Objective Bayesian Analysis of Geometrically Anisotropic Spatial Data

Anisotropic models are often used in spatial statistics to analyze spatially referenced data. Within a Bayesian framework we develop default priors for the anisotropic Gaussian random field model with and without including a nugget parameter accounting for the effects of microscale variations and measurement errors. We present Jeffreys priors and a reference prior and study their posterior propriety. Moreover, we obtain that the predictive distributions at ungauged locations have finite variance. We also show that the seemingly uninformative uniform prior for the anisotropy parameters, ratio and angle, yields an improper posterior. Finally, we find that the proposed priors have good frequentist properties and we illustrate our approach by analyzing two data sets for which we discuss model choice as well as predictions and uncertainty estimates.     

Another Look at Bayesian Analysis of AMMI Models for Genotype-Environment Data

Linear–bilinear models are frequently used to analyze two-way data such as genotype-by-environment data. A well-known example of this class of models is the additive main effects and multiplicative interaction effects model (AMMI). We propose a new Bayesian treatment of such models offering a proper way to deal with the major problem of overparameterization. The rationale is to ignore the issue at the prior level and apply an appropriate processing at the posterior level to be able to arrive at easily interpretable inferences. Compared to previous attempts, this new strategy has the great advantage of being directly implementable in standard software packages devoted to Bayesian statistics such as WinBUGS/OpenBUGS/JAGS. The method is assessed using simulated datasets and a real dataset from plant breeding. We discuss the benefits of a Bayesian perspective to the analysis of genotype-by-environment interactions, focusing on practical questions related to general and local adaptation and stability of genotypes. We also suggest a new solution to the estimation of the risk of a genotype not exceeding a given threshold.     

Bayesian Semiparametric Model for Pathway-Based Analysis with Zero-Inflated Clinical Outcomes

In this paper, we propose a semiparametric regression approach for identifying pathways related to zero-inflated clinical outcomes, where a pathway is a gene set derived from prior biological knowledge. Our approach is developed by using a Bayesian hierarchical framework. We model the pathway effect nonparametrically into a zero-inflated Poisson hierarchical regression model with an unknown link function. Nonparametric pathway effect was estimated via a kernel machine, and the unknown link function was estimated by transforming a mixture of the beta cumulative density function. Our approach provides flexible nonparametric settings to describe the complicated association between gene expressions and zero-inflated clinical outcomes. The Metropolis-within-Gibbs sampling algorithm and Bayes factor were adopted to make statistical inferences. Our simulation results support that our semiparametric approach is more accurate and flexible than zero-inflated Poisson regression with the canonical link function, which is especially true when the number of genes is large. The usefulness of our approach is demonstrated through its applications to the Canine data set from Enerson et al. (Toxicol Pathol 34:27–32, 2006). Our approach can also be applied to other settings where a large number of highly correlated predictors are present.Supplementary materials accompanying this paper appear on-line.     

Bayesian Hierarchical Analysis for Multiple Health Endpoints in a Toxicity Study

Bayesian hierarchical models are built to fit multiple health endpoints from a dose-response study of a chemical contaminant, perchlorate. Perchlorate exposure results in iodine uptake inhibition in the thyroid, with health effects manifested by changes in blood hormone concentrations and histopathological effects on the thyroid. We propose empirical models to fit blood hormone concentration and thyroid histopathology data for rats exposed to perchlorate in the 90-day study of Springborn Laboratories Inc. (1998), based upon a mechanistic model derived from the assumed toxicological relationships between dose and the various endpoints. All of the models are fit in a Bayesian framework, and predictions about each endpoint in response to dose are simulated based on the posterior predictive distribution. A hierarchical model tries to exploit possible similarities between different combinations of sex and exposure duration, and it allows us to produce more stable estimates of dose-response curves. We also illustrate how the Bayesian model specification allows us to address additional questions that arise after the analysis.     

