Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

Bayesian hierarchical models for the prediction of volleyball results

Statistical modelling of sports data has become more and more popular in the recent years and different types of models have been proposed to achieve a variety of objectives: from identifying the key characteristics which lead a team to win or lose to predicting the outcome of a game or the team rankings in national leagues. Although not as popular as football or basketball, volleyball is a team sport with both national and international level competitions in almost every country. However, there is almost no study investigating the prediction of volleyball game outcomes and team rankings in national leagues. We propose a Bayesian hierarchical model for the prediction of the rankings of volleyball national teams, which also allows to estimate the results of each match in the league. We consider two alternative model specifications of different complexity which are validated using data from the women''s volleyball Italian Serie A1 2017–2018 season.     

Bayesian hierarchical regression models for detecting QTLs in plant experiments

Quantitative trait loci (QTL) mapping is a growing field in statistical genetics. In plants, QTL detection experiments often feature replicates or clones within a specific genetic line. In this work, a Bayesian hierarchical regression model is applied to simulated QTL data and to a dataset from the Arabidopsis thaliana plants for locating the QTL mapping associated with cotyledon opening. A conditional model search strategy based on Bayesian model averaging is utilized to reduce the computational burden.     

A general class of hierarchical ordinal regression models with applications to correlated roc analysis

The authors discuss a general class of hierarchical ordinal regression models that includes both location and scale parameters, allows link functions to be selected adaptively as finite mixtures of normal cumulative distribution functions, and incorporates flexible correlation structures for the latent scale variables. Exploiting the well‐known correspondence between ordinal regression models and parametric ROC (Receiver Operating Characteristic) curves makes it possible to use a hierarchical ROC (HROC) analysis to study multilevel clustered data in diagnostic imaging studies. The authors present a Bayesian approach to model fitting using Markov chain Monte Carlo methods and discuss HROC applications to the analysis of data from two diagnostic radiology studies involving multiple interpreters.     

An iterative Monte Carlo method for nonconjugate Bayesian analysis

The Gibbs sampler has been proposed as a general method for Bayesian calculation in Gelfand and Smith (1990). However, the predominance of experience to date resides in applications assuming conjugacy where implementation is reasonably straightforward. This paper describes a tailored approximate rejection method approach for implementation of the Gibbs sampler when nonconjugate structure is present. Several challenging applications are presented for illustration.     

Bayesian modelling of the mean and covariance matrix in normal nonlinear models

An important problem in statistics is the study of longitudinal data taking into account the effect of other explanatory variables such as treatments and time. In this paper, a new Bayesian approach for analysing longitudinal data is proposed. This innovative approach takes into account the possibility of having nonlinear regression structures on the mean and linear regression structures on the variance–covariance matrix of normal observations, and it is based on the modelling strategy suggested by Pourahmadi [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterizations, Biometrika, 87 (1999), pp. 667–690.]. We initially extend the classical methodology to accommodate the fitting of nonlinear mean models then we propose our Bayesian approach based on a generalization of the Metropolis–Hastings algorithm of Cepeda [E.C. Cepeda, Variability modeling in generalized linear models, Unpublished Ph.D. Thesis, Mathematics Institute, Universidade Federal do Rio de Janeiro, 2001]. Finally, we illustrate the proposed methodology by analysing one example, the cattle data set, that is used to study cattle growth.     

Statistical profiling methods with hierarchical logistic regression for healthcare providers with binary outcomes

Within the context of California's public report of coronary artery bypass graft (CABG) surgery outcomes, we first thoroughly review popular statistical methods for profiling healthcare providers. Extensive simulation studies are then conducted to compare profiling schemes based on hierarchical logistic regression (LR) modeling under various conditions. Both Bayesian and frequentist's methods are evaluated in classifying hospitals into ‘better’, ‘normal’ or ‘worse’ service providers. The simulation results suggest that no single method would dominate others on all accounts. Traditional schemes based on LR tend to identify too many false outliers, while those based on hierarchical modeling are relatively conservative. The issue of over shrinkage in hierarchical modeling is also investigated using the 2005–2006 California CABG data set. The article provides theoretical and empirical evidence in choosing the right methodology for provider profiling.     

