A particular diffusion model for incomplete longitudinal data: application to the multicenter AIDS cohort study

Longitudinal studies, in which individuals are measured repeatedly in time, are often incomplete. We model continuous-time longitudinal data from the Multicenter AIDS Cohort Study using a diffusion model in which the diffusion parameters are functions of the covariates. These data are jointly modeled with the process of time-to-death due to AIDS. We show that, even for large data sets with a large number of missing variables, a Bayesian analysis is feasible using Gibbs sampling and compare a complete case analysis with a Bayesian treatment of missing values.     

Analysis of melanoma onset: assessing familial aggregation by using estimating equations and fitting variance components via Bayesian random effects models.

We investigate whether relative contributions of genetic and shared environmental factors are associated with an increased risk in melanoma. Data from the Queensland Familial Melanoma Project comprising 15,907 subjects arising from 1912 families were analyzed to estimate the additive genetic, common and unique environmental contributions to variation in the age at onset of melanoma. Two complementary approaches for analyzing correlated time-to-onset family data were considered: the generalized estimating equations (GEE) method in which one can estimate relationship-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modeled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov Chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the free ware package BUGS. In addition, we also used a Bayesian model to investigate the relative contribution of genetic and environmental effects on the expression of naevi and freckles, which are known risk factors for melanoma.     

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.     

Multivariate Bayesian analysis of Gaussian,right censored Gaussian,ordered categorical and binary traits using Gibbs sampling

A fully Bayesian analysis using Gibbs sampling and data augmentation in a multivariate model of Gaussian, right censored, and grouped Gaussian traits is described. The grouped Gaussian traits are either ordered categorical traits (with more than two categories) or binary traits, where the grouping is determined via thresholds on the underlying Gaussian scale, the liability scale. Allowances are made for unequal models, unknown covariance matrices and missing data. Having outlined the theory, strategies for implementation are reviewed. These include joint sampling of location parameters; efficient sampling from the fully conditional posterior distribution of augmented data, a multivariate truncated normal distribution; and sampling from the conditional inverse Wishart distribution, the fully conditional posterior distribution of the residual covariance matrix. Finally, a simulated dataset was analysed to illustrate the methodology. This paper concentrates on a model where residuals associated with liabilities of the binary traits are assumed to be independent. A Bayesian analysis using Gibbs sampling is outlined for the model where this assumption is relaxed.     

Bayesian analysis of wildlife age-at-harvest data

SUMMARY: State and federal natural resource management agencies often collect age-structured harvest data. These data represent finite realizations of stochastic demographic and sampling processes and have long been used by biologists to infer population trends. However, different sources of data have been combined in ad hoc ways and these methods usually failed to incorporate sampling error. In this article, we propose a "hidden process" (or state-space) model for estimating abundance, survival, recovery rate, and recruitment from age-at-harvest data that incorporate both demographic and sampling stochasticity. To this end, a likelihood for age-at-harvest data is developed by embedding a population dynamics model within a model for the sampling process. Under this framework, the identification of abundance parameters can be achieved by conducting a joint analysis with an auxiliary data set. We illustrate this approach by conducting a Bayesian analysis of age-at-harvest and mark-recovery data from black bears (Ursus americanus) in Pennsylvania. Using a set of reasonable prior distributions, we demonstrate a substantial increase in precision when posterior summaries of abundance are compared to a bias-corrected Lincoln-Petersen estimator. Because demographic processes link consecutive abundance estimates, we also obtain a more realistic biological picture of annual changes in abundance. Because age-at-harvest data are often readily obtained, we argue that this type of analysis provides a valuable strategy for wildlife population monitoring.     

Bayesian Linkage Analysis of Categorical Traits for Arbitrary Pedigree Designs

Background

Pedigree studies of complex heritable diseases often feature nominal or ordinal phenotypic measurements and missing genetic marker or phenotype data.

Methodology

We have developed a Bayesian method for Linkage analysis of Ordinal and Categorical traits (LOCate) that can analyze complex genealogical structure for family groups and incorporate missing data. LOCate uses a Gibbs sampling approach to assess linkage, incorporating a simulated tempering algorithm for fast mixing. While our treatment is Bayesian, we develop a LOD (log of odds) score estimator for assessing linkage from Gibbs sampling that is highly accurate for simulated data. LOCate is applicable to linkage analysis for ordinal or nominal traits, a versatility which we demonstrate by analyzing simulated data with a nominal trait, on which LOCate outperforms LOT, an existing method which is designed for ordinal traits. We additionally demonstrate our method''s versatility by analyzing a candidate locus (D2S1788) for panic disorder in humans, in a dataset with a large amount of missing data, which LOT was unable to handle.

