Michael A Proschan K.K Gordon Lan Janet Turk Wittes A Unified Approach ~Springer Statistics for Biology and Health Series Editors M Gail, K Krickeberg, J Samet, A Tsiatis, W Wong Statistics for Biology and Health Borchers/Buckland/Zucchini: Estimating Animal Abundance: Closed Populations Burzykowski/Molenberghs/Buyse: The Evaluation of Surrogate Endpoints Everitt/Rabe-Hesketh: Analyzing Medical Data Using S-PLUS Evens/Grant: Statistical Methods in Bioinformatics: An Introduction Gentleman/Carey/Huber/Izirarry/Dudoit: Bioinformatics and Computational Biology Solutions Using R and Bioconductor Hougaard: Analysis of Multivariate Survival Data Keyfitz/Caswell: Applied Mathematical Demography 3rd ed Klein/Moeschberger: Survival Analysis: Techniques for Censored and Truncated Data, 2nd ed Kleinbaum: Survival Analysis: A Self-Learning Text, 2nd ed Kleinbaum/Klein: Logistic Regression: A Self-Learning Text, 2nd ed Lange: Mathematical and Statistical Methods for Genetic Analysis, 2nd ed Manton/Singer/Suzman: Forecasting the Health of Elderly Populations Martinussen/Scheike: Dynamic Regression Models for Survival Data Moyé: Multiple Analyses in Clinical Trials: Fundamentals for Investigators Nielsen: Statistical Methods in Molecular Evolution Parmigiani/Garrett/Irizarry/Zeger: The Analysis of Gene Expression Data: Methods and Software Proschan/Lan/Wittes: Statistical Monitoring of Clinical Trials: A Unified Approach Salsburg: The Use of Restricted Significance Tests in Clinical Trials Simon/Korn/McShane/Radmacher/Wright/Zhao: Design and Analysis of DNA Microarray Investigations Sorensen/Gianola: Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics Stallard/Manton/Cohen: Forecasting Product Liability Claims: Epidemiology and Modeling in the Manville Asbestos Case Therneau/Grambsch: Modeling Survival Data: Extending the Cox Model Ting: Dose Finding in Drug Development Vittinghoff/Glidden/Shiboski/McCulloch: Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models Zhang/Singer: Recursive Partitioning in the Health Sciences Michael A Proschan K.K Gordan Lan Janet Turk Wittes Statistical Monitoring of Clinical Trials A Unified Approach Michael A Proschan Biostatistics Research Branch, NIAID Bethesda, MD 20892 USA ProschaM@mail.nih.gov K.K Gordon Lan Johnson & Johnson Raritan, NJ 08869 glan@prdus.jnj.com Janet Turk Wittes Statistics Collaborative Washington, DC 20036 USA Series Editors M Gail National Cancer Institute Rockville, MD 20892 USA K Krickeberg Le Chatelet F-63270 Manglieu France A Tsiatis Department of Statistics North Carolina State University Raleigh, NC 27695 USA J Sarnet Department of Epidemiology School of Public Health Johns Hopkins University 615 Wolfe Street Baltimore, MD 21205-2103 USA W Wong Department of Statistics Stanford University Stanford, CA 94305-4065 USA ISBN-13: 978-0-387-30059-7 Library of Congress Control Number: 2005939187 ©2006 Springer Science+Business Media, LLC All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed in the United States of America Printed on acid-free paper springer.com To the National Heart, Lung, and Blood Institute, which allowed biostatisticians to learn, to contribute, and to flourish Preface We statisticians, especially those of us who work with randomized clinical trials within a regulatory environment, typically operate within the constraints of careful prespecification of analyses We worry lest ad hoc response to data that we see affect the integrity of our inference When we are involved in interim monitoring of clinical trials, however, we must have the latitude to respond with intellectual agility to unexpected findings Perhaps that very mixture of careful prespecification—to protect the scientific integrity of the study—and data-driven