Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the Metropolis-Hastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, high-dimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. Rosenthal (1995b) presents a method to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical values needed in Rosenthal's theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems...

This paper considers the class of sequential probit models in relation to other models for ordinal data. Hierarchical and other extensions of the model are proposed for applications involving discrete time (grouped) survival data. Computationally practical Markov chain Monte Carlo algorithms are developed for the fitting of these models. The ideas and methods are illustrated in detail with a real data example on the length of hospital stay for patients undergoing heart surgery. A notable aspect of this analysis is the comparison, based on marginal likelihoods and training sample priors, of several non-nested models, such as the sequential model, the cumulative ordinal model and Weibull and log-logistic models. Keywords: Bayes factor; Discrete hazard function; Gibbs sampling; Marginal likelihood; Metropolis-Hastings algorithm; Non-nested models; Sequential probit; Training sample prior; Model comparison. 1 Introduction Ordinal response data is generally analyzed using the cumulative o...

Data from industrial experiments often involve an ordered categorical response, such as a qualitative rating. ANOVA based analyses may be inappropriate for such data, suggesting the use of Generalized Linear Models (GLMs). When the data are observed from a fractionated experiment, likelihood-based GLM estimates may be innite, especially when factors have large eects. These diculties are overcome with a Bayesian GLM, which is implemented via the Gibbs sampling algorithm. Techniques for modeling data and for subsequently using the identied model to optimize the process are outlined. An important advantage in the optimization stage is that uncertainty in the parameter estimates is accounted for in the model. For robust design experiments, the Bayesian approach easily incorporates the variability of the noise factors using the response modeling approach (Welch, Yu, Kang and Sacks 1990 and Shoemaker, Tsui and Wu 1991). This approach and its techniques are used to analyze two...

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. R ESUM E Les auteurs s'interessent a une classe assez vaste de modeles de regression ordinale avec parametres de localisation et d'echelle, laquelle permet la selection adaptative de fonctions...

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This paper considers the fitting, criticism and comparison of three ordinal regression models -- the cumulative, sequential and two-step models. Efficient algorithms based on Markov chain Monte Carlo methods are developed for each model. In the case of the cumulative model, a new Metropolis-Hastings procedure to sample the cut points is proposed. This procedure relies on a simple transformation of the cut-points that leaves the transformed cut-points unordered. For comparing these models, we develop a coherent approach based on marginal likelihoods and Bayes factors. To help in the assignment of prior distributions to regression parameters and the cut-points, different methods for forming and representing prior beliefs are provided. One set of methods is based on the idea of a training sample and a prior imaginary sample. Another method is based on the direct assessment of distributions on the multinomial response, followed by change of variable to a distribution on the parameters of t...

Many databases involve ordered discrete responses in a temporal and spatial context, including, for example, land development intensity levels, vehicle ownership, and pavement conditions. An appreciation of such behaviors requires rigorous statistical methods, recognizing spatial effects and dynamic processes. This study develops a dynamic spatial ordered probit (DSOP) model in order to capture patterns of spatial and temporal autocorrelation in ordered categorical response data. This model is estimated in a Bayesian framework using Gibbs sampling and data augmentation, in order to generate all autocorrelated latent variables. It is found that the DSOP model yields much more accurate estimates than standard, non-spatial techniques. As for model selection, the DSOP model is clearly preferred to standard OP, dynamic OP and spatial OP models. These methods are then used to analyze land use changes over an 18-year period in Austin, Texas. In this analysis, temporal and spatial autocorrelation effects are found to be significantly positive. In addition, increases in travel times to the region’s central business district (CBD) are estimated to substantially reduce land development intensity. The proposed and tested DSOP model is felt to be a significant contribution to the field of spatial econometrics, where binary applications (for discrete response data) have been seen as the cutting edge. The Bayesian framework and Gibbs sampling techniques used here permit such complexity, in world of twodimensional autocorrelation.

Ministe`re de l’Education Nationale, de l’Enseignement Supe´rieur

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 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 di#erent link functions. The approach is illustrated by considering data from a pediatric radiology study. R ESUM E L'auteur propose une nouvelle parametrisation du modelederegression ordinale bayesien a lien cumulatif dont il se sert pour explorer la relation entre des covariables et des "points de coupure." Cette reparametrisation permet d'ajuster les modeles par une methode de Monte-Carlo a saute-mouton modifiee, evitant ainsi le besoin d'augmentation de donnees de la variable latente et la lenteur...

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. R ESUM E Les auteurs s'interessent a une classe assez vaste de modeles de regression ordinale avec parametres de localisation et d'echelle, laquelle permet la selection adaptative de fonctions...