Table Of Content

Sample Size Calculations in Clinical Research Third Edition 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 Sample Size Calculations in Clinical Research Third Edition Shein-Chung Chow, Jun Shao, Hansheng Wang, and Yuliya Lokhnygina CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-74098-3 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, 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 pho- tocopying, 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 Names: Chow, Shein-Chung, 1955- editor. | Shao, Jun (Statistician) editor. | Wang, Hansheng, 1977- editor. | Lokhnygina, Yuliya, editor. Title: Sample size calculations in clinical research / [edited by] Shein-Chung Chow, Jun Shao, Hansheng Wang, Yuliya Lokhnygina. Description: Third edition. | Boca Raton : Taylor & Francis, 2017. | Series: Chapman & Hall/CRC biostatistics series | “A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.” | Includes bibliographical references. Identifiers: LCCN 2017011239 | ISBN 9781138740983 (hardback) Subjects: LCSH: Clinical medicine--Research--Statistical methods. | Drug development--Statistical methods. | Sampling (Statistics) Classification: LCC R853.S7 S33 2017 | DDC 610.72/7--dc23 LC record available at https://lccn.loc.gov/2017011239 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface ..........................................................................................................................................xvii 1. Introduction .............................................................................................................................1 1.1 Regulatory Requirement ..............................................................................................2 1.1.1 Adequate and Well-Controlled Clinical Trials ............................................2 1.1.2 Substantial Evidence .......................................................................................3 1.1.3 Why At Least Two Studies? ............................................................................3 1.1.4 Substantial Evidence with a Single Trial ......................................................4 1.1.5 Sample Size .......................................................................................................5 1.2 Basic Considerations .....................................................................................................5 1.2.1 Study Objectives...............................................................................................6 1.2.2 Study Design ....................................................................................................6 1.2.3 Hypotheses .......................................................................................................7 1.2.3.1 Test for Equality ................................................................................7 1.2.3.2 Test for Noninferiority .....................................................................8 1.2.3.3 Test for Superiority ...........................................................................8 1.2.3.4 Test for Equivalence .........................................................................8 1.2.3.5 Relationship among Noninferiority, Superiority, and Equivalence ..........................................................9 1.2.4 Primary Study Endpoint .................................................................................9 1.2.5 Clinically Meaningful Difference ...............................................................10 1.3 Procedures for Sample Size Calculation ..................................................................11 1.3.1 Type I and Type II Errors ..............................................................................11 1.3.2 Precision Analysis..........................................................................................12 1.3.3 Power Analysis ...............................................................................................13 1.3.4 Probability Assessment .................................................................................15 1.3.5 Reproducibility Probability ..........................................................................16 1.3.6 Sample Size Reestimation without Unblinding ........................................18 1.4 Aims and Structure of this Book ..............................................................................18 1.4.1 Aim of this Book ............................................................................................18 1.4.2 Structure of this Book ...................................................................................19 2. Considerations Prior to Sample Size Calculation ..........................................................21 2.1 Confounding and Interaction ....................................................................................21 2.1.1 Confounding ..................................................................................................21 2.1.2 Interaction .......................................................................................................22 2.1.3 Remark ............................................................................................................22 2.2 One-Sided Test versus Two-Sided Test ....................................................................23 2.2.1 Remark ............................................................................................................24 2.3 Crossover Design versus Parallel Design ................................................................24 2.3.1 Intersubject and Intrasubject Variabilities .................................................25 2.3.2 Crossover Design ...........................................................................................25 2.3.3 Parallel Design ...............................................................................................26 2.3.4 Remark ............................................................................................................26 v vi Contents 2.4 Subgroup/Interim Analyses ......................................................................................26 2.4.1 Group Sequential Boundaries ......................................................................27 2.4.2 Alpha Spending Function.............................................................................28 2.5 Data Transformation ...................................................................................................29 2.5.1 Remark ............................................................................................................31 2.6 Practical Issues ............................................................................................................31 2.6.1 Unequal Treatment Allocation .....................................................................31 2.6.2 Adjustment for Dropouts or Covariates .....................................................32 2.6.3 Mixed-Up Randomization Schedules .........................................................33 2.6.4 Treatment or Center Imbalance ...................................................................35 2.6.5 Multiplicity .....................................................................................................37 2.6.6 Multiple-Stage Design for Early Stopping..................................................37 2.6.7 Rare Incidence Rate .......................................................................................38 3. Comparing Means ................................................................................................................39 3.1 One-Sample Design ....................................................................................................39 3.1.1 Test for Equality .............................................................................................40 3.1.2 Test for Noninferiority/Superiority.............................................................42 