Book Review Lemuel A. Moye´ (2003) Multiple Analyses in Clinical Trials: Fundamentals for Investigators. Springer: New York. ISBN: 0-387-00727-X, XXIII + 436pp, £ 89.95 DOI: 10.1007/s00184-005-0371-6 Multiple testing has become a serious matter in interpreting clinical trials. Classical examples include the interpretation of subgroup results and the evaluation of secondary or exploratory endpoints. It is furthermore of importance in the quick emerging area of molecular medicine, where thousands of gene expression levels or genetic polymorphisms are measured simultaneously. Though many hundred articles and several excellent textbooks have been published on this topic, most of them focus on the statistical aspects and procedures. Hence, the understanding of this material can be difficult or even impossible for readers with weaker statistical background. Therefore, the book by L.A. Moye´ fills an important gap in the literature: It is essentially a nonmathematical discussion of multiple analyses in clinical trials. It concentrates on the rationale for the analyses, the difficulties posed by their interpretation, and easily understood solutions. Hence, the book is written for advanced medical students, clinical investigators, research groups within the pharmaceutical industry, regulators and biostatisticians. Only a basic background in health care and introductory statistics is required of the reader. Since the reader has to understand the basic principles of a clinical trial, chapter 1 provides a short review of the important principles in clinical trial design and execution, including randomization, blinding, interim monitoring, and intention to treat analyses. Without formulae the basic ideas of statistical testing, type I error and power, as well as the considerations for sample size computations are discussed. Chapter 2 demonstrates the need for prospective planning of statistical analyses in clinical trials by a series of examples. It shows that drawing conclusions can be difficult if a data-driven approach is taken. Chapter 3 discusses in detail why multiple analyses are common in clinical trials and the difficulties in interpreting the results. In addition, some basic terms, e.g. family-wise error rate are defined. The author also sketches a few procedures to adjust for multiple testing, including Bonferroni and Bonferroni-Holm. Chapters 4 to 6 each discuss a separate dilemma in the application of multiple analyses in clinical trials. Chapter 4 covers the issue of multiple analyses and multiple endpoints. In an excellent way, the process of endpoint triage is illustrated. Furthermore, the differences between primary, secondary and exploratory endpoints are discussed in detail. In addition, the differential allocation of alpha levels is introduced and nicely illustrated. Chapters 5 and 6 focus on multiple endpoints when the statistical tests are dependent. Chapter 5 introduces the concept of a dependency parameter and provides guidelines and cautions in estimating the degree of dependence between different hypothesis tests. Chapter 6 extends the treatise to three, four and K dependent primary analyses. A rigorous analytical treatise of this topic is given in an Appendix. Chapters 7 to 13 apply the methodologies for multiple analyses that have been derived in chapters 3 to 6. Chapters 7 and 8 focus on composite endpoints, and chapters 9 to 11 deal with subgroup analyses. Chapter 12

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examines the many active treatments to one control situation. Finally, chapter 13 combines different approaches for multiple analyses, in detail: differential alpha allocation, combined endpoints, prospective requirements of different levels of efficacy, and confirmatory subgroup analyses. In summary, I am convinced that the book by Moye´ closes an important gap about approaches for dealing with the problem of multiple testing in clinical trials. It excellently illustrates the various issues arising in this area and shows how these can be addressed adequately. It will therefore be not only a fundamental book for clinical investigators but also a resource of marvellous examples for biostatisticians. Andreas Ziegler, Lu¨beck

Barndorff-Nielsen, Ole E.; Mikosch, Thomas; Resnick, Sidney I. (Editors): ‘‘Le´vy Processes: Theory and Applications’’, Birkha¨user-Verlag 2001, hardcover, x + 415 pages, $84.95, ISBN 0-8176-4167-X. DOI: 10.1007/s00184-005-0390-3 Although non-Gaussian Le´vy processes have been well understood for a long time, they have become increasingly popular for describing phenomena in various fields of applications, such as finance, meteorology, physics or telecommunications over the last decade or so. This is due to the fact that Le´vy processes are often a suitable model, e.g., for extremal behavior or for clustering and that computational power nowadays allows for simulation as well as for inference in such models. This volume presents a useful overview of recent developments concerning Le´vy processes in both theory and applications. It originated from the First MaPhySto Conference on Le´vy Processes: Theory and Applications, held at the University of Aarhus, Denmark, 1999. The book is divided into six parts and begins with a compact tutorial on important definitions and properties of Le´vy processes (Part I). Part II is dedicated to distributional, pathwise and structural results for Le´vy processes, and includes, e.g., an article on exponential functionals which play an important role in both mathematical finance and mathematical physics. It should be pointed out that two out of the four theoretical articles of Part II contain various open questions and problems to generate further research in this area. Part III on extensions and generalizations of Le´vy processes comprises three theoretical survey articles, one on Le´vy processes in stochastic differential geometry, one on pseudodifferential operators, and one on semistable distributions generalizing the well known class of stable distributions. Although Part IV is dedicated to applications in finance, two out of the four articles in this part are theoretical; one reviews Le´vy processes and fields in quantum theory and the other one describes a Le´vy process approach to continuous quantum measurement processes. The remaining two articles are applied and rather non-technical. In particular, Woyczyn´ski’s paper on using a-stable processes to model physical phenomena from various fields such as fluid mechanics and polymer chemistry is delightful to read.