modifications—to protect the interest of the participants in the trial—explains why so many of us enjoy the challenge of interim monitoring of clinical trials Of course we must, even in that context, carefully describe the analyses we plan to conduct and the nature of the inference to which various outcomes will lead us; on the other hand, if our analyses lead to a premature—in contrast to an early—stopping of the clinical trial, there is no putting the train back on the track The past half century has seen an explosion of methods for statistical monitoring of ongoing clinical trials with the view toward stopping the trial if the interim data show unequivocal evidence of benefit, worrisome evidence of harm, or a strong indication that the completed trial will likely show equivocal results The methods appear to come from a variety of different underlying statistical frameworks In this book we stress that a common mathematical unifying formulation—Brownian motion—underlies most of the basic methods We aim to show when and how the statistician can use that framework and when the statistician must modify it to produce valid inference We hope that our presentation will help the reader understand the relationships among commonly used methods of group-sequential analysis, conditional power, and futility analysis The level of the book is appropriate to graduate students in biostatistics and to statisticians involved in clinical trials One of our goals is to provide biostatisticians with tools not only to perform the necessary calculations but to be able to explain the methodology to our clinical colleagues When the process of statistical decision-making becomes too opaque, the clinicians with whom we work tune out and leave important parts of the discussion to the statisticians VIII Preface We believe the stark separation of clinical and biostatistical thinking cannot be healthy to intelligent, thoughtful decision-making, especially when it occurs in the middle of a trial The book represents our distillation of years of collaboration with many colleagues, both from the clinical and biostatistical worlds All three of us spent formative years at the National Heart, Lung, and Blood Institute where Claude Lenfant, Director, encouraged the growth of biostatistics We learned much from the many lively discussions we had there with coworkers as we grappled collectively with issues related to ongoing monitoring of clinical trials Especially useful was the opportunity we had to attend as many Data Safety Monitoring Board meetings as we desired; those experiences formed the basis for our view of data monitoring We hope that the next generation of biostatisticians will find themselves in an organization that recognizes the value of training by apprenticeship We particularly want to acknowledge the insights we gained from other members of the biostatistics group—Kent Bailey, Erica Brittain, Dave DeMets, Dean Follmann, Max Halperin, Marian Fisher, Nancy Geller, Ed Lakatos, Joel Verter, Margaret Wu, and David Zucker Physician colleagues who, while they were at NHBLI and in later years, have been especially influential have been the two Bills (William Friedewald and William Harlan), as well as Larry Friedman, Curt Furberg (who pointed out to us the distinction between premature and early stopping of trials), Gene Passamani, and Salim Yusuf One of us (it is not hard to guess which one) is especially indebted to insights gained from Robert Wittes, who for four decades has provided thoughtful balanced judgment to a variety of issues related to clinical trials (and many other topics) And then there have been so many others with whom we have had fruitful discussions about monitoring trials over the years Of particular