Conclusion

LOCate''s accuracy and applicability to both ordinal and nominal traits will prove useful to researchers interested in mapping loci for categorical traits.     

Analysis of rabbit doe longevity using a semiparametric log-Normal animal frailty model with time-dependent covariates

Data on doe longevity in a rabbit population were analysed using a semiparametric log-Normal animal frailty model. Longevity was defined as the time from the first positive pregnancy test to death or culling due to pathological problems. Does culled for other reasons had right censored records of longevity. The model included time dependent covariates associated with year by season, the interaction between physiological state and the number of young born alive, and between order of positive pregnancy test and physiological state. The model also included an additive genetic effect and a residual in log frailty. Properties of marginal posterior distributions of specific parameters were inferred from a full Bayesian analysis using Gibbs sampling. All of the fully conditional posterior distributions defining a Gibbs sampler were easy to sample from, either directly or using adaptive rejection sampling. The marginal posterior mean estimates of the additive genetic variance and of the residual variance in log frailty were 0.247 and 0.690.     

QUANTITATIVE GENETIC MODELING AND INFERENCE IN THE PRESENCE OF NONIGNORABLE MISSING DATA

Natural selection is typically exerted at some specific life stages. If natural selection takes place before a trait can be measured, using conventional models can cause wrong inference about population parameters. When the missing data process relates to the trait of interest, a valid inference requires explicit modeling of the missing process. We propose a joint modeling approach, a shared parameter model, to account for nonrandom missing data. It consists of an animal model for the phenotypic data and a logistic model for the missing process, linked by the additive genetic effects. A Bayesian approach is taken and inference is made using integrated nested Laplace approximations. From a simulation study we find that wrongly assuming that missing data are missing at random can result in severely biased estimates of additive genetic variance. Using real data from a wild population of Swiss barn owls Tyto alba, our model indicates that the missing individuals would display large black spots; and we conclude that genes affecting this trait are already under selection before it is expressed. Our model is a tool to correctly estimate the magnitude of both natural selection and additive genetic variance.     

Monte Carlo EM for missing covariates in parametric regression models

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.     

Bayesian analysis for generalized linear models with nonignorably missing covariates

We propose Bayesian methods for estimating parameters in generalized linear models (GLMs) with nonignorably missing covariate data. We show that when improper uniform priors are used for the regression coefficients, phi, of the multinomial selection model for the missing data mechanism, the resulting joint posterior will always be improper if (i) all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism, or (ii) at least one of the covariates is continuous and unbounded. This impropriety will result regardless of whether proper or improper priors are specified for the regression parameters, beta, of the GLM or the parameters, alpha, of the covariate distribution. To overcome this problem, we propose a novel class of proper priors for the regression coefficients, phi, in the selection model for the missing data mechanism. These priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a Gibbs sampling scheme that leads to accelerated convergence. In addition, we extend the model assessment criterion of Chen, Dey, and Ibrahim (2004a, Biometrika 91, 45-63), called the weighted L measure, to GLMs and missing data problems as well as extend the deviance information criterion (DIC) of Spiegelhalter et al. (2002, Journal of the Royal Statistical Society B 64, 583-639) for assessing whether the missing data mechanism is ignorable or nonignorable. A novel Markov chain Monte Carlo sampling algorithm is also developed for carrying out posterior computation. Several simulations are given to investigate the performance of the proposed Bayesian criteria as well as the sensitivity of the prior specification. Real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods.     

Hierarchical Bayes estimation of hunting success rates with spatial correlations

A Bayesian hierarchical generalized linear model is used to estimate hunting success rates at the subarea level for postseason harvest surveys. The model includes fixed week effects, random geographic effects, and spatial correlations between neighboring subareas. The computation is done by Gibbs sampling and adaptive rejection sampling techniques. The method is illustrated using data from the Missouri Turkey Hunting Survey in the spring of 1996. Bayesian model selection methods are used to demonstrate that there are significant week differences and spatial correlations of hunting success rates among counties. The Bayesian estimates are also shown to be quite robust in terms of changes of hyperparameters.     

