A bayesian approach to crop Model calibration under unknown error covariance

Bayesian methods seem well adapted to dynamic system models in general and to crop models in particular, because there is in general prior information about parameter values. The usefulness of a Bayesian approach has often been pointed out, but actual applications are rather rare. A major difficulty is including the elements of the covariance matrix of model errors in the treatment. We treat the specific case of balanced data and an unstructured covariance matrix. In our particular case this is a 3 × 3 matrix. We illustrate two methods for deriving a sample from the joint posterior density for the crop model parameters and the error covariance matrix parameters. The first method is based on importance sampling, the second on Metropolis within Gibbs sampling. We derive an instrumental density for the former and a proposal density for the latter which are adapted to this type of model and data. Both algorithms work well and they give very similar results. The example concerns a model for sunflowers during rapid leaf growth. The ultimate goal is to use the model as a decision aid in predicting disease risk.     

Bayesian spatial modeling of data from avian point count surveys

We present a unified framework for modeling bird survey data collected at spatially replicated survey sites in the form of repeated counts or detection history counts, through which we model spatial dependence in bird density and variation in detection probabilities due to changes in covariates across the landscape. The models have a complex hierarchical structure that makes them suited to Bayesian analysis using Markov chain Monte Carlo (MCMC) algorithms. For computational efficiency, we use a form of conditional autogressive model for modeling spatial dependence. We apply the models to survey data for two bird species in the Great Smoky Mountains National Park. The algorithms converge well for the more abundant and easily detected of the two species, but some simplification of the spatial model is required for convergence for the second species. We show how these methods lead to maps of estimated relative density which are an improvement over those that would follow from past approaches that ignored spatial dependence. This work also highlights the importance of good survey design for bird species mapping studies.     

Analyzing designed experiments in distance sampling

Distance sampling is a survey technique for estimating the abundance or density of wild animal populations. Detection probabilities of animals inherently differ by species, age class, habitats, or sex. By incorporating the change in an observer’s ability to detect a particular class of animals as a function of distance, distance sampling leads to density estimates that are comparable across different species, ages, habitats, sexes, and so on. Increasing interest in evaluating the effects of management practices on animal populations in an experimental context has led to a need for suitable methods of analyzing distance sampling data. We outline a two-stage approach for analyzing distance sampling data from designed experiments, in which a two-step bootstrap is used to quantify precision and identify treatment effects. We illustrate this approach using data from a before—after control-impact experiment designed to assess the effects of large-scale prescribed fire treatments on bird densities in ponderosa pine forests of the southwestern United States.     

Bayesian Spatial Point Process Modeling of Line Transect Data

This paper develops a Bayesian approach for spatial inference on animal density from line transect survey data. We model the spatial distribution of animals within a geographical area of interest by an inhomogeneous Poisson process whose intensity function incorporates both covariate effects and spatial smoothing of residual variation. Independently thinning the animal locations according to their estimated detection probabilities results into another spatial Poisson process for the sightings (the observations). Prior distributions are elicited for all unknown model parameters. Due to the sparsity of data in the application we consider, eliciting sensible prior distributions is important in order to get meaningful estimation results. A reversible jump Markov Chain Monte Carlo (MCMC) algorithm for simulation of the posterior distribution is developed. We present results for simulated data and a real data set of minke whale pods from Antarctic waters. The main advantages of our method compared to design-based analyses are that it can use data arising from sources other than specifically designed surveys and its ability to link covariate effects to variation of animal density. The Bayesian paradigm provides a coherent framework for quantifying uncertainty in estimation results.     

A Varying Coefficients Model For Estimating Finite Population Totals: A Hierarchical Bayesian Approach

In some finite sampling situations, there is a primary variable that is sampled, and there are measurements on covariates for the entire population. A Bayesian hierarchical model for estimating totals for finite populations is proposed. A nonparametric linear model is assumed to explain the relationship between the dependent variable of interest and covariates. The regression coefficients in the linear model are allowed to vary as a function of a subset of covariates nonparametrically based on B-splines. The generality of this approach makes it robust and applicable to data collected using a variety of sampling techniques, provided the sample is representative of the finite population. A simulation study is carried out to evaluate the performance of the proposed model for the estimation of the population total. Results indicate accurate estimation of population totals using the approach. The modeling approach is used to estimate the total production of avocado for a large group of groves in Mexico.     

Model-Based Distance Sampling

Conventional distance sampling adopts a mixed approach, using model-based methods for the detection process, and design-based methods to estimate animal abundance in the study region, given estimated probabilities of detection. In recent years, there has been increasing interest in fully model-based methods. Model-based methods are less robust for estimating animal abundance than conventional methods, but offer several advantages: they allow the analyst to explore how animal density varies by habitat or topography; abundance can be estimated for any sub-region of interest; they provide tools for analysing data from designed distance sampling experiments, to assess treatment effects. We develop a common framework for model-based distance sampling, and show how the various model-based methods that have been proposed fit within this framework.     

