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...

INTRODUCTION Markov chain Monte Carlo methods have enjoyed a surge of interest since Gelfand and Smith (1990) described the Gibbs sampler and its effectiveness in providing approximate Bayesian solutions for models that had previously been approachable only with great difficulty, or that had been discarded as being too difficult to work with. Ongoing research in this area includes widening the applications to ever more detailed and difficult problems, alteration and improvement of the algorithm, and improvement of estimates based on the Markov chain. See Besag and Green (1993) and Smith and Roberts (1993). One of the extraordinary features of the Gibbs sampler is that the theory behind it can be presented at an elementary level (Casella and George, 1992), giving upper level undergraduate or beginning graduate students a glimpse Steve MacEachern is Assistant Professor, Department of Statistics, Ohio State University, and Visiting Assistant Professor, Institute o

This paper proposes five concepts for discussion at the beginning of an introductory statistics course: (1) entities, (2) properties of entities (which are roughly equivalent to variables), (3) a major goal of empirical research: to predict and control the values of variables, (4) relationships between variables as a key to prediction and control, and (5) statistical techniques for studying relationships between variables as a means to accurate prediction and control. After students have learned the five concepts they learn standard statistical topics in terms of the concepts. It is recommended that students learn the material through numerous practical examples. It is argued that the approach gives students a lasting appreciation of the vital role of the field of statistics in empirical research

Some key ideas in statistics and probability are hard for students, including sampling distributions. Computer simulation lets students gain experience with and intuition for these concepts. Bootstrapping can reinforce that learning, and provide a way for students (and future practitioners!) to estimate sampling distributions when they have data but do not know the underlying distribution. Bootstrapping also frees us from the requirement to teach inference only for statistics for which simple formulas are available|we can bootstrap robust statistics like the median as easily as the mean. For the promise of simulation and bootstrapping to be realized, they must be available and easy to use in general-purpose statistical software, complete with the exploratory data analysis and inferential capabilities required in teaching and practice. We discuss some of the available software for simulation and bootstrapping, in particular software built on S-Plus. Key words: bootstrap, resampling, simulation,

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Traditional statistical methods attempt to provide objective information about treatment effects through the use of easily computed P –values. However, much controversy surrounds the use of P –values, including statistical vs. clinical significance, artificiality of null hypotheses, 1–tailed vs. 2–tailed tests, difficulty in interpreting confidence intervals, falsely interpreting non– informative studies as ”negative”, arbitrariness in testing for equivalence, trading off type I and type II error, using P –values to quantify evidence, which statistical test should be used for 2 × 2 frequency tables, α–spending and adjusting for multiple comparisons, whether to adjust final P –values for the intention of terminating a trial early even though it completed as planned, complexity of group sequential monitoring procedures, and whether a promising but statistically insignificant trial can be extended. Bayesian methods allow calculation of probabilities that are usually of more interest to consumers, e.g. the probability that treatment A is similar to treatment B or the probability that treatment A is at least 5 % better than treatment B, and these methods are

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