The growth of biostatistics has been phenomenal in recent years andhas been marked by considerable technical innovation in bothmethodology and computational practicality. One area that hasexperienced significant growth is Bayesian methods. The growing useof Bayesian methodology has taken place partly due to an increasingnumber of practitioners valuing the Bayesian paradigm as matchingthat of scientific discovery. In addition, computational advanceshave allowed for more complex models to be fitted routinely torealistic data sets. Through examples, exercises and a combination of introductoryand more advanced chapters, this book provides an invaluableunderstanding of the complex world of biomedical statisticsillustrated via a diverse range of applications taken fromepidemiology, exploratory clinical studies, health promotionstudies, image analysis and clinical trials. Key Features: * Provides an authoritative account of Bayesian methodology, fromits most basic elements to its practical implementation, with anemphasis on healthcare techniques. * Contains introductory explanations of Bayesian principlescommon to all areas of application. * Presents clear and concise examples in biostatisticsapplications such as clinical trials, longitudinal studies,bioassay, survival, image analysis and bioinformatics. * Illustrated throughout with examples using software includingWinBUGS, OpenBUGS, SAS and various dedicated Rprograms. * Highlights the differences between the Bayesian and classicalapproaches. * Supported by an accompanying website hosting free softwareand case study guides. Bayesian Biostatistics introduces the reader smoothlyinto the Bayesian statistical methods with chapters that graduallyincrease in level of complexity. Master students in biostatistics,applied statisticians and all researchers with a good background inclassical statistics who have interest in Bayesian methods willfind this book useful.
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Notation, terminology and some guidance for reading the book
Notation and terminology
Guidance for reading the book
Part 1: BASIC CONCEPTS IN BAYESIAN METHODS
Chapter 1: Modes of statistical inference
1.1 The frequentist approach: A critical reflection
1.2 Statistical inference based on the likelihood function
1.3 The Bayesian approach: Some basic ideas
Chapter 2: Bayes theorem: Computing the posterior distribution
2.2 Bayes theorem – the binary version
2.3 Probability in a Bayesian context
2.4 Bayes theorem – the categorical version
2.5 Bayes theorem – the continuous version
2.6 The binomial case
2.7 The Gaussian case
2.8 The Poisson case
2.9 The prior and posterior distribution of (θ)
2.10 Bayesian versus likelihood approach
2.11 Bayesian versus frequentist approach
2.12 The different modes of the Bayesian approach
2.13 An historical note on the Bayesian approach
2.14 Closing remarks
Chapter 3: Introduction to Bayesian inference
3.2 Summarizing the posterior by probabilities
3.3 Posterior summary measures
3.4 Predictive distributions
3.6 A normal approximation to the posterior
3.7 Numerical techniques to determine the posterior
3.8 Bayesian hypothesis testing
3.9 Closing remarks
Chapter 4: More than one parameter
4.2 Joint versus marginal posterior inference
4.3 The normal distribution with μ and unknown
4.4 Multivariate distributions
4.5 Frequentist properties of Bayesian inference
4.6 Sampling from the posterior distribution: The Method of Composition
4.7 Bayesian linear regression models
4.8 Bayesian generalized linear models
4.9 More complex regression models
4.10 Closing remarks
Chapter 5: Choosing the prior distribution
5.2 The sequential use of Bayes theorem
5.3 Conjugate prior distributions
5.4 Noninformative prior distributions
5.5 Informative prior distributions
5.6 Prior distributions for regression models
5.7 Modeling priors
5.8 Other regression models
5.9 Closing remarks
Chapter 6: Markov chain Monte Carlo sampling
6.2 The Gibbs sampler
6.3 The Metropolis(–Hastings) algorithm
6.4 Justification of the MCMC approaches*
6.5 Choice of the sampler
6.6 The Reversible Jump MCMC algorithm*
6.7 Closing remarks
Chapter 7: Assessing and improving convergence of the Markov chain
7.2 Assessing convergence of a Markov chain
7.3 Accelerating convergence
7.4 Practical guidelines for assessing and accelerating convergence
7.5 Data augmentation
7.6 Closing remarks
Chapter 8: Software
8.1 WinBUGS and related software
8.2 Bayesian analysis using SAS
8.3 Additional Bayesian software and comparisons
8.4 Closing remarks
Part II: BAYESIAN TOOLS FOR STATISTICAL MODELING