Stochastic models for greenhouse gas emission rate estimation from hydroelectric reservoirs: a Bayesian hierarchical approach

Herein, we propose a fully Bayesian approach to the greenhouse gas emission problem. The goal of this work is to estimate the emission rate of polluting gases from the area flooded by hydroelectric reservoirs. We present models for gas concentration evolution in two ways: first, by proposing them from ordinary differential equation solutions and, second, by using stochastic differential equations with a discretization scheme. Finally, we present techniques to estimate the emission rate for the entire reservoir. In order to carry out the inference, we use the Bayesian framework with Monte Carlo via Markov Chain methods. Discretization schemes over continuous differential equations are used when necessary. These models applied to greenhouse gas emission and Bayesian inference for this purpose are completely new in statistical literature, as far as we know, and contribute to estimate the amount of polluting gases released from hydroelectric reservoirs in Brazil. The proposed models are applied in a real data set and results are presented.     

Bayesian hierarchical model for the prediction of football results

The problem of modelling football data has become increasingly popular in the last few years and many different models have been proposed with the aim of estimating the characteristics that bring a team to lose or win a game, or to predict the score of a particular match. We propose a Bayesian hierarchical model to fulfil both these aims and test its predictive strength based on data about the Italian Serie A 1991–1992 championship. To overcome the issue of overshrinkage produced by the Bayesian hierarchical model, we specify a more complex mixture model that results in a better fit to the observed data. We test its performance using an example of the Italian Serie A 2007–2008 championship.     

Bayesian factor models in characterizing molecular adaptation

Assessing the selective influence of amino acid properties is important in understanding evolution at the molecular level. A collection of methods and models has been developed in recent years to determine if amino acid sites in a given DNA sequence alignment display substitutions that are altering or conserving a prespecified set of amino acid properties. Residues showing an elevated number of substitutions that favorably alter a physicochemical property are considered targets of positive natural selection. Such approaches usually perform independent analyses for each amino acid property under consideration, without taking into account the fact that some of the properties may be highly correlated. We propose a Bayesian hierarchical regression model with latent factor structure that allows us to determine which sites display substitutions that conserve or radically change a set of amino acid properties, while accounting for the correlation structure that may be present across such properties. We illustrate our approach by analyzing simulated data sets and an alignment of lysin sperm DNA.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

Predictive comparisons in ordinal models

Bayesian predictive probability function approximations are derived and compared for ordinal logistic regression models. Classification and variable selection problems are also discussed. The methods are illustrated on a large data set of head injury patients.     

Parallel Markov chain Monte Carlo for Bayesian hierarchical models with big data,in two stages

Due to the escalating growth of big data sets in recent years, new Bayesian Markov chain Monte Carlo (MCMC) parallel computing methods have been developed. These methods partition large data sets by observations into subsets. However, for Bayesian nested hierarchical models, typically only a few parameters are common for the full data set, with most parameters being group specific. Thus, parallel Bayesian MCMC methods that take into account the structure of the model and split the full data set by groups rather than by observations are a more natural approach for analysis. Here, we adapt and extend a recently introduced two-stage Bayesian hierarchical modeling approach, and we partition complete data sets by groups. In stage 1, the group-specific parameters are estimated independently in parallel. The stage 1 posteriors are used as proposal distributions in stage 2, where the target distribution is the full model. Using three-level and four-level models, we show in both simulation and real data studies that results of our method agree closely with the full data analysis, with greatly increased MCMC efficiency and greatly reduced computation times. The advantages of our method versus existing parallel MCMC computing methods are also described.     