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Part V deals with applications in finance and stresses differences and generalizations compared to the use of Brownian motion. The first two articles, by Barndorff-Nielsen and Shephard, who use Le´vy processes to model time series from financial econometrics, and by Eberlein, who replaces Brownian motion by generalized hyperbolic Le´vy motions to achieve a better fit to financial data, are driven by applications and are rather non-technical. Both papers do not require knowledge on mathematical finance for reading. The remaining two articles of this section, one on explicit form and path regularity of martingale representations and one on Brownian and Bessel meanders are theoretical and lack the connection to applications in finance. Part VI on numerical and statistical aspects is particularly important. However, with only two articles this part is too short. The first article by Nolan is dedicated to inference. He describes a program of maximum likelihood estimation of stable distributions and discusses – giving many applications – diagnostics for assessing the stability of data sets. The second article by Rosin´ski presents several series representations of jump processes which could in principle be used for the simulation of linear and non-linear functionals of Le´vy processes. However, the author only alludes to numerical applications and simulations are clearly missed. All articles of this volume are well written. Although most articles require advanced knowledge in probability and stochastic processes, some of the more applied papers are easily accessible with basic knowledge in these areas. Impressive is the volume’s extensive and very useful bibliography for Le´vy process articles in both theory and applications. However, opportunities for cross-referencing, e.g., among articles in Parts II and V of the book have been missed. The book is dominated by the theory of Le´vy processes, theoretical articles outweigh the applied ones. In particular, more papers dedicated to inference and simulation, key aspects for a successful application using the class of Le´vy processes, would have been welcome to achieve a more balanced view. However, the book provides more than a good start to link the theory and the applications in this research area, and it clearly stimulates further use of non-Gaussian Le´vy processes for modelling. Therefore, I certainly recommend this volume for graduate students and researchers interested in Le´vy processes. Katja Ickstadt, Dortmund

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Egbert Dettweiler: Risk Processes. Edition am Gutenbergplatz, Leipzig, (2004) 207 S., 19.50, ISBN 3-937219-08-0 DOI: 10.1007/s00184-005-0391-2 One of the origins of the theory of risk processes is Ove Lundberg’s famous thesis On Random Processes and Their Applications to Sickness and Accident Statistics published in 1940. Since then, the theory has been developed into various directions and it is still growing, mainly because of its mathematical interest and in spite of its rather limited impact on actuarial practice. The monograph by Egbert Dettweiler is a most original contribution to the theory of risk processes. In the terminology of his book, a risk process is a two-dimensional sequence ðTn ; Xn Þn2N of strictly positive random variables which is strictly increasing in the first coordinate. The claim arrivalPsequence ðTn Þn2N defines the claim number process ðNt Þt2Rþ given by Nt :¼ n2N vð0;t
Tn and can be recovered from the latter. In the collective model of risk theory it is, in addition, assumed that the claim size sequence ðXn Þn2N is independent and identically distributed and that it is also independent of the claim arrival sequence (or the claim number process); in the present book, however, none of these assumptions is being made. Therefore, the general risk processes considered here allow to model a portfolio of risks of an insurance company in which the size of a claim may depend on the claim sizes and the occurrence times of claims incurred in the past. The book is divided into five chapters. Chapter 1 provides some generalities on stochastic processes, filtrations, stopping times and martingales in continuous time. Chapter 2 studies claim number processes with special emphasis on Poisson, Markov and mixed Poisson claim number processes, and Chapter 3 studies risk processes and their special cases resulting from the particular claim number processes considered before. Chapter 4 considers, among other topics, the problems of thinning and superposition of risk processes; these problems are of interest with regard to applications in reinsurance. Chapter 5 is devoted to the ruin problem; a particularity here is the consideration of a random premium process and an adjustment process which generalize the deterministic premium intensity and the adjustment coefficient used in classical risk theory. The dominating subject of Chapters 2 and 3 is the construction of claim number or risk processes from real- or measure-valued intensity processes, respectively. In both cases, general existence and uniqueness theorems are provided, and existence theorems are also given for the special claim number or risk processes mentioned before. The monograph by Egbert Dettweiler focusses on theoretical aspects of risk processes but is rather scarce with regard to applications. The reader is supposed to have a solid background in measure theoretic probability theory (as suggested by the author) and he would also benefit from at least some basic knowledge on martingales, Poisson processes and Poisson risk processes. Modulo its advanced mathematical level, the book is a valuable and highly welcome contribution to the theory of risk processes. Klaus D. Schmidt, Dresden