Estimating Distance Sampling Detection Functions When Distances Are Measured With Errors

Distance sampling methods assume that distances are known but in practice there are often errors in measuring them. These can have substantial impact on the bias and precision of distance sampling estimators. In this paper we develop methods that accommodate both systematic and stochastic measurement errors. We use the methods to estimate detection probability in two surveys with substantial measurement error. The first is a shipboard line transect survey in the North Sea in which information on measurement error comes from photographically measured distances to a subset of detections. The second is an aerial cue-counting survey off Iceland in which information on measurement error comes from pairs of independently estimated distances to a subset of detections. Different methods are required for measurement error estimation in the two cases. We investigate by simulation the properties of the new estimators and compare them to conventional estimators. They are found to perform better than conventional estimators in the presence of measurement error, more so in the case of cue-counting and point transect estimators than line transect estimators. An appendix on the asymptotic distributions of conditional and full likelihood estimators is available online.     

A hierarchical model for compositional data analysis

This article introduces a hierarchical model for compositional analysis. Our approach models both source and mixture data simultaneously, and accounts for several different types of variation: these include measurement error on both the mixture and source data; variability in the sample from the source distributions; and variability in the mixing proportions themselves, generally of main interest. The method is an improvement on some existing methods in that estimates of mixing proportions (including their interval estimates) are sure to lie in the range [0, 1]; in addition, it is shown that our model can help in situations where identification of appropriate source data is difficult, especially when we extend our model to include a covariate. We first study the likelihood surface of a base model for a simple example, and then include prior distributions to create a Bayesian model that allows analysis of more complex situations via Markov chain Monte Carlo sampling from the likelihood. Application of the model is illustrated with two examples using real data: one concerning chemical markers in plants, and another on water chemistry.     

Likelihood-based inference for clustered line transect data

The uncertainty in estimation of spatial animal density from line transect surveys depends on the degree of spatial clustering in the animal population. To quantify the clustering we model line transect data as independent thinnings of spatial shot-noise Cox processes. Likelihood-based inference is implemented using Markov chain Monte Carlo (MCMC) methods to obtain efficient estimates of spatial clustering parameters. Uncertainty is addressed using parametric bootstrap or by consideration of posterior distributions in a Bayesian setting. Maximum likelihood estimation and Bayesian inference are compared in an example concerning minke whales in the northeast Atlantic.     

On modeling repeated binary responses and time-dependent missing covariates

We develop a novel modeling strategy for analyzing data with repeated binary responses over time as well as time-dependent missing covariates. We assume that covariates are missing at random (MAR). We use the generalized linear mixed logistic regression model for the repeated binary responses and then propose a joint model for time-dependent missing covariates using information from different sources. A Monte Carlo EM algorithm is developed for computing the maximum likelihood estimates. We propose an extended version of the AIC criterion to identify the important factors that m a y explain the binary responses. A real plant dataset is used to motivate and illustrate the proposed methodology.     

Dynamic Models of Animal Movement with Spatial Point Process Interactions

When analyzing animal movement, it is important to account for interactions between individuals. However, statistical models for incorporating interaction behavior in movement models are limited. We propose an approach that models dependent movement by augmenting a dynamic marginal movement model with a spatial point process interaction function within a weighted distribution framework. The approach is flexible, as marginal movement behavior and interaction behavior can be modeled independently. Inference for model parameters is complicated by intractable normalizing constants. We develop a double Metropolis–Hastings algorithm to perform Bayesian inference. We illustrate our approach through the analysis of movement tracks of guppies (Poecilia reticulata).     

Full Open Population Capture–Recapture Models With Individual Covariates

Traditional analyses of capture–recapture data are based on likelihood functions that explicitly integrate out all missing data. We use a complete data likelihood (CDL) to show how a wide range of capture–recapture models can be easily fitted using readily available software JAGS/BUGS even when there are individual-specific time-varying covariates. The models we describe extend those that condition on first capture to include abundance parameters, or parameters related to abundance, such as population size, birth rates or lifetime. The use of a CDL means that any missing data, including uncertain individual covariates, can be included in models without the need for customized likelihood functions. This approach also facilitates modeling processes of demographic interest rather than the complexities caused by non-ignorable missing data. We illustrate using two examples, (i) open population modeling in the presence of a censored time-varying individual covariate in a full robust design, and (ii) full open population multi-state modeling in the presence of a partially observed categorical variable. Supplemental materials for this article are available online.     