Analysis of Clustered Binary Data With Unequal Cluster Sizes: A Semiparametric Bayesian Approach

The analysis of clustered binary data is a common task in many areas of application. Parametric approaches to the analysis of such data are numerous, but there has been much recent interest in nonparametric and semiparametric approaches. When cluster sizes are unequal, an assumption is often made of compatibility of marginal distributions in order for semiparametric approaches to be developed when there is little replication for different cluster sizes. Here, we use the marginal compatibility assumption to extend flexible semiparametric Bayesian methods able to shrink towards a “parametric backbone” to the situation where there are few replicated observations for distinct cluster sizes and each distinct value of a covariate. A motivating application is the analysis of developmental toxicology data where pregnant laboratory animals are exposed to a dose of some potentially toxic compound and interest lies in describing the distribution, as a function of the dose level, of the number of fetuses exhibiting some characteristic abnormality. Flexible semiparametric methods are required here, as the data typically exhibit overdispersion and complex structure. We also consider a further extension appropriate to the analysis of clustered binary data in the situation where there is little or no replication for distinct covariate values.     

Radiocarbon Dating of Organic Matter Fractions of a Chinese Mollisol

Humic acids, humins, and the total organic matter of a Mollisol from Inner Mongolia in China were investigated regarding ¹⁴C age and δ¹³C. The ¹⁴C age of the total organic matter increases with 69.3 years per cm depth which corresponds to an extremely slow soil development. The humic acids are generally older than the humins. Humins and humic acids are older than the total organic matter, except in the deepest horizon. δ¹³C does not deviate from that of fresh plant material throughout the profile indicating early protection against further degradation.     

Bayesian analysis of semiparametric linear-circular models

In many environmental and agricultural studies, data are collected on both linear and circular random variables, with possible dependence between the variables. Classically, the analysis of such data has been carried out in a classical regression framework. We propose a Bayesian hierarchical framework to handle all forms of uncertainty arising in a linear-circular data set. One novelty of our multivariate linear-circular model is that, marginally, the circular component is assumed to be a mixture model with an unknown number of von Mises (or circular normal) distributions. We use the Dirichlet process to introduce variability in the model dimensionality, and develop a simple Gibbs sampling algorithm for simulating the mixture components. Although we illustrate our methodology on von Mises mixtures, it is widely applicable. We thus avoid complicated reversible-jump Markov chain Monte Carlo methods, which are considered ideal for analyzing mixtures of unknown number of distributions. We illustrate our methodologies with simulated and real data sets. Using pseudo-Bayes factors, we also compare different models associated with both fixed and variable numbers of von Mises distributions. Our findings suggest that models associated with varying numbers of mixture components perform at least as well as those with known numbers of mixture components. We tentatively argue that model averaging associated with variable number of mixture components improves the model’s predictive power, which compensates for the lack of knowledge of the actual number of mixture components.     

Bayesian approach in agrochemical surveys

In the scheme of agrochemical surveys, sampling is one of the most costly stages. The use of a priori information about the distributions of properties (Bayesian approach) allows one to reduce the number of samples to be taken by 5–10% without deteriorating the accuracy of the estimation. The maintenance of regional soil-mapping databases is a necessary condition for applying the Bayesian approach.     

Bayesian estimation of survival from mark-recapture data

An understanding of survival patterns is a fundamental component of animal population biology. Mark-recapture models are often used in the estimation of animal survival rates. Maximum likelihood estimation, via either analytic solution or numerical approximation, has typically been used for inference in these models throughout the literature. In this article, a Bayesian approach is outlined and an easily applicable implementation via Markov chain Monte Carlo is described. The method is illustrated using 13 years of mark-recapture data for fulmar petrels on an island in Orkney. Point estimates of survival are similar to the maximum likelihood estimates (MLEs), but the posterior variances are smaller than the corresponding asymptotic variances of the MLEs. The Bayesian approach yields point estimates of 0.9328 for the average annual survival probability and 14.37 years for the expected lifetime of the fulmar petrels. A simple modification that accounts for missing data is also described. The approach is easier to apply than augmentation methods in this case, and simulations indicate that the performance of the estimators is not significantly diminished by the missing data.     

Bayesian methods in plant conservation biology

We provide an introduction to Bayesian methods in conservation biology, illustrating inferences, prediction and decision making issues. After presenting the basic framework with a recovery plan evaluation problem, we illustrate more complex issues related to forecasting trends of structured populations using matrix population models and, finally, describe relevant topics in spatial and logistic regression problems. Computational and other implementation difficulties are also discussed.     

