A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler conver...

Functional linear models are useful in longitudinal data analysis. They include many classical and recently proposed statistical models for longitudinal data and other functional data. Recently, smoothing spline and kernel methods have been proposed for estimating their coefficient functions nonparametrically but these methods are either intensive in computation or inefficient in performance. Toovercome these drawbacks, in this paper, a simple and powerful two-step alternativeis proposed. In particular, the implementation of the proposed approach via local polynomial smoothing is discussed. Methods for estimating standard deviations of estimated coefficient functions are also proposed. Some asymptotic results for the local polynomial estimators are established. Two longitudinal data sets, one of which involves time-dependent covariates, are used to demonstrate the proposed approach. Simulation studies show that our two-step approach improves the kernel method proposed in Hoover, et al...

The term data augmentation refers to methods for constructing iterative optimization or sampling algorithms via the introduction of unobserved data or latent variables. For deterministic algorithms,the method was popularizedin the general statistical community by the seminal article by Dempster, Laird, and Rubin on the EM algorithm for maximizing a likelihood function or, more generally, a posterior density. For stochastic algorithms, the method was popularized in the statistical literature by Tanner and Wong’s Data Augmentation algorithm for posteriorsampling and in the physics literatureby Swendsen and Wang’s algorithm for sampling from the Ising and Potts models and their generalizations; in the physics literature,the method of data augmentationis referred to as the method of auxiliary variables. Data augmentationschemes were used by Tanner and Wong to make simulation feasible and simple, while auxiliary variables were adopted by Swendsen and Wang to improve the speed of iterative simulation. In general,however, constructing data augmentation schemes that result in both simple and fast algorithms is a matter of art in that successful strategiesvary greatlywith the (observed-data) models being considered.After an overview of data augmentation/auxiliary variables and some recent developments in methods for constructing such

this paper we construct several (partially and fully blocked) MCMC algorithms for minimizing the autocorrelation in MCMC samples arising from important classes of longitudinal data models. We exploit an identity used by Chib (1995) in the context of Bayes factor computation to show how the parameters in a general linear mixed model may be updated in a single block, improving convergence and producing essentially independent draws from the posterior of the parameters of interest. We also investigate the value of blocking in non-Gaussian mixed models, as well as in a class of binary response data longitudinal models. We illustrate the approaches in detail with three real-data examples.

8> R f(`)d`. To marginalize, say for ` i ; requires h(` i ) = R f(`)d` (i) = R f(`)d` where ` (i) denotes all components of ` save ` i : To obtain Eg(` i ) requires similar integration; to obtain the marginal distribution of say g(`) or its expectation requires similar integration. When p is large (as it will be in the applications we envision) such integration is analytically infeasible (the so-called curse of dimensionality*). Gibbs sampling provides a Monte Carlo approach for carrying out such integrations. In what sorts of settings would we have need to mar

this article, we review some important general methods for Markov chain Monte Carlo

An approach for developing Bayesian outlier and goodness of fit statistics is presented for the linear model and extended to a hierarchical random effects model for repeated measures data. Diagnostics for univariate outliers, missing covariates, multivariate outliers and global goodness of fit are developed. Distribution theory for the posterior of the residuals is worked out. A local approach is used to show how omitted covariates and fixed and random effects affect residual summaries. Standard plots are interpreted in light of these understandings. Key Words: Bayesian Data Analysis, Goodness-of-Fit, Hierarchical Models, Longitudinal Data, Outlier, Philosophy of Statistics, Shrinkage. 1 Introduction. This paper develops a Bayesian approach to residual analysis and extends the approach to the random effects model (REM) used to model repeated Robert E. Weiss is Assistant Professor, Department of Biostatistics, Box 177220; UCLA School of Public Health; Los Angeles CA 90095-1772 U.S....

Common repeated measures random effects models contain two random components, a random person effect and time-varying errors. An observation can be an outlier due to either an extreme person effect or an extreme time varying error. Outlier statistics are presented that can distinguish between these types of outliers. For each person there is one statistic per observation, plus one statistic per random effect. Methodology is developed to reduce the explosion of statistics to two summary outlier statistics per person; one for the random effects and one for the time varying errors. If either of these screening statistics are large, then individual statistics for each observation or random effect can be inspected. Multivariate, targeted outlier statistics and goodness-of-fit tests are also developed. Distribution theory is given, along with some geometric intuition. Key Words: Bayesian Data Analysis, Goodness-of-Fit, Hierarchical Models, Observed Errors, Repeated Measures. 1 Introduction...

The purpose of this research is the development of an accurate model of cycle time based on data collected at a complex manufacturing process. In this paper, we model average cycle time during a period as a two-segment piecewise linear function of the work-in-process during that period. This model allows us to capture changes in cycle time mean as well as variance with different levels of work-in-process. The challenge is to determine the breakpoint, intercept, slope and corresponding variance associated with each line segment. To accomplish this, we propose a Bayesian procedure which makes use of the Gibbs sampler and Metropolis-Hastings algorithms. We compare our model and results with those of an analytical nonlinear model. The results help us understand the relationship between work-in-process and cycle time mean and variance.

We investigate the importance of the assumed covariance structure for longitudinal modelling of CD4 counts. We examine how individual predictions of future CD4 counts are affected by the covariance structure. We consider four covariance structures: one based on an integrated Ornstein-Uhlenbeck stochastic process, one based on Brownian motion and two derived from standard linear and quadratic random e#ects models. Using data from the Multicenter AIDS Cohort Study and from a simulation study,we show that there is a noticeable deterioration in the coverage rate of confidence intervals if we assume the wrong covariance. There is also a loss in e#ciency. The quadratic random effects model is found to be the best in terms of correctly calibrated prediction intervals, but is substantially less efficient than the others. Incorrectly specifying the covariance structure as linear random effects gives too narrow prediction intervals with poor coverage rates. Fitting using the model based on th...