Chapter 9: Hierarchical models
9.2 The Poisson-gamma hierarchical model
9.3 Full versus empirical Bayesian approach
9.4 Gaussian hierarchical models
9.5 Mixed models
9.6 Propriety of the posterior
9.7 Assessing and accelerating convergence
9.8 Comparison of Bayesian and frequentist hierarchical models
9.9 Closing remarks
Chapter 10: Model building and assessment
10.2 Measures for model selection
10.3 Model checking
10.4 Closing remarks
Chapter 11: Variable selection
11.2 Classical variable selection
11.3 Bayesian variable selection: Concepts and questions
11.4 Introduction to Bayesian variable selection
11.5 Variable selection based on Zellner's -prior
11.6 Variable selection based on Reversible Jump Markov chain Monte Carlo
11.7 Spike and slab priors
11.8 Bayesian regularization
11.9 The many regressors case
11.10 Bayesian model selection
11.11 Bayesian model averaging
11.12 Closing remarks
Part III: BAYESIAN METHODS IN PRACTICAL APPLICATIONS
Chapter 12: Bioassay
12.1 Bioassay essentials
12.2 A generic in vitro example
12.3 Ames/Salmonella mutagenic assay
12.4 Mouse lymphoma assay (L5178Y TK+/−)
12.5 Closing remarks
Chapter 13: Measurement error
13.1 Continuous measurement error
13.2 Discrete measurement error
13.3 Closing remarks
Chapter 14: Survival analysis
14.1 Basic terminology
14.2 The Bayesian model formulation
14.4 Closing remarks
Chapter 15: Longitudinal analysis
15.1 Fixed time periods
15.2 Random event times
15.3 Dealing with missing data
15.4 Joint modeling of longitudinal and survival responses
15.5 Closing remarks
Chapter 16: Spatial applications: Disease mapping and image analysis
16.2 Disease mapping
16.3 Image analysis
Chapter 17: Final chapter
17.1 What this book covered
17.2 Additional Bayesian developments
17.3 Alternative reading
A.2 Continuous univariate distributions
A.3 Discrete univariate distributions
A.4 Multivariate distributions
End User License Agreement
Statistics in Practice
Human and Biological SciencesStephen Senn CRP-Santé, Luxembourg
Earth and Environmental SciencesMarian Scott University of Glasgow, UK
Industry, Commerce and FinanceWolfgang Jank University of Maryland, USA
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Library of Congress Cataloging-in-Publication Data
Lesaffre, Emmanuel. Bayesian biostatistics / Emmanuel Lesaffre, Andrew Lawson. p. ; cm. Includes bibliographical references and index. ISBN 978-0-470-01823-1 (cloth) 1. Biometry–Methodology. 2. Bayesian statistical decision theory. I. Lawson, Andrew (Andrew B.) II. Title. [DNLM: 1. Biostatistics–methods. 2. Bayes Theorem. QH 323.5] QH323.5.L45 2012 570.1′5195–dc23 2012004237
A catalogue record for this book is available from the British Library.
The growth of biostatistics as a subject has been phenomenal in recent years and has been marked by a considerable technical innovation in methodology and computational practicality. One area that has a significant growth is the class of Bayesian methods. This growth has taken place partly because a growing number of practitioners value the Bayesian paradigm as matching that of scientific discovery. But the computational advances in the last decade that have allowed for more complex models to be fitted routinely to realistic data sets have also led to this growth.
In this book, we explore the Bayesian approach via a great variety of medical application areas. In effect, from the elementary concepts to the more advanced modeling exercises, the Bayesian tools will be exemplified using a diversity of applications taken from epidemiology, exploratory clinical studies, health promotion studies and clinical trials.
This book grew out of a course that the first author has taught for many years (especially) in the Master programs in (bio)statistics at the universities of Hasselt and Leuven, both in Belgium. The course material was the inspiration for two out of three parts in the book. Therefore, the intended readership of this book are Master program students in (bio)statistics, but we hope that applied researchers with a good statistical background will also find the book useful. The structure of the book allows it to be used as course material for a course in Bayesian methods at an undergraduate or early stage postgraduate level. The aim of the book is to introduce the reader smoothly into Bayesian statistical methods with chapters that gradually increase in the level of complexity. The book consists of three parts. The first two parts of this work were the chapters primarily covered by the first author, while the last five chapters were primarily covered by the second author.