Lead-210 Age Dating of Three Peat Cores in the Jura Mountains,Switzerland

Peat cores collected from three sites in the Jura region of Switzerland, La Tourbière des Genevez, Etang de la Gruère and Praz Rodet, were analysed for the fallout radionuclides 210Pb, 137Cs, 134Cs and 241Am, and the minerogenic radionuclide 226Ra. Unsupported 210Pb inventories of the cores were all in the range 3900-4784 Bq m-2 and are consistent with their having largely retained the atmospheric flux (ca. 130 Bq m-2y-1). In contrast, comparisons with earlier (1986) cores from these sites suggest that there have been significant losses of fallout 137Cs. Samples from all three sites had low but significant levels of supported 210Pb (226Ra) activity, the origin of which is presumably wind blown soil dust. 210Pb dates calculated using the CRS model were independently validated by 241Am and pollen stratigraphy. The core from La Tourbière des Genevez had a fairly conventional 210Pb activity versus depth profile indicating more or less constant net accumulation during the past 100 years. At the other two sites however the 210Pb profiles contained significant deviations from simple exponential decline that may record episodes in the bog during which there were major variations in the net dry mass accumulation rate. These episodes are dated 1930-50 (Etang de la Gruère) and 1960-80 (Praz Rodet). Although the depths at which total 210Pb reaches equilibrium with the supporting 226Ra range widely, from ca.37 cm in Etang de la Gruére to ca.60 cm in Praz Rodet, mean net dry mass accumulation rates for the past 100 years are remarkably similar, the values for all three sites are in the range 0.023-0.027 g cm-2y-1.     

Bayesian Methods for Hierarchical Distance Sampling Models

The few distance sampling studies that use Bayesian methods typically consider only line transect sampling with a half-normal detection function. We present a Bayesian approach to analyse distance sampling data applicable to line and point transects, exact and interval distance data and any detection function possibly including covariates affecting detection probabilities. We use an integrated likelihood which combines the detection and density models. For the latter, densities are related to covariates in a log-linear mixed effect Poisson model which accommodates correlated counts. We use a Metropolis-Hastings algorithm for updating parameters and a reversible jump algorithm to include model selection for both the detection function and density models. The approach is applied to a large-scale experimental design study of northern bobwhite coveys where the interest was to assess the effect of establishing herbaceous buffers around agricultural fields in several states in the US on bird densities. Results were compared with those from an existing maximum likelihood approach that analyses the detection and density models in two stages. Both methods revealed an increase of covey densities on buffered fields. Our approach gave estimates with higher precision even though it does not condition on a known detection function for the density model.     

贝叶斯最大熵地统计方法研究与应用进展

以克里格估算为基础的插值和随机模拟为代表的经典地统计方法是目前研究地理属性空间分布的主要方法,但仍存在精度不高及不能有效利用其他有价值信息的缺陷。近年来贝叶斯最大熵地统计方法在国外逐渐流行,该方法能够在有效利用多源数据的基础上,提高空间分布研究精度,是一种新的非线性方法。本文详细阐述了贝叶斯最大熵方法的数据内容、实施步骤、一般算法及计算结果,并介绍了该方法的应用情况,最后对该方法的优点和不足作出了评价。     

Dating chicks: Calibration and discrimination in a nonlinear multivariate hierarchical growth model

The motivation for this work was to investigate the possibility of accurately determining the age of a tern chick using easily obtained body measurements. We describe the construction of a nonlinear multivariate hierarchical model for chick growth and show how it can be estimated using Markov chain Monte Carlo techniques. A simple extension of the analysis allows for estimation of the ages of unknown chicks. Posterior distributions of the unknown ages are derived, so that the accuracy of age determination can be examined. We further extend our model and analysis to include the possibility that chicks fall into distinct groups with different growth characteristics. The technique is illustrated using data on the weight and wing length of black-fronted terns from the Ohau River, New Zealand. It is found that dating to within one day is possible, but only in some areas of the data space. The concept of “braiding” of multivariate growth curves is introduced to explain the varying accuracy of age determination.     