note are Jonas Ellenberg, Susan Ellenberg, Tom Fleming, Genell Knatterud, and Scott Emerson Dave DeMets has kindly agreed to maintain a constant free version of his software so that readers of this book would have access to it We thank Mary Foulkes, Tony Lachenbruch, Jon Turk, and Joe Shih for their helpful comments on earlier versions of the book Their suggestions helped strengthen the presentations It goes without saying that any errors or lapses of clarity remaining are our fault Without further ado, we stop this preface early Michael A Proschan K.K Gordon Lan Janet Turk Wittes Washington D.C 3/2006 Contents Introduction A General Framework 2.1 Hypothesis Testing: The Null Distribution of Test Statistics Over Time 2.1.1 Continuous Outcomes 2.1.2 Dichotomous Outcomes 2.1.3 Survival Outcomes 2.1.4 Summary of Sums 2.2 An Estimation Perspective 2.2.1 Information 2.2.2 Summary of Treatment Effect Estimators 2.3 Connection Between Estimators, Sums, Z-Scores, and Brownian Motion 2.4 Maximum Likelihood Estimation 2.5 Other Settings Leading to E-Processes and Brownian Motion 2.5.1 Minimum Variance Unbiased Estimators 2.5.2 Complete Sufficient Statistics 2.6 The Normal Linear and Mixed Models 2.6.1 The Linear Model 2.6.2 The Mixed Model 2.7 When Is Brownian Motion Not Appropriate? 2.8 Summary 2.9 Appendix 2.9.1 Asymptotic Validity of Using Estimated Standard Errors 2.9.2 Proof of Result 2.1 2.9.3 Proof that for the Logrank Test, Di = Oi − Ei Are Uncorrelated Under H0 2.9.4 A Rigorous Justification of Brownian Motion with Drift: Local Alternatives 10 10 14 15 17 18 18 21 21 24 28 28 29 30 30 31 36 38 39 39 40 41 41 X Contents 2.9.5 Basu’s Theorem 42 Power: Conditional, Unconditional, and Predictive 3.1 Unconditional Power 3.2 Conditional Power for Futility 3.3 Varied Uses of Conditional Power 3.4 Properties of Conditional Power 3.5 A Bayesian Alternative: Predictive Power 3.6 Summary 3.7 Appendix 3.7.1 Proof of Result 3.1 3.7.2 Formula for corr{B(t), θ} and var{θ | B(t) = b} 3.7.3 Simplification of Formula (3.8) 43 43 45 53 57 60 63 64 64 65 66 Historical Monitoring Boundaries 4.1 How Bad Can the Naive Approach Be? 4.2 The Pocock Procedure 4.3 The Haybittle Procedure and Variants 4.4 The O’Brien-Fleming Procedure 4.5 A Comparison of the Pocock and O’Brien-Fleming Boundaries 4.6 Effect of Monitoring on Power 4.7 Appendix: Computation of Boundaries Using Numerical Integration 67 67 69 69 71 72 75 Spending Functions 5.1 Upper Boundaries 5.1.1 Using a Different Time Scale for Spending 5.1.2 Data-Driven Looks 5.2 Upper and Lower Boundaries 5.3 Summary 5.4 Appendix 5.4.1 Proof of Result 5.1 5.4.2 Proof of Result 5.2 5.4.3 An S-Plus or R Program to Compute Boundaries 81 81 87 89 90 92 92 92 93 93 Practical Survival Monitoring 99 6.1 Introduction 99 6.2 Survival Trials with Staggered Entry 99 6.3 Stochastic Process Formulation and Linear Trends 101 6.4 A Real Example 102 6.5 Nonlinear Trends of the Statistics: Analogy with Monitoring a t-Test 103 6.6 Considerations for Early Termination 104 6.7 The Information Fraction with Survival Data 105 77 ... for ti = I(τi )/ I( 1) ≤ tj = I(τj )/ I( 1), cov{B(ti ), B(tj )} = = = = cov[SI(τi ) /{I( 1)} 1/2, SI(τj ) /{I( 1)} 1/2] {I( 1)} −1 cov(SI(τi ) , SI(τj ) ) {I( 1)} −1 I(τi ) ti The mean of B(t) is different... value of the z-score at the end of the trial, as follows: ˆ )/ {I( 1)} 1/2 E to B : B(t) = I(τ )? ?(? ? S to B : B(t) = S(τ )/ {I( 1)} 1/2 Z to B : B(t) = t1/2Z(t) Sum S(τ ) ˆ ) I(τ )? ?(? ? ˆ ) {I(τ )} 1/2 ? ?(? ?... 0= = = = ˆ j ), ? ?(? ? ˆ i )} 1/I(τj ) − cov{? ?(? ? ˆ j ), ? ?(? ? ˆ i )} ˆ j )} − cov{? ?(? ? var{? ?(? ? ˆ ˆ ˆ j ), ? ?(? ? ˆ i )} cov{? ?(? ?j ), ? ?(? ?j )} − cov{? ?(? ? ˆ ˆ ˆ cov{? ?(? ?j ), ? ?(? ?j ) − ? ?(? ?i )} (2 .2 0) Thus, E3 is