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Biostatistics Series Sample Size Calculations in Clinical Research Second Edition © 2008 by Taylor & Francis Group, LLC Biostatistics Series Editor-in-Chief Shein-Chung Chow, Ph.D. Professor Department of Biostatistics and Bioinformatics Duke University School of Medicine Durham, North Carolina, U.S.A. Series Editors Byron Jones Jen-pei Liu Senior Director Professor Statistical Research and Consulting Centre Division of Biometry (IPC 193) Department of Agronomy Pfizer Global Research and Development National Taiwan University Sandwich, Kent, UK Taipei, Taiwan Karl E. Peace Director, Karl E. Peace Center for Biostatistics Professor of Biostatistics Georgia Cancer Coalition Distinguished Cancer Scholar Georgia Southern University, Statesboro, GA © 2008 by Taylor & Francis Group, LLC Biostatistics Series Published Titles 1. Design and Analysis of Animal Studies in Pharmaceutical Development, Shein-Chung Chow and Jen-pei Liu 2. Basic Statistics and Pharmaceutical Statistical Applications, James E. De Muth 3. Design and Analysis of Bioavailability and Bioequivalence Studies, Second Edition, Revised and Expanded, Shein-Chung Chow and Jen-pei Liu 4. Meta-Analysis in Medicine and Health Policy, Dalene K. Stangl and Donald A. Berry 5. Generalized Linear Models: A Bayesian Perspective, Dipak K. Dey, Sujit K. Ghosh, and Bani K. Mallick 6. Difference Equations with Public Health Applications, Lemuel A. Moyé and Asha Seth Kapadia 7. Medical Biostatistics, Abhaya Indrayan and Sanjeev B. Sarmukaddam 8. Statistical Methods for Clinical Trials, Mark X. Norleans 9. Causal Analysis in Biomedicine and Epidemiology: Based on Minimal Sufficient Causation, Mikel Aickin 10. Statistics in Drug Research: Methodologies and Recent Developments, Shein-Chung Chow and Jun Shao 11. Sample Size Calculations in Clinical Research, Shein-Chung Chow, Jun Shao, and Hansheng Wang 12. Applied Statistical Design for the Researcher, Daryl S. Paulson 13. Advances in Clinical Trial Biostatistics, Nancy L. Geller 14. Statistics in the Pharmaceutical Industry, 3rd Edition, Ralph Buncher and Jia-Yeong Tsay 15. DNA Microarrays and Related Genomics Techniques: Design, Analysis, and Interpretation of Experiments, David B. Allsion, Grier P. Page, T. Mark Beasley, and Jode W. Edwards 16. Basic Statistics and Pharmaceutical Statistical Applications, Second Edition, James E. De Muth 17. Adaptive Design Methods in Clinical Trials, Shein-Chung Chow and Mark Chang 18. Handbook of Regression and Modeling: Applications for the Clinical and Pharmaceutical Industries, Daryl S. Paulson 19. Statistical Design and Analysis of Stability Studies, Shein-Chung Chow 20. Sample Size Calculations in Clinical Research, Second Edition, Shein-Chung Chow, Jun Shao, and Hansheng Wang © 2008 by Taylor & Francis Group, LLC Biostatistics Series Sample Size Calculations in Clinical Research Second Edition Shein-Chung Chow Duke University School of Medicine Duham, North Carolina, U.S.A. Jun Shao University of Wisconsin Madison, U.S.A. Hansheng Wang Peking University Beijing, China Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑58488‑982‑3 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Sample size calculations in clinical research / [edited by] Shein‑Chung Chow, Jun Shao, and Hansheng Wan. ‑‑ 2nd ed. p. ; cm. ‑‑ (Chapman & Hall/CRC biostatistics series ; 20) “A CRC title.” Includes bibliographical references and index. ISBN 978‑1‑58488‑982‑3 (alk. paper) 1. Clinical medicine‑‑Research‑‑Statistical methods. 2. Drug development‑‑Statistical methods. 3. Sampling (Statistics) I. Chow, Shein‑Chung, 1955‑ II. Shao, Jun. III. Wang, Hansheng, 1977‑ IV. Title. V. Series. [DNLM: 1. Sample Size. 2. Biometry‑‑methods. WA 950 S192 2008] R853.S7S33 2008 615.5072’4‑‑dc22 2007009660 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2008 by Taylor & Francis Group, LLC Series Introduction The primary objectives of the Biostatistics Book Series are to provide use- fulreferencebooksforresearchersandscientistsinacademia,industry,and government, and also to offer textbooks for undergraduate and/or gradu- ate courses in the area of biostatistics. This book series will provide comprehensive and unified presentations of statistical designs and analyses of important applications in biostatistics, such as those in biopharmaceuti- cals. A well-balanced summary will be given of current and recently de- veloped