A Bayesian multivariate partially linear single-index probit model for ordinal responses

Combining the multivariate probit models with the multivariate partially linear single-index models, we propose new semiparametric latent variable models for multivariate ordinal response data. Based on the reversible jump Markov chain Monte Carlo technique, we develop a fully Bayesian method with free-knot splines to analyse the proposed models. To address the problem that the ordinary Gibbs sampler usually converges slowly, we make use of the partial-collapse and parameter-expansion techniques in our algorithm. The proposed methodology are demonstrated by simulated and real data examples.     

Latent variable models with ordinal categorical covariates

We propose a general latent variable model for multivariate ordinal categorical variables, in which both the responses and the covariates are ordinal, to assess the effect of the covariates on the responses and to model the covariance structure of the response variables. A?fully Bayesian approach is employed to analyze the model. The Gibbs sampler is used to simulate the joint posterior distribution of the latent variables and the parameters, and the parameter expansion and reparameterization techniques are used to speed up the convergence procedure. The proposed model and method are demonstrated by simulation studies and a real data example.     

Univariate and multirater ordinal cumulative link regression with covariate specific cutpoints

The author considers a reparameterized version of the Bayesian ordinal cumulative link regression model as a tool for exploring relationships between covariates and “cutpoint” parameters. The use of this parameterization allows one to fit models using the leapfrog hybrid Monte Carlo method, and to bypass latent variable data augmentation and the slow convergence of the cutpoints which it usually entails. The proposed Gibbs sampler is not model specific and can be easily modified to handle different link functions. The approach is illustrated by considering data from a pediatric radiology study.     

A Bayesian conditional model for bivariate mixed ordinal and skew continuous longitudinal responses using quantile regression

In this paper, we develop a conditional model for analyzing mixed bivariate continuous and ordinal longitudinal responses. We propose a quantile regression model with random effects for analyzing continuous responses. For this purpose, an Asymmetric Laplace Distribution (ALD) is allocated for continuous response given random effects. For modeling ordinal responses, a cumulative logit model is used, via specifying a latent variable model, with considering other random effects. Therefore, the intra-association between continuous and ordinal responses is taken into account using their own exclusive random effects. But, the inter-association between two mixed responses is taken into account by adding a continuous response term in the ordinal model. We use a Bayesian approach via Markov chain Monte Carlo method for analyzing the proposed conditional model and to estimate unknown parameters, a Gibbs sampler algorithm is used. Moreover, we illustrate an application of the proposed model using a part of the British Household Panel Survey data set. The results of data analysis show that gender, age, marital status, educational level and the amount of money spent on leisure have significant effects on annual income. Also, the associated parameter is significant in using the best fitting proposed conditional model, thus it should be employed rather than analyzing separate models.     

Modeling a mixture of ordinal and continuous repeated measures

We study the correlation structure for a mixture of ordinal and continuous repeated measures using a Bayesian approach. We assume a multivariate probit model for the ordinal variables and a normal linear regression for the continuous variables, where latent normal variables underlying the ordinal data are correlated with continuous variables in the model. Due to the probit model assumption, we are required to sample a covariance matrix with some of the diagonal elements equal to one. The key computational idea is to use parameter-extended data augmentation, which involves applying the Metropolis-Hastings algorithm to get a sample from the posterior distribution of the covariance matrix incorporating the relevant restrictions. The methodology is illustrated through a simulated example and through an application to data from the UCLA Brain Injury Research Center.     

Modeling individual migraine severity with autoregressive ordered probit models

This paper considers the problem of modeling migraine severity assessments and their dependence on weather and time characteristics. We take on the viewpoint of a patient who is interested in an individual migraine management strategy. Since factors influencing migraine can differ between patients in number and magnitude, we show how a patient’s headache calendar reporting the severity measurements on an ordinal scale can be used to determine the dominating factors for this special patient. One also has to account for dependencies among the measurements. For this the autoregressive ordinal probit (AOP) model of Müller and Czado (J Comput Graph Stat 14: 320–338, 2005) is utilized and fitted to a single patient’s migraine data by a grouped move multigrid Monte Carlo (GM-MGMC) Gibbs sampler. Initially, covariates are selected using proportional odds models. Model fit and model comparison are discussed. A comparison with proportional odds specifications shows that the AOP models are preferred.     