Estimation of detection probability in aerial surveys of antarctic pack-ice seals

We use line transect detection functions together with generalized linear and additive models to estimate detection probability when detection on the line (“g(0)”) may not be certain. The methods provide a flexible way of modeling detection probability for independent observer surveys, and for investigating the effects of explanatory variables. Analysis of data from an aerial survey of pack-ice seals produced g(0) estimates substantially below 1 for some observers (it varied from 0.80 to 0.98), demonstrated a fairly complex dependence of detection probability on covariates, and showed negative correlation between observers’ search width and their g(0). In addition to illustrating the utility of generalized additive models for capturing the effect of covariates on detection probability, the analysis suggests that detection functions may be sufficiently variable that use of g(0) correction factors obtained from other surveys would be inadvisable. We recommend that estimation of g(0) be considered for all aerial surveys; if g(0) is found to be very close to 1, estimation from subsequent surveys under the assumption that it is 1 may be reasonable, but without any estimation of g(0), the assumption that it is 1 is a matter of faith.     

Evaluating the performance of a block updating MCMC sampler in a simple genetic application

Markov chain Monte Carlo (MCMC) methods have provided an enormous break-through in the analysis of large complex problems such as those which frequently arise in genetic applications. The richness of the inference and the flexibility of an MCMC Bayesian approach in terms of design, data structure that can be analyzed, and models that can be posed, is indisputable. However, despite the strengths of the Bayesian approach, it is important to acknowledge that there are other, often easier, ways of tackling a problem. This is so, especially when simpler, qualitative answers are sought, such as presence or absence of a quantitative trait locus. We critically evaluate the behavior of a Bayesian aCMC block sampler for the detection of a quantitative trait locus by linkage with marker data, and compare it with a traditional least squares method. Some practical issues are illustrated by discussing the pros and cons of a Bayesian block updating sampling scheme versus the least squares method in the context of a simple genetic mapping problem. Depending on the focus of analysis, we show that the MCMC sampler does not always outperform the simpler approach from a frequentist perspective, and, more to the point, may not always perform appropriately in any particular replication.     

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.     

A Bayesian Approach to Fitting Gibbs Processes with Temporal Random Effects

We consider spatial point pattern data that have been observed repeatedly over a period of time in an inhomogeneous environment. Each spatial point pattern can be regarded as a ??snapshot?? of the underlying point process at a series of times. Thus, the number of points and corresponding locations of points differ for each snapshot. Each snapshot can be analyzed independently, but in many cases there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analyzing all snapshots within a single robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns can be regarded as independent replicates, given spatial covariates. We develop a joint mixed effects Gibbs point process model for the replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effects (or hierarchical) component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a data set of musk oxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.     

Spatial designs and properties of spatial correlation: Effects on covariance estimation

In a spatial regression context, scientists are often interested in a physical interpretation of components of the parametric covariance function. For example, spatial covariance parameter estimates in ecological settings have been interpreted to describe spatial heterogeneity or “patchiness” in a landscape that cannot be explained by measured covariates. In this article, we investigate the influence of the strength of spatial dependence on maximum likelihood (ML) and restricted maximum likelihood (REML) estimates of covariance parameters in an exponential-with-nugget model, and we also examine these influences under different sampling designs—specifically, lattice designs and more realistic random and cluster designs—at differing intensities of sampling (n=144 and 361). We find that neither ML nor REML estimates perform well when the range parameter and/or the nugget-to-sill ratio is large—ML tends to underestimate the autocorrelation function and REML produces highly variable estimates of the autocorrelation function. The best estimates of both the covariance parameters and the autocorrelation function come under the cluster sampling design and large sample sizes. As a motivating example, we consider a spatial model for stream sulfate concentration.     

Autologistic regression analysis of spatial-temporal binary data via Monte Carlo maximum likelihood

This article considers logistic regression analysis of binary data that are measured on a spatial lattice and repeatedly over discrete time points. We propose a spatial-temporal autologistic regression model and draw statistical inference via maximum likelihood. Due to an unknown normalizing constant in the likelihood function, we use Monte Carlo to obtain maximum likelihood estimates of the model parameters and predictive distributions at future time points. We also use path sampling to estimate the unknown normalizing constant and approximate an information criterion for model assessment. The methodology is illustrated by the analysis of a dataset of mountain pine beetle outbreaks in western Canada.     

Point-Based Mark-Recapture Distance Sampling

Avian surveys using point sampling for abundance estimation have either focused on distance sampling or more commonly mark-recapture to correct for detection bias. Combining mark-recapture and distance sampling (MRDS) has become an effective tool for line transects, but it has been largely ignored in point sampling literature. We describe MRDS and show that the previously published methods for point sampling are special cases. Using simulated data and golden-cheeked warbler (Dendroica chrysoparia) survey data from Texas, we demonstrate large differences in abundance estimates resulting from different independence assumptions. Data and code are provided in supplementary materials.     

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.