statistical methods and interpretations for both statisticians and researchers/scientists with minimal statistical knowledge who are engaged in the field of applied biostatistics. The series is committed to providing easy-to-understand, state-of-the-art references and textbooks. In each vol- ume,statisticalconceptsandmethodologieswillbeillustratedthroughreal world examples. Clinicaldevelopmentisanintegralpartofpharmaceuticaldevelopment. Sample size calculation plays an important role for providing accurate and reliable assessment of the efficacy and safety of the pharmaceutical entities under investigation. Sample size calculation is usually conducted based on a pre-study power analysis for achieving a desired power for detection of a clinically meaningful difference at a given level of significance. In practice, however,samplesizerequiredforanintendedclinicaltrialisoftenobtained using inappropriate test statistic for correct hypotheses, appropriate test statistic for wrong hypotheses, or inappropriate test statistic for wrong hypotheses. Consequently, the validity and integrity of the clinical study is questionable. For good clinical practice (GCP), it is then required that samplesizecalculationbeperformedusingappropriatestatisticsforcorrect hypotheses that will address the scientific/clinical questions regarding the pharmaceutical entities under investigation. Sample size calculation is one of the keys to the success of studies conducted at various phases of clinical development. It not only ensures the validity of the clinical trials, but also assures that the intended trials will have a desired power for correctly detecting a clinically meaningful difference of the pharmaceutical entity under study if such a difference truly exists. vii © 2008 by Taylor & Francis Group, LLC viii Series Introduction Thisbookprovidesacomprehensiveandunifiedpresentationofvarious test statistics and formulas/procedures for sample size calculation that are commonly employed at various phses of clinical development. It also provides a challenge to clinical scientists especially biostatisticians regarding current regulatory requirements, methodologies and recent developments for those issues that remain unsolved such as testing equivalence/non- inferiority in active control trials and comparing variabilities (or repro- ducibilities) in clinical development. This second edition would be beneficial to biostatisticians, medical researchers, and pharmaceutical scientists who are engaged in the areas of medical and pharmaceutical research. Shein-Chung Chow © 2008 by Taylor & Francis Group, LLC Preface Clinical development is an integral part of pharmaceutical development, which is a lengthy and costly process for providing accurate and reliable assessmentoftheefficacyandsafetyofpharmaceuticalentitiesunderinves- tigation. Samplesizecalculationplaysanimportantrole,whichensuresthe success of studies conducted at various phases of clinical development. It not only ensures the validity of the clinical trials, but also assures that the intended trials will have a desired power for correctly detecting a clinically meaningful difference of the pharmaceutical entity under study if such a difference truly exists. Sample size calculationisusually conductedthroughapre-study power analysis. The purpose is toselect a sample size such that theselectedsam- ple size will achieve a desired power for correctly detecting a pre-specified clinically meaningful difference at a given level of significance. In clinical research, however, it is not uncommon to perform sample size calculation with inappropriate test statistics for wrong hypotheses regardless of the studydesignemployed. Thisbookprovidesformulasand/orproceduresfor determinationofsamplesizerequirednotonlyfortestingequality,butalso fortestingnon-inferiority/superiority,andequivalence(similarity)basedon both untransformed (raw) data and log-transformed data under a parallel- groupdesignoracrossoverdesignwithequalorunequalratiooftreatment allocations. It provides not only a comprehensive and unified presenta- tion of various statistical procedures for sample size calculation that are commonly employed at various phases of clinical development, but also a well-balanced summary of current regulatory requirements, methodology for design and analysis in clinical research and recent developments in the area of clinical development. This book is a useful reference