Bayesian factor models for multivariate categorical data obtained from questionnaires

Factor analysis is a flexible technique for assessment of multivariate dependence and codependence. Besides being an exploratory tool used to reduce the dimensionality of multivariate data, it allows estimation of common factors that often have an interesting theoretical interpretation in real problems. However, standard factor analysis is only applicable when the variables are scaled, which is often inappropriate, for example, in data obtained from questionnaires in the field of psychology, where the variables are often categorical. In this framework, we propose a factor model for the analysis of multivariate ordered and non-ordered polychotomous data. The inference procedure is done under the Bayesian approach via Markov chain Monte Carlo methods. Two Monte Carlo simulation studies are presented to investigate the performance of this approach in terms of estimation bias, precision and assessment of the number of factors. We also illustrate the proposed method to analyze participants'' responses to the Motivational State Questionnaire dataset, developed to study emotions in laboratory and field settings.     

A BETA-BINOMIAL MODEL FOR ESTIMATING THE SIZE OF A HETEROGENEOUS POPULATION

This paper compares the properties of various estimators for a beta‐binomial model for estimating the size of a heterogeneous population. It is found that maximum likelihood and conditional maximum likelihood estimators perform well for a large population with a large capture proportion. The jackknife and the sample coverage estimators are biased for low capture probabilities. The performance of the martingale estimator is satisfactory, but it requires full capture histories. The Gibbs sampler and Metropolis‐Hastings algorithm provide reasonable posterior estimates for informative priors.     

Sensitivity analysis of longitudinal count responses: a local influence approach and application to medical data

Longitudinal count responses are often analyzed with a Poisson mixed model. However, under overdispersion, these responses are better described by a negative binomial mixed model. Estimators of the corresponding parameters are usually obtained by the maximum likelihood method. To investigate the stability of these maximum likelihood estimators, we propose a methodology of sensitivity analysis using local influence. As count responses are discrete, we are unable to perturb them with the standard scheme used in local influence. Then, we consider an appropriate perturbation for the means of these responses. The proposed methodology is useful in different applications, but particularly when medical data are analyzed, because the removal of influential cases can change the statistical results and then the medical decision. We study the performance of the methodology by using Monte Carlo simulation and applied it to real medical data related to epilepsy and headache. All of these numerical studies show the good performance and potential of the proposed methodology.     

Equation-solving estimator based on Metropolis-Hastings algorithm with delayed rejection

Numerous works have recently attempted to develop more efficient estimators for MCMC inference than classical ones. In this perspective and approximate nonstandard discrete distributions, Liang and Liu proposed the equation solving estimator as an alternative to the conventional frequency estimator. The specific MCMC method used is the Metropolis-Hastings (M-H) algorithm. In this work, we propose to adapt the equation-solving estimator to the context of simulation using the Metropolis-Hastings algorithm with delayed rejection (MHDR). Developed originally by Mira, this algorithm is considered an improved version of the standard M-H sampler which aims to reduce the variance of MCMC estimators. An application to a Bayesian hypothesis test problem shows the superiority of the equation-solving estimator, based on MHDR sampling, over the one introduced by Liang and Liu.     

Analysis of generalized linear mixed models via a stochastic approximation algorithm with Markov chain Monte-Carlo method

In recent years much effort has been devoted to maximum likelihood estimation of generalized linear mixed models. Most of the existing methods use the EM algorithm, with various techniques in handling the intractable E-step. In this paper, a new implementation of a stochastic approximation algorithm with Markov chain Monte Carlo method is investigated. The proposed algorithm is computationally straightforward and its convergence is guaranteed. A simulation and three real data sets, including the challenging salamander data, are used to illustrate the procedure and to compare it with some existing methods. The results indicate that the proposed algorithm is an attractive alternative for problems with a large number of random effects or with high dimensional intractable integrals in the likelihood function.     

Bayesian inference for smooth transition autoregressive (STAR) model: A prior sensitivity analysis

The main aim of this paper is to perform sensitivity analysis to the specification of prior distributions in a Bayesian analysis setting of STAR models. To achieve this aim, the joint posterior distribution of model order, coefficient, and implicit parameters in the logistic STAR model is first being presented. The conditional posterior distributions are then shown, followed by the design of a posterior simulator using a combination of Metropolis-Hastings, Gibbs Sampler, RJMCMC, and Multiple Try Metropolis algorithms, respectively. Following this, simulation studies and a case study on the prior sensitivity for the implicit parameters are being detailed at the end.     

A Computational Bayesian Method for Generalized Semiparametric Regression Models

To better understand the power shift and the U.S. role compared to China and others regional actors, the Chicago Council on Global Affairs and the East Asia Institute (EAI) surveyed people in six countries - China, Japan, South Korea, Vietnam, Indonesian, and the United States - in the first half of 2008 about regional security and economic integration in Asia and about how these nations perceive each other (Bouton et al., 2010 Bouton, M., Steven, K., Benjamin, P., and Gregory, H. (2010). Soft power in Asia survey, 2008. ICPSR25342-v1. Ann Arbor, MI: Inter-university Consortium for Political and Social Research [distributor], 2010-04-05. doi:10.3886/ICPSR25342.v1[Crossref] , [Google Scholar]). There exists latent variance that cannot be adequately explained by parametric models. This is, in large part, due to the hidden structures and latent stories that from in unexpected ways. Therefore, a new Gibbs sampler is developed here in order to reveal preciously unseen structures and latent variances found in the survey dataset of Bouton et al. This new sampler is based upon the semiparametric regression, a well-known tool frequently utilized in order to capture the functional dependence between variables with fixed effect parametric and nonlinear regression. This is then extended to a generalized semiparametric regression for binary responses with logit and probit link function. The new sampler is then developed for the generalized linear mixed model with a nonparametric random effect. It is expressed as nonparametric regression with the multinomial-Dirichlet distribution for the number and positions of knots.     

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X₍₁₎, X₍₂₎, …, X_(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.     

Bayesian quantile regression for longitudinal data models

In this paper, we discuss a fully Bayesian quantile inference using Markov Chain Monte Carlo (MCMC) method for longitudinal data models with random effects. Under the assumption of error term subject to asymmetric Laplace distribution, we establish a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at τ-th level. We overcome the current computational limitations using two approaches. One is the general MCMC technique with Metropolis–Hastings algorithm and another is the Gibbs sampling from the full conditional distribution. These two methods outperform the traditional frequentist methods under a wide array of simulated data models and are flexible enough to easily accommodate changes in the number of random effects and in their assumed distribution. We apply the Gibbs sampling method to analyse a mouse growth data and some different conclusions from those in the literatures are obtained.     

A Bayesian Approach for Nonlinear Regression Models with Continuous Errors

Abstract

In this paper we develop a Bayesian analysis for the nonlinear regression model with errors that follow a continuous autoregressive process. In this way, unequally spaced observations do not present a problem in the analysis. We employ the Gibbs sampler, (see Gelfand, A., Smith, A. (1990 Gelfand, A. and Smith, A. 1990. Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc., 85: 398–409. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85:398–409.), as the foundation for making Bayesian inferences. We illustrate these Bayesian inferences with an analysis of a real data-set. Using these same data, we contrast the Bayesian approach with a generalized least squares technique.