zero means. Approximate formulas for sample size calculation are also obtained for achieving a desired power for detecting a clinically meaningful difference under the corresponding alternative hypotheses. An example concerning the assessment of the gastrointestinal (GI) safety in terms of the number of erosion counts of a newly developed compound for the treatment of osteoarthritis and rheumatoid arthritis is given for illustration purpose (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zero‐Inflated Negative Binomial Mixed Regression Modeling of Over‐Dispersed Count Data with Extra Zeros

In many biometrical applications, the count data encountered often contain extra zeros relative to the Poisson distribution. Zero‐inflated Poisson regression models are useful for analyzing such data, but parameter estimates may be seriously biased if the nonzero observations are over‐dispersed and simultaneously correlated due to the sampling design or the data collection procedure. In this paper, a zero‐inflated negative binomial mixed regression model is presented to analyze a set of pancreas disorder length of stay (LOS) data that comprised mainly same‐day separations. Random effects are introduced to account for inter‐hospital variations and the dependency of clustered LOS observations. Parameter estimation is achieved by maximizing an appropriate log‐likelihood function using an EM algorithm. Alternative modeling strategies, namely the finite mixture of Poisson distributions and the non‐parametric maximum likelihood approach, are also considered. The determination of pertinent covariates would assist hospital administrators and clinicians to manage LOS and expenditures efficiently.     

Elucidating the spatio‐temporal dynamics of an emerging wildlife pathogen using approximate Bayesian computation

Emerging pathogens constitute a severe threat for human health and biodiversity. Determining the status (native or non‐native) of emerging pathogens, and tracing back their spatio‐temporal dynamics, is crucial to understand the eco‐evolutionary factors promoting their emergence, to control their spread and mitigate their impacts. However, tracing back the spatio‐temporal dynamics of emerging wildlife pathogens is challenging because (i) they are often neglected until they become sufficiently abundant and pose socio‐economical concerns and (ii) their geographical range is often little known. Here, we combined classical population genetics tools and approximate Bayesian computation (i.e. ABC) to retrace the dynamics of Tracheliastes polycolpus, a poorly documented pathogenic ectoparasite emerging in Western Europe that threatens several freshwater fish species. Our results strongly suggest that populations of T. polycolpus in France emerged from individuals originating from a unique genetic pool that were most likely introduced in the 1920s in central France. From this initial population, three waves of colonization occurred into peripheral watersheds within the next two decades. We further demonstrated that populations remained at low densities, and hence undetectable, during 10 years before a major demographic expansion occurred, and before its official detection in France. These findings corroborate and expand the few historical records available for this emerging pathogen. More generally, our study demonstrates how ABC can be used to determine the status, reconstruct the colonization history and infer key evolutionary parameters of emerging wildlife pathogens with low data availability, and for which samples from the putative native area are inaccessible.     

Pattern‐Mixture Zero‐Inflated Mixed Models for Longitudinal Unbalanced Count Data with Excessive Zeros

Analysis of longitudinal data with excessive zeros has gained increasing attention in recent years; however, current approaches to the analysis of longitudinal data with excessive zeros have primarily focused on balanced data. Dropouts are common in longitudinal studies; therefore, the analysis of the resulting unbalanced data is complicated by the missing mechanism. Our study is motivated by the analysis of longitudinal skin cancer count data presented by Greenberg, Baron, Stukel, Stevens, Mandel, Spencer, Elias, Lowe, Nierenberg, Bayrd, Vance, Freeman, Clendenning, Kwan, and the Skin Cancer Prevention Study Group[New England Journal of Medicine 323 , 789–795]. The data consist of a large number of zero responses (83% of the observations) as well as a substantial amount of dropout (about 52% of the observations). To account for both excessive zeros and dropout patterns, we propose a pattern‐mixture zero‐inflated model with compound Poisson random effects for the unbalanced longitudinal skin cancer data. We also incorporate an autoregressive of order 1 correlation structure in the model to capture longitudinal correlation of the count responses. A quasi‐likelihood approach has been developed in the estimation of our model. We illustrated the method with analysis of the longitudinal skin cancer data.     

