Clinical trials are prospective studies to evaluate the effect of interventions in humans under prespecified conditions. They have become a standard and an integral part of modern medicine. A properly planned and executed clinical trial is the most definitive tool for evaluating the effect and applicability of new treatment modalities (Pocock, 1983; Piantadosi, 2005; Cook and Demets, 2008). The standard statistical approach to designing and analyzing clinical tri-

als and other medical experiments is frequentist. A primary purpose of this book is to describe an alternative approach called the Bayesian approach. The eponym originates from a mathematical theorem derived by Thomas Bayes (1763), an English clergyman who lived from 1702 to 1761. Bayes’ theorem plays a fundamental role in the inferential and calculational aspects of the Bayesian approach. The Bayesian approach can be applied separately from frequentist methodology, as a supplement to it, or as a tool for designing efficient clinical trials that have good frequentist properties. The two approaches have rather different philosophies, although both deal with empirical evidence and both use probability. Because of the similarities, the distinction between them is often poorly understood by nonstatisticians. A major difference is flexibility, in both design and analysis. In the

Bayesian approach, experiments can be altered in midcourse, disparate sources of information can be combined, and expert opinion can play a role in inferences. This is not to say that “anything goes.” For example, even though nonrandomized trials can be used in a Bayesian analysis, biases that can creep into some such trials can, in effect, make legitimate conclusions impossible. Another major difference is that the Bayesian approach can be decision-oriented, with experimental designs tailored to maximize objective functions, such as company profits or overall public health benefit. Much of the material in this book is accessible to nonstatisticians. How-

ever, to ensure that statisticians can follow the arguments and reproduce the results, we also include technical details. Not all of this technical development will be accessible to all readers. Readers who are not interested in

technicalities may skim or skip the mathematics and still profitably focus on the ideas. Certain subjects presented in this book are treated in a rather cursory

fashion. References written from the same perspective as the current report but that are somewhat more comprehensive in certain regards include (Berry, 1991; 1993). The text by Berry (1996) and its companion computing supplement by Albert (1996) explain and illustrate Bayesian statistics in very elementary terms and may be helpful to readers who are not statisticians. Other readers may find more advanced Bayesian texts accessible. These texts include Box and Tiao (1973), Berger (1985), DeGroot (1970), Bernardo and Smith (1994), Lee (1997), Robert (2001), Gelman, Carlin, Stern, and Rubin (2004), and Carlin and Louis (2009). Berry and Stangl (1996) is a collection of case studies in Bayesian biostatistics; it gives applications of modern Bayesian methodology. Finally, the lovely text by Spiegelhalter et al. (2004) is an outstanding introduction to Bayesian thinking in many problems important to biostatisticians and medical professionals generally, one of which is clinical trials. Turning to the area of computing, Gilks, Richardson, and Spiegelhalter

(1996) is a collection of papers dealing with modern Bayesian computer simulation methodology that remains relevant since it was so many years ahead of its time at publication. Two other recent Bayesian computing books by Albert (2007) and Marin and Robert (2007) are also important. Both books adopt the R language as their sole computing platform; indeed, both include R tutorials in their first chapters. Albert (2007) aims at North American first-year graduate or perhaps advanced undergraduate students, building carefully from first principles and including an R package, LearnBayes, for implementing many standard methods. By contrast, the level of formality and mathematical rigor in Marin and Robert (2007) is at least that of its fairly mature stated audience of second-year master’s students. In the present book, we also use R as our “base” computing platform, consistent with its high and accelerating popularity among statisticians. However, we also take advantage of other, mostly freely available packages when they offer the most sensible solutions. In particular, we rely on WinBUGS, both by itself and as called from R by the BRugs library. This popular software has emerged as the closest thing to an “industry standard” that exists in the applied Bayesian statistical community. We now offer a simple example to help show some of the primary features

of the frequentist perspective. We will return to this setting in Example 2.2 to show the corresponding Bayesian solution and its features.