In Part I, we first review the fundamental concepts of the significance test and the associated P-value and note that frequentist methods, although proved to be quite useful over many years, are not without conceptual flaws. We also note that there are other methods on the market, such as the likelihood approach, but more importantly, the Bayesian approach. In addition, we introduce (an embryonic version of) the Bayes theorem. In Chapter 2, we derive the general expression of Bayes theorem and illustrate extensively the analytical computations to arrive at the posterior distribution on the binomial, the Gaussian and the Poisson case. For this, simple textbook examples are used. In Chapter 3, the reader is introduced to various posterior summary measures and the predictive distributions. Since sampling is fundamental to contemporary Bayesian approaches, sampling algorithms are introduced and exemplified in this chapter. While these sampling procedures will not yet prove their usefulness in practice, we believe that the early introduction of relatively simple sampling techniques will prepare the reader better for the advanced algorithms seen in later chapters. In this chapter, approaches to Bayesian hypothesis testing are treated and we introduce the Bayes factor. In Chapter 4, we extend all the concepts and computations seen in the first three chapters for univariate problems to the multivariate case, introducing also Bayesian regression models. It is then seen that, in general, no analytical methods to derive the posterior distribution are available, neither are the classical numerical approaches to integration sufficient. A new approach is then needed. Before addressing the solution to the problem, we treat in Chapter 5 the choice of the prior distribution. The prior distribution is the keystone to the Bayesian methodology. Yet, the appropriate choice of the prior distribution has been the topic of extensive discussions between non-Bayesians and Bayesians, but also among Bayesians. In this chapter, we extensively treat the various ways of specifying prior knowledge. In Chapters 6 and 7, we treat the basics of the Markov chain Monte Carlo methodologies. In Chapter 6, the Gibbs and the Metropolis–Hastings samplers are introduced again illustrated using a variety of medical examples. Chapter 7 is devoted to assessing and accelerating the convergence of the Markov chains. In addition, we cover the extension of the EM-algorithm to the Bayesian context, i.e. the data augmentation approach is exemplified. It is then time to see how Bayesian analyses can be done in practice. For this reason, we review in Chapter 8 the Bayesian software. We focus on two software packages: the most popular WinBUGS and the recently released Bayesian SAS® procedures. In both cases, a simple regression analysis serves as a guiding example helping the readers in their first analyzes with these packages. We end this chapter with a review of other Bayesian software, such as the packages OpenBUGS and JAGS, and also various R packages written to perform specific analyses.
In Part II, we develop Bayesian tools for statistical modeling. We start in Chapter 9 with reviewing hierarchical models. To fix ideas, we focus first on two simple two-level hierarchical models. The first is the Poisson-gamma model applied to a spatial data set on lip cancer cases in former East Germany. This example serves to introduce the concepts of hierarchical modeling. Then, we turn to the Gaussian hierarchical model as an introduction to the more general mixed models. A variety of mixed models are explored and amply exemplified. Also in this chapter comparisons between frequentist and Bayesian solutions aim to help the reader to see the differences between the two approaches. Model building and assessment are the topics of Chapter 10. The aim of this chapter is to see how statistical modeling could be performed entirely within the Bayesian framework. To this end, we reintroduce the Bayes factor (and its variants) to select between two statistical models. The Bayes factor is an important tool in model selection but is also fraught with serious computational difficulties. We then move to the Deviance Information Criterion (DIC). For a better understanding of this popular model selection criterion, we introduce at length the classical model selection criteria, i.e. AIC and BIC and relate them to DIC. The part on model checking describes all classical actions one would take to construct and evaluate a model, such as checking the residuals for outliers and influential observations, finding the correct scale of the response and the covariates, choosing the correct link function, etc. In this chapter, we also elaborate on the posterior predictive check as a general tool for goodness-of-fit testing. The final chapter, Chapter 11, in Part II handles Bayesian variable selection. This is a rapidly evolving topic that received a great impetus from the developments in bioinformatics. A broad overview of possible variable and model selection approaches and the associated software is given. While in the previous two chapters, the WinBUGS software and to a lesser extent the Bayesian SAS procedures were dominant, in this chapter we focus on software packages in R.
In Part III, we address particular application areas for Bayesian modeling. We include the most important areas from a practical biostatistical standpoint. In Chapter 12, we examine bioassay, where we consider preclinical testing methods: both Ames and Mouse Lymphoma in vitro assays and the famous Beetles LD50 toxicity assay are considered. In Chapter 13, we consider the important and pervasive problem of measurement error and also the misclassification in biostatistical studies. We discuss Berkson and classical joint models, bias such as attenuation and the problem of discrete error in the form of misclassification. In Chapter 14, we examine survival analysis from a Bayesian perspective. In this chapter, we cover basic survival time models and risk-set-based approaches and extend models to consider contextual effects within hazards. In Chapter 15, longitudinal analysis is considered in greater depth. Correlated prior distributions for parameters and also temporally correlated errors are considered. Missingness mechanisms are discussed and a nonrandom missingness example is explored. In Chapter 16, two important spatial biostatistical application areas are then considered: disease mapping and image analysis. In disease mapping, basic Poisson convolution models that include spatially structured random effects are examined for risk estimation, while in image analysis a focus on Bayesian fMRI analysis with correlated prior distributions is presented.
In Chapter 17, we end the book with a brief review of the topics that we did not cover in this book and give some key references for further reading. Finally, in the appendix, we provide an overview of the characteristics of most popular distributions used in Bayesian analyses.
Throughout the book there are numerous examples. In Parts I and II, explicit reference is made to the programs associated with the examples. These programs can be found at the website www.wiley.com/go/bayesian_methods_biostatistics. The programs used in Part III can also be found at this website.