Likelihood-free approximate Gibbs sampling

Likelihood-free methods such as approximate Bayesian computation (ABC) have extended the reach of statistical inference to problems with computationally intractable likelihoods. Such approaches perform well for small-to-moderate dimensional problems, but suffer a curse of dimensionality in the number of model parameters. We introduce a likelihood-free approximate Gibbs sampler that naturally circumvents the dimensionality issue by focusing on lower-dimensional conditional distributions. These distributions are estimated by flexible regression models either before the sampler is run, or adaptively during sampler implementation. As a result, and in comparison to Metropolis-Hastings-based approaches, we are able to fit substantially more challenging statistical models than would otherwise be possible. We demonstrate the sampler’s performance via two simulated examples, and a real analysis of Airbnb rental prices using a intractable high-dimensional multivariate nonlinear state-space model with a 36-dimensional latent state observed on 365 time points, which presents a real challenge to standard ABC techniques.     

Conditional Heteroscedasticity in Qualitative Response Models of Time Series: A Gibbs-Sampling Approach to the Bank Prime Rate

Previous time series applications of qualitative response models have ignored features of the data, such as conditional heteroscedasticity, that are routinely addressed in time series econometrics of financial data. This article addresses this issue by adding Markov-switching heteroscedasticity to a dynamic ordered probit model of discrete changes in the bank prime lending rate and estimating via the Gibbs sampler. The dynamic ordered probit model of Eichengreen, Watson, and Grossman allows for serial autocorrelation in probit analysis of a time series, and this article demonstrates the relative simplicity of estimating a dynamic ordered probit using the Gibbs sampler instead of the Eichengreen et al. maximum likelihood procedure. In addition, the extension to regime-switching parameters and conditional heteroscedasticity is easy to implement under Gibbs sampling. The article compares tests of goodness of fit between dynamic ordered probit models of the prime rate that have constant variance and conditional heteroscedasticity.     

Reparameterizing the generalized linear model to accelerate gibbs sampler convergence

Albert and Chib introduced a complete Bayesian method to analyze data arising from the generalized linear model in which they used the Gibbs sampling algorithm facilitated by latent variables. Recently, Cowles proposed an alternative algorithm to accelerate the convergence of the Albert-Chib algorithm. The novelty in this latter algorithm is achieved by using a Hastings algorithm to generate latent variables and bin boundary parameters jointly instead of individually from their respective full conditionals. In the same spirit, we reparameterize the cumulative-link generalized linear model to accelerate the convergence of Cowles’ algorithm even further. One important advantage of our method is that for the three-bin problem it does not require the Hastings algorithm. In addition, for problems with more than three bins, while the Hastings algorithm is required, we provide a proposal density based on the Dirichlet distribution which is more natural than the truncated normal density used in the competing algorithm. Also, using diagnostic procedures recommended in the literature for the Markov chain Monte Carlo algorithm (both single and multiple runs) we show that our algorithm is substantially better than the one recently obtained. Precisely, our algorithm provides faster convergence and smaller autocorrelations between the iterates. Using the probit link function, extensive results are obtained for the three-bin and the five-bin multinomial ordinal data problems.     

Model-based clustering of Gaussian copulas for mixed data

Clustering of mixed data is important yet challenging due to a shortage of conventional distributions for such data. In this article, we propose a mixture model of Gaussian copulas for clustering mixed data. Indeed copulas, and Gaussian copulas in particular, are powerful tools for easily modeling the distribution of multivariate variables. This model clusters data sets with continuous, integer, and ordinal variables (all having a cumulative distribution function) by considering the intra-component dependencies in a similar way to the Gaussian mixture. Indeed, each component of the Gaussian copula mixture produces a correlation coefficient for each pair of variables and its univariate margins follow standard distributions (Gaussian, Poisson, and ordered multinomial) depending on the nature of the variable (continuous, integer, or ordinal). As an interesting by-product, this model generalizes many well-known approaches and provides tools for visualization based on its parameters. The Bayesian inference is achieved with a Metropolis-within-Gibbs sampler. The numerical experiments, on simulated and real data, illustrate the benefits of the proposed model: flexible and meaningful parameterization combined with visualization features.     

Bayesian spatial prediction of skew and censored data via a hybrid algorithm

A correct detection of areas with excess of pollution relies first on accurate predictions of pollutant concentrations, a task that is usually complicated by skewed histograms and the presence of censored data. The unified skew-Gaussian (SUG) random field proposed by Zareifard and Jafari Khaledi [19] offers a more flexible class of sampling spatial models to account for skewness. In this paper, we adopt a Bayesian framework to perform prediction for the SUG model in the presence of censored data. Owing to the presence of many latent variables with strongly dependent components in the model, we encounter convergence issues when using Monte Carlo Markov Chain algorithms. To overcome this obstacle, we use a computationally efficient inverse Bayes formulas sampling procedure to obtain approximately independent samples from the posterior distribution of latent variables. Then they are applied to update parameters in a Gibbs sampler scheme. This hybrid algorithm provides effective samples, resulting in some computational advantages and precise predictions. The proposed approach is illustrated with a simulation study and applied to a spatial data set which contains right censored data.     

