Probability Theory With Applications

This book is a revised and expanded edition of a successful graduate and reference text. The material in the book is designed for a standard graduate course on probability theory, including some important applications. This new edition contains a detailed treatment of the core area of probability, and both structural and limit results are presented in full detail. Compared to the first edition, the material and presentation are better highlighted with several (small and large) alterations made to each chapter. Key features of the book include:

- Indicating the need for abstract theory even in applications and showing the inadequacy of existing results for certain apparently simple real-world problems

- Attempting to deal with the existence problems for various classes of random families that figure in the main results of the subject

- Providing a treatment of conditional expectations and of conditional probabilities that is more complete than in other existing textbooks

Since this is a textbook, essentially all proofs are given in complete detail (even at the risk of repetition), and some key results are given multiple proofs when each argument has something to contribute.

Doody Review Services

Reviewer: Michael J. Schrift, DO(University of Illinois at Chicago College of Medicine)
Description: One of the ethical principles, beneficence, as described in the Belmont Report and codified in The Common Rule ( 45 C.F.R.Part 46), is an obligation to do no harm and to "maximize possible benefits and minimize possible harms" to the individual research participant. In research this implies that "one should not injure one person regardless of the benefits that might come to others." At times, though, one cannot know that something is harmful until it is tried and in the process of trying, or experimentation, persons may be exposed to risk of harm. Investigators and their institutions have to plan to maximize benefits and minimize risks and since it is not possible to remove all risks from the research process, the research needs to have continuing review and monitoring of data to ensure that the study does not continue after the emergence of reliable evidence of reduced efficacy and/or safety. That's where this essential book comes in: what are the statistics and underlying mathematics of the so-called "stopping rules" that data safety monitoring committees employ? Written by a physician-scientist who also happens to be a biostatistician, this book should be required reading for all principal investigators, data safety monitoring committee members, and research ethics committee members.
Purpose: The author correctly notes that clinical investigators with little quantitative background frequently have difficulty communicating with biostatisticians and research methodologists. This book is intended to bring clinical investigators up to speed and toremedy their statistical/mathematical challenges. As he notes, "If you know nothing about monitoring guidelines in clinical trials, then this book is for you."
Audience: The intended audience includes clinical investigators and data monitoring committee members.
Features: The first of the book's eight chapters is an overview of the history of monitoring procedures, understanding randomized blinded controlled trials, and the ethical concerns of this methodology. Chapter 2 covers basic statistics in research including the principles of probability. Chapter 3 summarizes the various monitoring procedures that are used in clinical trials. Chapters 4 and 5 focus on path analysis and group sequential analysis respectively. Chapter 6 helps the reader understand the circumstances in which a clinical trial may be stopped early for a beneficial finding based on a conditional power approach. The monitoring procedures to identify the harmful effects of the tested intervention are detailed in Chapter 7. Lastly, Bayesian statistics are introduced in chapter 8 in the use of monitoring procedures. The appendixes were difficult for me to understand mathematically, but for those so inclined, they include such topics as boundary values for normal mean, conditional Brownian motion, boundary values and conditional power, supporting Bayesian computations, and standard normal probabilities. The index was useful.
Assessment: Written by a clinical researcher/physician/biostatistician, this book is an excellent way to learn the statistics and mathematics underlying data safety monitoring. I highly recommend it.