When writing the early chapters, the first author benefitted from discussions with master students at Leuven, Hasselt and Leiden who pointed out various typos and ambiguities in earlier versions of the book. In addition, thanks go to the colleagues and former/current PhD students at L-Biostat at KU Leuven and at the Department of Biostatistics, Erasmus MC, Rotterdam, for illustrative discussions, critical remarks and help with software. In this respect, special thanks go to Susan Bryan, Silvia Cecere, Luwis Diya, Alejandro Jara, Arnošt Komárek, Marek Molas, Mukendi Mbuyi, Timothy Mutsvari, Veronika Rockova, Robin Van Oirbeek and Sten Willemsen. Software support was received from Sabanés Bové on the R package glmBfp, Fang Chen on the Bayesian SAS programs, Robert Gramacy on the R program monomvn, David Hastie and Peter Green on an R program for RJMCMC, David Lunn on the Jump interface in WinBUGS, Elizabeth Slate and Karen Vines. Thanks also go to those who provided data for the book or who gave permission to use their data, i.e. Steven Boonen, Elly Den Hondt, Jolanda Luime, the Signal-Tandmobiel® team, Bako Topal and Vincent van Weel. The authors also wish to thank, for interesting discussions, their colleagues in Leuven, Rotterdam and Charleston especially Dipankar Bandyopadhyay, Paul Eilers, Steffen Fieuws, Mulugeta Gebregziabher, Dimitris Rizopoulos and Elizabeth Slate. Finally, insightful conversations with George Casella, James Hodges, Helmut Küchenhoff and Paul Schmitt were much appreciated.
Last, the first author especially wishes to thank his wife Lieve Sels for her patience not only during the preparation of the ‘book’ but also during his whole professional career. To his children, Annemie and Kristof, he apologizes for the many times that their father was present but ‘absent’.
Emmanuel Lesaffre (Rotterdam and Leuven) Andrew B. Lawson (Charleston)
Notation, terminology and some guidance for reading the book
Notation and terminology
In this section, we review some notation used in the book. We limit ourselves to outline some general principles; for precise definitions, we refer to the text.
First, both the random variable and its realization are denoted in this book as y. The vector of covariates is most often denoted as . The distribution of a discrete y as well as the density of a continuous variable will be denoted as p(y), unless otherwise stated. In the text, we make clear which of the two meanings applies. A sample of observations is denoted as , but also, a d-dimensional vector will be denoted in bold, i.e. y. We make it clear from the context what is implied. Further, independent, identically distributed random variables are indicated as i.i.d. A distribution (density) depending on a parameter vector is denoted as . The joint distribution of y and z is denoted as and the conditional distribution of y, given z will be denoted as . Alternatively, we use the notation . The probability that an event happens will occasionally be denoted as P for reasons of clarity.
A particular distribution will be addressed in two ways. For example, indicates that the random variable y has a gamma distribution with parameters α and β, but to indicate that the distribution is evaluated in y we will use the notation . When parameters have been given almost the same notation, say , then the notation is used where * is a place holder for 1, 2, 3.
In the normal case, some Bayesian textbooks use the precision notation while others use the variance notation. More specifically, if y has a normal distribution with mean μ and variance , then instead of (classical notation) the alternative notation (or even ) with the precision is used by some. This alternative notation is inspired by the fact that in Bayesian statistics some key results are better expressed in terms of the precision rather than the variance. This is also the notation used by WinBUGS. In this book, we frequently refer to classical, frequentist statistics. The use of precision would then be more confusing, rather than illuminating. In addition, when it comes to summarizing the results of a statistical analysis, the standard deviation is a better tool than the precision. For these reasons, we primarily used in this book the variance notation. But, throughout the book (especially in the later chapters) we regularly make the transition from one notation to the other.
Finally, we use some generally accepted standard notations, such as always denotes a column vector, |A| denotes the determinant of a matrix A, tr(A) is the trace of a matrix and AT denotes the transpose of a matrix A. The sample mean of is denoted as and their standard deviation as s, sy or simply SD.
Guidance for reading the book
No particular guidance is needed to read this book. The flow in the book is natural: starting from elementary Bayesian concepts we gradually introduce the more complex topics. This book deals, basically, only with parametric Bayesian methods. This means that all our random variables are assumed to have a particular distribution with a finite number of parameters. In the Bayesian world, many more distributions are used than in classical statistics. So for the classical reader (whatever meaning this may have), many of the distributions that pop-up in the book will be new. A brief characterization of these distributions, together with a graphical display, can be found in the appendix of the book.
Finally, some of the sections indicated by ‘*’ are technical and may be skipped at first reading.
BASIC CONCEPTS IN BAYESIAN METHODS
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