含不可忽略缺失数据非线性再生散度模型参数的Bayes估计

给出协变量带有不可忽略缺失数据的非线性再生散度模型的Bayes方法,缺失数据机制由Logistic回归模型来确定.Gibbs抽样技术和Metropolis-Hastings算法(简称MH算法)用来得到模型参数、缺失数据机制中回归系数的联合Bayes估计,并用实例加以说明.     

Bayesian estimation in animal breeding using the Dirichlet process prior for correlated random effects

In the case of the mixed linear model the random effects are usually assumed to be normally distributed in both the Bayesian and classical frameworks. In this paper, the Dirichlet process prior was used to provide nonparametric Bayesian estimates for correlated random effects. This goal was achieved by providing a Gibbs sampler algorithm that allows these correlated random effects to have a nonparametric prior distribution. A sampling based method is illustrated. This method which is employed by transforming the genetic covariance matrix to an identity matrix so that the random effects are uncorrelated, is an extension of the theory and the results of previous researchers. Also by using Gibbs sampling and data augmentation a simulation procedure was derived for estimating the precision parameter M associated with the Dirichlet process prior. All needed conditional posterior distributions are given. To illustrate the application, data from the Elsenburg Dormer sheep stud were analysed. A total of 3325 weaning weight records from the progeny of 101 sires were used.     

Maximum likelihood estimation in random effects cure rate models with nonignorable missing covariates

We introduce a method of parameter estimation for a random effects cure rate model. We also propose a methodology that allows us to account for nonignorable missing covariates in this class of models. The proposed method corrects for possible bias introduced by complete case analysis when missing data are not missing completely at random and is motivated by data from a pair of melanoma studies conducted by the Eastern Cooperative Oncology Group in which clustering by cohort or time of study entry was suspected. In addition, these models allow estimation of cure rates, which is desirable when we do not wish to assume that all subjects remain at risk of death or relapse from disease after sufficient follow-up. We develop an EM algorithm for the model and provide an efficient Gibbs sampling scheme for carrying out the E-step of the algorithm.     

Open population maximum likelihood spatial capture‐recapture

Open population capture‐recapture models are widely used to estimate population demographics and abundance over time. Bayesian methods exist to incorporate open population modeling with spatial capture‐recapture (SCR), allowing for estimation of the effective area sampled and population density. Here, open population SCR is formulated as a hidden Markov model (HMM), allowing inference by maximum likelihood for both Cormack‐Jolly‐Seber and Jolly‐Seber models, with and without activity center movement. The method is applied to a 12‐year survey of male jaguars (Panthera onca) in the Cockscomb Basin Wildlife Sanctuary, Belize, to estimate survival probability and population abundance over time. For this application, inference is shown to be biased when assuming activity centers are fixed over time, while including a model for activity center movement provides negligible bias and nominal confidence interval coverage, as demonstrated by a simulation study. The HMM approach is compared with Bayesian data augmentation and closed population models for this application. The method is substantially more computationally efficient than the Bayesian approach and provides a lower root‐mean‐square error in predicting population density compared to closed population models.     

A Bayesian approach for subgroup analysis

Several penalization approaches have been developed to identify homogeneous subgroups based on a regression model with subject-specific intercepts in subgroup analysis. These methods often apply concave penalty functions to pairwise comparisons of the intercepts, such that the subjects with similar intercept values are assigned to the same group, which is very similar to the procedure of the penalization approaches for variable selection. Since the Bayesian methods are commonly used in variable selection, it is worth considering the corresponding approaches to subgroup analysis in the Bayesian framework. In this paper, a Bayesian hierarchical model with appropriate prior structures is developed for the pairwise differences of intercepts based on a regression model with subject-specific intercepts, which can automatically detect and identify homogeneous subgroups. A Gibbs sampling algorithm is also provided to select the hyperparameter and estimate the intercepts and coefficients of the covariates simultaneously, which is computationally efficient for pairwise comparisons compared to the time-consuming procedures for parameter estimation of the penalization methods (e.g., alternating direction method of multiplier) in the case of large sample sizes. The effectiveness and usefulness of the proposed Bayesian method are evaluated through simulation studies and analysis of a Cleveland Heart Disease Dataset.     