forclinical scientists and biostatisticians in the pharmaceutical industry, regulatory agencies, and academia, and other scientists who are in the related fields of clinical development. The primaryfocusofthisbookisonstatisticalproceduresforsamplesizecalcu- lation and/or justification that are commonly employed at various phases of clinical research and development. This book provides clear, illustrated ix © 2008 by Taylor & Francis Group, LLC x Preface explanations of how the derived formulas and/or statistical procedures for sample size calculation and/or justification can be used at various phases of clinical research and development. The book contains 15 chapters, which cover various important topics in clinical research and development, such as comparing means, comparing proportions, comparing time-to-event data, tests for independence and goodness-of-fit in contingency tables, comparing variabilities in clinical research, sample size adjustment and/or re-estimation in interim analysis, proceduresforsample size calculationforoptimalorflexible multiple-stage designs for phase II cancer trials, sample size calculation based on rank statistics,samplesizecalculationforstandard,higher-order,andreplicated crossover designs, sample size calculationfordose response studies and mi- croarraystudies, Bayesiansample sizecalculation, andsample size calcula- tioninotherareassuchasQT/QTcstudieswithtime-dependentreplicates, propensity score analysis in non-randomized studies, analysis of variance with repeated measures, quality of life studies, bridging studies, and vac- cine clinical trials. Each chapter provides a brief history or background, regulatory requirements (if any), statistical design and methods for data analysis, recent development, and related references. From Taylor & Francis, we thank Acquisitions Editor David Crubbs for providing us with the opportunity to work on this project, and the ProductionEditorforhis/heroutstandingeffortsinpreparingthisbookfor publication. WearedeeplyindebtedtoDukeUniversityandtheUniversity of Wisconsin for their support. We would like to express our gratitude to many friends from the academia, industry and government for their input, support and encouragement during the preparation of this edition. Finally, we are fully responsible for any errors remaining in this book. The views expressed are those of the authors and are not necessarily those oftheirrespectivecompanyanduniversity. Anycommentsandsuggestions that youmay have are very much appreciatedforthe preparation of future editions of this book. Shein-Chung Chow Jun Shao Hansheng Wang © 2008 by Taylor & Francis Group, LLC Contents 1 Introduction 1 1.1 Regulatory Requirement . . . . . . . . . . . . . . . . . . . . 2 1.2 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Procedures for Sample Size Calculation . . . . . . . . . . . 13 1.4 Aims and Structure of the Book . . . . . . . . . . . . . . . 21 2 Considerations Prior to Sample Size Calculation 25 2.1 Confounding and Interaction . . . . . . . . . . . . . . . . . 26 2.2 One-Sided Test Versus Two-Sided Test . . . . . . . . . . . . 28 2.3 Crossover Design Versus Parallel Design . . . . . . . . . . . 30 2.4 Subgroup/Interim Analyses . . . . . . . . . . . . . . . . . . 32 2.5 Data Transformation . . . . . . . . . . . . . . . . . . . . . . 36 2.6 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Comparing Means 49 3.1 One-Sample Design . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Two-Sample Parallel Design . . . . . . . . . . . . . . . . . . 57 3.3 Two-Sample Crossover Design . . . . . . . . . . . . . . . . . 65 3.4 Multiple-Sample One-Way ANOVA . . . . . . . . . . . . . . 70 3.5 Multiple-Sample Williams Design . . . . . . . . . . . . . . . 74 3.6 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Large Sample Tests for Proportions 83 4.1 One-Sample Design . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Two-Sample Parallel Design . . . . . . . . . . . . . . . . . . 89 4.3 Two-Sample Crossover Design . . . . . . . . . . . . . . . . . 95 4.4 One-Way Analysis of Variance . . . . . . . . . . . . . . . . 99 4.5 Williams Design . . . . . . . . . . . . . . . . . . . . . . . . 101 4.6 Relative Risk—Parallel Design . . . . . . . . . . . . . . . . 104 4.7 Relative Risk—Crossover Design . . . . . . . . . . . . . . . 109 4.8 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 111 xi © 2008 by Taylor & Francis Group, LLC