A Bayesian analysis of directional data using the projected normal distribution

This paper presents a Bayesian analysis of the projected normal distribution, which is a flexible and useful distribution for the analysis of directional data. We obtain samples from the posterior distribution using the Gibbs sampler after the introduction of suitably chosen latent variables. The procedure is illustrated using simulated data as well as a real data set previously analysed in the literature.     

General location multivariate latent variable models for mixed correlated bounded continuous,ordinal, and nominal responses with non-ignorable missing data

Using a multivariate latent variable approach, this article proposes some new general models to analyze the correlated bounded continuous and categorical (nominal or/and ordinal) responses with and without non-ignorable missing values. First, we discuss regression methods for jointly analyzing continuous, nominal, and ordinal responses that we motivated by analyzing data from studies of toxicity development. Second, using the beta and Dirichlet distributions, we extend the models so that some bounded continuous responses are replaced for continuous responses. The joint distribution of the bounded continuous, nominal and ordinal variables is decomposed into a marginal multinomial distribution for the nominal variable and a conditional multivariate joint distribution for the bounded continuous and ordinal variables given the nominal variable. We estimate the regression parameters under the new general location models using the maximum-likelihood method. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms of the model on the maximal normal curvature. The proposed models are applied to two data sets: BMI, Steatosis and Osteoporosis data and Tehran household expenditure budgets.     

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.     

Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables

We demonstrate the use of auxiliary (or latent) variables for sampling non-standard densities which arise in the context of the Bayesian analysis of non-conjugate and hierarchical models by using a Gibbs sampler. Their strategic use can result in a Gibbs sampler having easily sampled full conditionals. We propose such a procedure to simplify or speed up the Markov chain Monte Carlo algorithm. The strength of this approach lies in its generality and its ease of implementation. The aim of the paper, therefore, is to provide an alternative sampling algorithm to rejection-based methods and other sampling approaches such as the Metropolis–Hastings algorithm.     

Bayesian estimation in the multidimensional three-parameter logistic model

The Gibbs sampler has a great potential to be an efficient and versatile estimation procedure in item response theory. In this article, based on a data augmentation scheme using the Gibbs sampler, we propose a Bayesian procedure to estimate the multidimensional three-parameter logistic model. With the introduction of the two latent variables, the full conditional distributions are tractable, and consequently the Gibbs sampling is easy to implement. Finally, the technique is illustrated by using simulated and real data, respectively.     

Efficient Markov chain Monte Carlo with incomplete multinomial data

We propose a block Gibbs sampling scheme for incomplete multinomial data. We show that the new approach facilitates maximal blocking, thereby reducing serial dependency and speeding up the convergence of the Gibbs sampler. We compare the efficiency of the new method with the standard, non-block Gibbs sampler via a number of numerical examples.     

On implementation of the Gibbs sampler for estimating the accuracy of multiple diagnostic tests

Implementation of the Gibbs sampler for estimating the accuracy of multiple binary diagnostic tests in one population has been investigated. This method, proposed by Joseph, Gyorkos and Coupal, makes use of a Bayesian approach and is used in the absence of a gold standard to estimate the prevalence, the sensitivity and specificity of medical diagnostic tests. The expressions that allow this method to be implemented for an arbitrary number of tests are given. By using the convergence diagnostics procedure of Raftery and Lewis, the relation between the number of iterations of Gibbs sampling and the precision of the estimated quantiles of the posterior distributions is derived. An example concerning a data set of gastro-esophageal reflux disease patients collected to evaluate the accuracy of the water siphon test compared with 24 h pH-monitoring, endoscopy and histology tests is presented. The main message that emerges from our analysis is that implementation of the Gibbs sampler to estimate the parameters of multiple binary diagnostic tests can be critical and convergence diagnostic is advised for this method. The factors which affect the convergence of the chains to the posterior distributions and those that influence the precision of their quantiles are analyzed.     

Optimal direction Gibbs sampler for truncated multivariate normal distributions

Generalized Gibbs samplers simulate from any direction, not necessarily limited to the coordinate directions of the parameters of the objective function. We study how to optimally choose such directions in a random scan Gibbs sampler setting. We consider that optimal directions will be those that minimize the Kullback–Leibler divergence of two Markov chain Monte Carlo steps. Two distributions over direction are proposed for the multivariate Normal objective function. The resulting algorithms are used to simulate from a truncated multivariate Normal distribution, and the performance of our algorithms is compared with the performance of two algorithms based on the Gibbs sampler.     

A full Bayesian analysis of circular data using the von Mises distribution

This paper presents a full Bayesian analysis of circular data, paying special attention to the von Mises distribution. We obtain samples from the posterior distribution using the Gibbs sampler which, after the introduction of strategic latent variables, has all full conditional distributions of known type.