Inferences about population dynamics from count data using multistate models: a comparison to capture–recapture approaches

Wildlife populations consist of individuals that contribute disproportionately to growth and viability. Understanding a population's spatial and temporal dynamics requires estimates of abundance and demographic rates that account for this heterogeneity. Estimating these quantities can be difficult, requiring years of intensive data collection. Often, this is accomplished through the capture and recapture of individual animals, which is generally only feasible at a limited number of locations. In contrast, N‐mixture models allow for the estimation of abundance, and spatial variation in abundance, from count data alone. We extend recently developed multistate, open population N‐mixture models, which can additionally estimate demographic rates based on an organism's life history characteristics. In our extension, we develop an approach to account for the case where not all individuals can be assigned to a state during sampling. Using only state‐specific count data, we show how our model can be used to estimate local population abundance, as well as density‐dependent recruitment rates and state‐specific survival. We apply our model to a population of black‐throated blue warblers (Setophaga caerulescens) that have been surveyed for 25 years on their breeding grounds at the Hubbard Brook Experimental Forest in New Hampshire, USA. The intensive data collection efforts allow us to compare our estimates to estimates derived from capture–recapture data. Our model performed well in estimating population abundance and density‐dependent rates of annual recruitment/immigration. Estimates of local carrying capacity and per capita recruitment of yearlings were consistent with those published in other studies. However, our model moderately underestimated annual survival probability of yearling and adult females and severely underestimates survival probabilities for both of these male stages. The most accurate and precise estimates will necessarily require some amount of intensive data collection efforts (such as capture–recapture). Integrated population models that combine data from both intensive and extensive sources are likely to be the most efficient approach for estimating demographic rates at large spatial and temporal scales.     

Integrated Population Modeling for White-Tailed Deer in Saskatchewan,Canada

Monitoring annual change and long-term trends in population structure and abundance of white-tailed deer (Odocoileus virginianus) is an important but challenging component of their management. Many monitoring programs consist of count-based indices of relative abundance along with a variety of population structure information. Analyzed separately these data can be difficult to interpret because of observation error in the data collection process, missing data, and the lack of an explicit biological model to connect the data streams while accounting for their relative imprecision. We used a Bayesian age-structured integrated population model to integrate data from a fall spotlight survey that produced a count-based index of relative abundance and a volunteer staff and citizen classification survey that generated a fall recruitment index. Both surveys took place from 2003–2018 in the parkland ecoregion of southeast Saskatchewan, Canada. Our approach modeled demographic processes for age-specific (0.5-, 1.5-, ≥2.5-year-old classes) populations and was fit to count and recruitment data via models that allowed for error in the respective observation processes. The Bayesian framework accommodated missing data and allowed aggregation of transects to act as samples from the larger management unit population. The approach provides managers with continuous time series of estimated relative abundance, recruitment rates, and apparent survival rates with full propagation of uncertainty and sharing of information among transects. We used this model to demonstrate winter severity effects on recruitment rates via an interaction between winter snow depth and minimum temperatures. In years with colder than average temperatures and above average snow depth, recruitment was depressed, whereas the negative effect of snow depth reversed in years with above average temperatures. This and other covariate information can be incorporated into the model to test relationships and provide predictions of future population change prior to setting of hunting seasons. Likewise, post hoc analysis of model output allows other hypothesis tests, such as determining the statistical support for whether population status has crossed a management trigger threshold. © 2020 The Wildlife Society.     

Bayesian inference of inbreeding depression in controlled crosses

Abstract. This study shows how a Gibbs sampling approach can be used for Bayesian inference of inbreeding depression. The method presented is mainly concerned with organisms that can be both selfed and outcrossed. Tests performed on simulated data with unequal variances and missing observations show that the method works well. Real data from the plant Scabiosa canescens is also analyzed.     

Bayesian estimation of movement and survival probabilities from capture-recapture data

The Arnason–Schwarz model is usually used for estimatingsurvival and movement probabilities of animal populations fromcapture-recapture data. The missing data structure of this capture-recapturemodel is exhibited and summarised via a directed graph representation.Taking advantage of this structure we implement a Gibbs samplingalgorithm from which Bayesian estimates and credible intervalsfor survival and movement probabilities are derived. Convergenceof the algorithm is proved using a duality principle. We illustrateour approach through a real example.