A Bayesian hierarchical model for risk assessment of methylmercury

This article uses a Bayesian hierarchical model to quantify the adverse health effects associated with in-utero exposure to methylmercury. By allowing for study-to-studyas well as outcome-to-outcome variability, the approach provides a useful meta-analytic tool for multi-outcome, multi-study environmental risk assessments. The analysis presented here expands on the findings of a National Academy of Sciences (NAS) committee, charged with advising the United States Environmental Protection Agency (EPA) on an appropriate approach to conducting a risk assessment for methylmercury. The NAS committee, for which the senior author (Ryan) was a committee member, reviewed the findings from several conflicting studies and reported the results from a Bayesian hierarchical model that synthesized information across several studies and for several outcomes. Although the NAS committee did not suggest that the hierarchical model be used as the actual basis for a methylmercury risk assessment, the results from the model were used to justify and support the final recommendation that the risk analysis be based on data from a study conducted in the Faroe Islands, which had found a positive association between in-utero exposure to methylmercury and impaired neurological development. We considera variety of statistical issues, but particularly sensitivity to model specification.     

Bayesian multivariate process modeling for prediction of forest attributes

This article investigates multivariate spatial process models suitable for predicting multiple forest attributes using a multisource forest inventory approach. Such data settings involve several spatially dependent response variables arising in each location. Not only does each variable vary across space, they are likely to be correlated among themselves. Traditional approaches have attempted to model such data using simplifying assumptions, such as a common rate of decay in the spatial correlation or simplified cross-covariance structures among the response variables. Our current focus is to produce spatially explicit, tree species specific, prediction of forest biomass per hectare over a region of interest. Modeling such associations presents challenges in terms of validity of probability distributions as well as issues concerning identifiability and estimability of parameters. Our template encompasses several models with different correlation structures. These models represent different hypotheses whose tenability are assessed using formal model comparisons. We adopt a Bayesian hierarchical approach offering a sampling-based inferential framework using efficient Markov chain Monte Carlo methods for estimating model parameters.     

Bayesian analysis of temperature sensitivity of plant phenology in Germany

The temperature sensitivity of 43 phenological phases was analysed in Germany within the period 1951–2006 with the help of a Bayesian approach. First a Bayesian model comparison of monthly temperatures and phenological phases throughout the year was conducted. We analysed the data as constant (mean onset date), as linear (constant trend over time) and as change point model (time varying change). The change point model involves the selection of two linear segments which match at a particular time. The matching point is estimated by an examination of all possible breaks weighted by their respective change point probability. Secondly a Bayesian coherence analysis was applied to investigate the relationship between phenological onset dates and an effective temperature generated as a weighted average of monthly means. Temperature weight coefficients were obtained from an optimization of a coherence factor by simulated annealing.Results reveal that late spring, summer and early autumn temperature months exhibit a clear preference for the change point model (>50%) indicating nonlinear change. The temperature development of April and August shows exceptionally high nonlinearities compared to the other months with change point model probabilities of 78% and 81% over the last five decades.For all phenophases a strong dependence of phenology on temperature is determined. We can classify two main temperature response patterns of the studied phenological phases: on the one hand spring phenophases are particularly sensitive to temperatures in April, exhibiting a prompt response. On the other hand summer phenophases are less influenced by temperature during or right before the month of the onset. They reveal a delayed response to nonlinear temperature changes mainly of April. Especially abrupt changes during the temperature sensitive stage of species cause a pronounced change in plant phenology regardless of the time of onset.     

The Bayesian Group Lasso for Confounded Spatial Data

Generalized linear mixed models for spatial processes are widely used in applied statistics. In many applications of the spatial generalized linear mixed model (SGLMM), the goal is to obtain inference about regression coefficients while achieving optimal predictive ability. When implementing the SGLMM, multicollinearity among covariates and the spatial random effects can make computation challenging and influence inference. We present a Bayesian group lasso prior with a single tuning parameter that can be chosen to optimize predictive ability of the SGLMM and jointly regularize the regression coefficients and spatial random effect. We implement the group lasso SGLMM using efficient Markov chain Monte Carlo (MCMC) algorithms and demonstrate how multicollinearity among covariates and the spatial random effect can be monitored as a derived quantity. To test our method, we compared several parameterizations of the SGLMM using simulated data and two examples from plant ecology and disease ecology. In all examples, problematic levels multicollinearity occurred and influenced sampling efficiency and inference. We found that the group lasso prior resulted in roughly twice the effective sample size for MCMC samples of regression coefficients and can have higher and less variable predictive accuracy based on out-of-sample data when compared to the standard SGLMM.Supplementary materials accompanying this paper appear online.