Bayesian Estimation of Hunting Success Rate and Harvest for Spatially Correlated Post‐stratified Data

We consider the estimation of success rate and harvest under post survey stratification at the sub‐domain (county) level. Often in this situation, the population size for the sub‐domain is unknown and the random mechanism that dictates the sample size for sub‐domains is ignored. Finding good estimators of success rate and harvest is very important for wildlife abundance. A Bayesian hierarchical model is developed to estimate both success rate and harvest simultaneously. The model includes a random sub‐domain sample size correlated with the number of successes in the sub‐domain, fixed week effects, random geographic effects, and spatial correlations between neighboring sub‐domains. The computation is done by Gibbs sampling and adaptive rejection sampling techniques. The method developed is illustrated using data from the Missouri Turkey Hunting Survey. The estimation of success rate is improved by treating the the sub‐domain sample size as a random variable instead of a fixed constant. The Bayesian model yields a reasonable harvest estimation. The spatial pattern of the estimated harvest matches the pattern of the check station data.     

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.     

A two-part mixed-effects pattern-mixture model to handle zero-inflation and incompleteness in a longitudinal setting

Two-part regression models are frequently used to analyze longitudinal count data with excess zeros, where the same set of subjects is repeatedly observed over time. In this context, several sources of heterogeneity may arise at individual level that affect the observed process. Further, longitudinal studies often suffer from missing values: individuals dropout of the study before its completion, and thus present incomplete data records. In this paper, we propose a finite mixture of hurdle models to face the heterogeneity problem, which is handled by introducing random effects with a discrete distribution; a pattern-mixture approach is specified to deal with non-ignorable missing values. This approach helps us to consider overdispersed counts, while allowing for association between the two parts of the model, and for non-ignorable dropouts. The effectiveness of the proposal is tested through a simulation study. Finally, an application to real data on skin cancer is provided.     

Empirical Bayes estimation of random effects parameters in mixed effects logistic regression models

We extend an approach for estimating random effects parameters under a random intercept and slope logistic regression model to include standard errors, thereby including confidence intervals. The procedure entails numerical integration to yield posterior empirical Bayes (EB) estimates of random effects parameters and their corresponding posterior standard errors. We incorporate an adjustment of the standard error due to Kass and Steffey (KS; 1989, Journal of the American Statistical Association 84, 717-726) to account for the variability in estimating the variance component of the random effects distribution. In assessing health care providers with respect to adult pneumonia mortality, comparisons are made with the penalized quasi-likelihood (PQL) approximation approach of Breslow and Clayton (1993, Journal of the American Statistical Association 88, 9-25) and a Bayesian approach. To make comparisons with an EB method previously reported in the literature, we apply these approaches to crossover trials data previously analyzed with the estimating equations EB approach of Waclawiw and Liang (1994, Statistics in Medicine 13, 541-551). We also perform simulations to compare the proposed KS and PQL approaches. These two approaches lead to EB estimates of random effects parameters with similar asymptotic bias. However, for many clusters with small cluster size, the proposed KS approach does better than the PQL procedures in terms of coverage of nominal 95% confidence intervals for random effects estimates. For large cluster sizes and a few clusters, the PQL approach performs better than the KS adjustment. These simulation results agree somewhat with those of the data analyses.     

Marginalized zero‐inflated Poisson models with missing covariates

Unlike zero‐inflated Poisson regression, marginalized zero‐inflated Poisson (MZIP) models for counts with excess zeros provide estimates with direct interpretations for the overall effects of covariates on the marginal mean. In the presence of missing covariates, MZIP and many other count data models are ordinarily fitted using complete case analysis methods due to lack of appropriate statistical methods and software. This article presents an estimation method for MZIP models with missing covariates. The method, which is applicable to other missing data problems, is illustrated and compared with complete case analysis by using simulations and dental data on the caries preventive effects of a school‐based fluoride mouthrinse program.     

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.     

