Hal S. Stern

Bayesian Data Analysis describes how to conceptualize, perform, and critique statistical analyses from a Bayesian perspective. Using examples largely from the authors' own experiences, the book focuses on modern computational tools and obtains inferences using computer simulations. Its unique features include thorough discussions of the methods for

Abstract: This paper considers Bayesian counterparts of the classical tests for goodness of fit and their use in judging the fit of a single Bayesian model to the observed data. We focus on posterior predictive assessment, in a framework that also includes conditioning on auxiliary statistics. The Bayesian formulation facilitates the construction and calculation of a meaningful reference distribution not only for any (classical) statistic, but also for any parameter-dependent “statistic ” or discrepancy. The latter allows us to propose the realized discrepancy assessment of model fitness, which directly measures the true discrepancy between data and the posited model, for any aspect of the model which we want to explore. The computation required for the realized discrepancy assessment is a straightforward byproduct of the posterior simulation used for the original Bayesian analysis. We illustrate with three applied examples. The first example, which serves mainly to motivate the work, illustrates the difficulty of classical tests in assessing the fitness of a Poisson model to a positron emission tomography image that is constrained to be nonnegative. The second and third examples illustrate the details of the posterior predictive approach in two problems: estimation in a model with inequality constraints on the parameters, and estimation in a mixture model. In all three examples, standard test statistics (either a χ 2 or a likelihood ratio) are not pivotal: the difficulty is not just how to compute the reference distribution for the test, but that in the classical framework no such distribution exists, independent of the unknown model parameters. Key words and phrases: Bayesian p-value, χ 2 test, discrepancy, graphical assessment, mixture model, model criticism, posterior predictive p-value, prior predictive

Missing data in clinical trials can have a major effect on the validity of the inferences that can be drawn from the trial. This article reviews methods for preventing missing data and, failing that, dealing with data that are missing.

It is common to summarize statistical comparisons by declarations of statistical significance or nonsignificance. Here we discuss one problem with such declarations, namely that changes in statistical significance are often not themselves statistically significant. By this, we are not merely making the commonplace observation that any particular threshold is arbitrary—for example, only a small change is required to move an estimate from a 5.1% significance level to 4.9%, thus moving it into statistical significance. Rather, we are pointing out that even large changes in significance levels can correspond to small, nonsignificant changes in the underlying quantities.The error we describe is conceptually different from other oft-cited problems—that statistical significance is not the same as practical importance, that dichotomization into significant and nonsignificant results encourages the dismissal of observed differences in favor of the usually less interesting null hypothesis of no difference, and that any particular threshold for declaring significance is arbitrary. We are troubled by all of these concerns and do not intend to minimize their importance. Rather, our goal is to bring attention to this additional error of interpretation. We illustrate with a theoretical example and two applied examples. The ubiquity of this statistical error leads us to suggest that students and practitioners be made more aware that the difference between “significant” and “not significant” is not itself statistically significant.

This article provides a comprehensive review of multiple imputation (MI), a technique for analyzing data sets with missing values. Formally, MI is the process of replacing each missing data point with a set of m > 1 plausible values to generate m complete data sets. These complete data sets are then analyzed by standard statistical software, and the results combined, to give parameter estimates and standard errors that take into account the uncertainty due to the missing data values. This article introduces the idea behind MI, discusses the advantages of MI over existing techniques for addressing missing data, describes how to do MI for real problems, reviews the software available to implement MI, and discusses the results of a simulation study aimed at finding out how assumptions regarding the imputation model affect the parameter estimates provided by MI.