Semiparametric Bayesian hierarchical models for heterogeneous population in nonlinear mixed effect model: application to gastric emptying studies

Gastric emptying studies are frequently used in medical research, both human and animal, when evaluating the effectiveness and determining the unintended side-effects of new and existing medications, diets, and procedures or interventions. It is essential that gastric emptying data be appropriately summarized before making comparisons between study groups of interest and to allow study the comparisons. Since gastric emptying data have a nonlinear emptying curve and are longitudinal data, nonlinear mixed effect (NLME) models can accommodate both the variation among measurements within individuals and the individual-to-individual variation. However, the NLME model requires strong assumptions that are often not satisfied in real applications that involve a relatively small number of subjects, have heterogeneous measurement errors, or have large variation among subjects. Therefore, we propose three semiparametric Bayesian NLMEs constructed with Dirichlet process priors, which automatically cluster sub-populations and estimate heterogeneous measurement errors. To compare three semiparametric models with the parametric model we propose a penalized posterior Bayes factor. We compare the performance of our semiparametric hierarchical Bayesian approaches with that of the parametric Bayesian hierarchical approach. Simulation results suggest that our semiparametric approaches are more robust and flexible. Our gastric emptying studies from equine medicine are used to demonstrate the advantage of our approaches.     

A Bayesian hierarchical model for categorical longitudinal data from a social survey of immigrants

Summary.  The paper investigates a Bayesian hierarchical model for the analysis of categorical longitudinal data from a large social survey of immigrants to Australia. Data for each subject are observed on three separate occasions, or waves, of the survey. One of the features of the data set is that observations for some variables are missing for at least one wave. A model for the employment status of immigrants is developed by introducing, at the first stage of a hierarchical model, a multinomial model for the response and then subsequent terms are introduced to explain wave and subject effects. To estimate the model, we use the Gibbs sampler, which allows missing data for both the response and the explanatory variables to be imputed at each iteration of the algorithm, given some appropriate prior distributions. After accounting for significant covariate effects in the model, results show that the relative probability of remaining unemployed diminished with time following arrival in Australia.     

Large-scale Bayesian spatial modelling of air pollution for policy support

The potential effects of air pollution are a major concern both in terms of the environment and in relation to human health. In order to support environmental policy, there is a need for accurate measurements of the concentrations of pollutants at high geographical resolution over large regions. However, within such regions, there are likely to be areas where the monitoring information will be sparse and so methods are required to accurately predict concentrations. Set within a Bayesian framework, models are developed which exploit the relationships between pollution and geographical covariate information, such as land use, climate and transport variables together with spatial structure. Candidate models are compared based on their ability to predict a set of validation sites. The chosen model is used to perform large-scale prediction of nitrogen dioxide at a 1×1 km resolution for the entire EU. The models allow probabilistic statements to be made with regard to the levels of air pollution that might be experienced in each area. When combined with population data, such information can be invaluable in informing policy by indicating areas for which improvements may be given priority.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

A Bayesian test of homogeneity of association parameter using transition modelling of longitudinal mixed responses

In this paper, a Bayesian framework using a joint transition model for analysing longitudinal mixed ordinal and continuous responses is considered. The joint model considers a multivariate mixed model for the responses in which a transitive cumulative logistic regression model and an autoregressive regression model are used to model ordinal and continuous responses, respectively. Also, to take into account the association between longitudinal ordinal and continuous responses, a dynamic association parameter is used. A test is conducted to see whether this parameter is time-invariant and another test is presented to see whether this parameter is equal to zero or significantly far from zero. Our approach is applied to longitudinal PIAT (Peabody Individual Achievement Test) data where the Bayesian estimates of parameters are obtained.     

Robustness of the posterior mean in normal hierarchical models

We consider the problem of robustness in hierarchical Bayes models. Let X = (X₁,X₂, … ,X_p)^τ be a random vector, the X₁ being independently distributed as N(θ₁,σ²) random variables (σ² known), while the θ₁ are thought to be exchangeable, modelled as i.i.d, N(μ,τ²). The hyperparameter µ is given a noninformative prior distribution π(μ) = 1 and τ² is assumed to be independent of µ having a distribution g(τ²) lying in a certain class of distributions g. For several g's, including e-contaminations classes and density ratio classes we determine the range of the posterior mean of θ₁ as g ranges over g.