Plant diversity increases spatio‐temporal niche complementarity in plant‐pollinator interactions

Ongoing biodiversity decline impairs ecosystem processes, including pollination. Flower visitation, an important indicator of pollination services, is influenced by plant species richness. However, the spatio‐temporal responses of different pollinator groups to plant species richness have not yet been analyzed experimentally. Here, we used an experimental plant species richness gradient to analyze plant–pollinator interactions with an unprecedented spatio‐temporal resolution. We observed four pollinator functional groups (honeybees, bumblebees, solitary bees, and hoverflies) in experimental plots at three different vegetation strata between sunrise and sunset. Visits were modified by plant species richness interacting with time and space. Furthermore, the complementarity of pollinator functional groups in space and time was stronger in species‐rich mixtures. We conclude that high plant diversity should ensure stable pollination services, mediated via spatio‐temporal niche complementarity in flower visitation.     

An integrated,spatio‐temporal modelling framework for analysing biological invasions

Aim

We develop a novel modelling framework for analysing the spatio‐temporal spread of biological invasions. The framework integrates different invasion drivers and disentangles their roles in determining observed invasion patterns by fitting models to historical distribution data. As a case study application, we analyse the spread of common ragweed (Ambrosia artemisiifolia).

Location

Central Europe.

Methods

A lattice system represents actual landscapes with environmental heterogeneity. Modelling covers the spatio‐temporal invasion sequence in this grid and integrates the effects of environmental conditions on local invasion suitability, the role of invaded cells and spatially implicit “background” introductions as propagule sources, within‐cell invasion level bulk‐up and multiple dispersal means. A modular framework design facilitates flexible numerical representation of the modelled invasion processes and customization of the model complexity. We used the framework to build and contrast increasingly complex models, and fitted them using a Bayesian inference approach with parameters estimated by Markov chain Monte Carlo (MCMC).

Results

All modelled invasion drivers codetermined the A. artemisiifolia invasion pattern. Inferences about individual drivers depended on which processes were modelled concurrently, and hence changed both quantitatively and qualitatively between models. Among others, the roles of environmental variables were assessed substantially differently subject to whether models included explicit source‐recipient cell relationships, spatio‐temporal variability in source cell strength and human‐mediated dispersal means. The largest fit improvements were found by integrating filtering effects of the environment and spatio‐temporal availability of propagule sources.

Main conclusions

Our modelling framework provides a straightforward means to build integrated invasion models and address hypotheses about the roles and mutual relationships of different putative invasion drivers. Its statistical nature and generic design make it suitable for studying many observed invasions. For efficient invasion modelling, it is important to represent changes in spatio‐temporal propagule supply by explicitly tracking the species’ colonization sequence and establishment of new populations.

     

Probabilistic spatio‐temporal assessment of vegetation vulnerability to climate change in Swaziland

In a spatially explicit climate change impact assessment, a Bayesian network (BN) model was implemented to probabilistically simulate future response of the four major vegetation types in Swaziland. Two emission scenarios (A2 and B2) from an ensemble of three statistically downscaled coupled atmosphere‐ocean global circulation models (CSIRO‐Mk3, CCCma‐CGCM3 and UKMO‐HadCM3) were used to simulate possible changes in BN‐based environmental envelopes of major vegetation communities. Both physiographic and climatic data were used as predictors representing the 2020s, 2050s and the 2080s periods. A comparison of simulated vegetation distribution and the expert vegetation map under baseline conditions showed an overall correspondence of 97.7% and a Kappa coefficient of 0.966. Although the ensemble projections showed comparable trends during the 2020s, the results from the A2 storyline were more drastic indicating that grassland and the Lebombo bushveld will be impacted negatively as early as the 2020s with about 1 °C temperature increase. The bioclimatically suitable areas of all but one vegetation type decline drastically after about 2 °C warming, more so under the more severe A2 scenario and in particular during the 2080s. The sour bushveld is the only vegetation type that initially responds positively to warming by possibly encroaching to the highly vulnerable grassland areas. Vulnerability of vegetation is increased by the limited ability to migrate into suitable climates due to close affinity to certain geological formations and the fragmentation of the landscape by agriculture and other land uses. This is expected to have serious impacts on biodiversity in the country. Under warmer climates, the likely vegetation types to emerge are uncertain due to future novel combinations of climate and bedrock lithology. The strengths and limitations of the BN approach are also discussed.