3.1.3 Test for Equivalence .......................................................................................44 3.1.4 An Example ....................................................................................................45 3.1.4.1 Test for Equality ..............................................................................45 3.1.4.2 Test for Noninferiority ...................................................................46 3.1.4.3 Test for Equivalence .......................................................................46 3.2 Two-Sample Parallel Design ......................................................................................47 3.2.1 Test for Equality .............................................................................................47 3.2.2 Test for Noninferority/Superiority ..............................................................50 3.2.3 Test for Equivalence .......................................................................................51 3.2.4 An Example ....................................................................................................52 3.2.4.1 Test for Equality ..............................................................................53 3.2.4.2 Test for Noninferiority ...................................................................53 3.2.4.3 Test for Equivalence .......................................................................54 3.2.5 Remarks ...........................................................................................................54 3.3 Two-Sample Crossover Design .................................................................................54 3.3.1 Test for Equality .............................................................................................55 3.3.2 Test for Noninferiority/Superiority.............................................................56 3.3.3 Test for Equivalence .......................................................................................57 3.3.4 An Example ....................................................................................................58 3.3.4.1 Therapeutic Equivalence ...............................................................58 3.3.4.2 Noninferiority .................................................................................58 3.3.5 Remarks ...........................................................................................................59 3.4 Multiple-Sample One-Way ANOVA .........................................................................59 3.4.1 Pairwise Comparison ....................................................................................60 3.4.2 Simultaneous Comparison ...........................................................................61 3.4.3 An Example ....................................................................................................61 3.4.4 Remarks ...........................................................................................................62 3.5 Multiple-Sample Williams Design ...........................................................................63 3.5.1 Test for Equality .............................................................................................64 3.5.2 Test for Noninferiority/Superiority.............................................................65 Contents vii 3.5.3 Test for Equivalence .......................................................................................65 3.5.4 An Example ....................................................................................................66 3.6 Practical Issues ............................................................................................................66 3.6.1 One-Sided versus Two-Sided Test ...............................................................67 3.6.2 Parallel Design versus Crossover Design ...................................................67 3.6.3 Sensitivity Analysis .......................................................................................68 4. Large Sample Tests for Proportions ..................................................................................71 4.1 One-Sample Design ....................................................................................................71 4.1.1 Test for Equality .............................................................................................72 4.1.2 Test for Noninferiority/Superiority.............................................................73 4.1.3 Test for Equivalence .......................................................................................74 4.1.4 An Example ....................................................................................................74 4.1.4.1 Test for Equality ..............................................................................75 4.1.4.2 Test for Noninferiority ...................................................................75 4.1.4.3 Test for Equivalence .......................................................................75 4.1.5 Remarks ...........................................................................................................75 4.2 Two-Sample Parallel Design ......................................................................................76 4.2.1 Test for Equality .............................................................................................77 4.2.2 Test for Noninferiority/Superiority.............................................................77 4.2.3 Test for Equivalence .......................................................................................78 4.2.4 An Example ....................................................................................................79 4.2.4.1 Test for Equality ..............................................................................80 4.2.4.2 Test for Noninferiority ...................................................................80 4.2.4.3 Test for Superiority .........................................................................80 4.2.4.4 Test for Equivalence .......................................................................81 4.2.5 Remarks ...........................................................................................................81 4.3 Two-Sample Crossover Design .................................................................................82 4.3.1 Test for Equality .............................................................................................83 4.3.2 Test for Noninferiority/Superiority.............................................................84 4.3.3 Test for Equivalence .......................................................................................84 4.3.4 An Example ....................................................................................................85 4.3.4.1 Test for Equality ..............................................................................85 4.3.4.2 Test for Noninferiority ...................................................................86 4.3.4.3 Test for Equivalence .......................................................................86 4.3.5 Remarks ...........................................................................................................86 4.4 One-Way Analysis of Variance..................................................................................86 4.4.1 Pairwise Comparison ....................................................................................87 4.4.2 An Example ....................................................................................................87 4.4.3 Remarks ...........................................................................................................88 4.5 Williams Design ..........................................................................................................88 4.5.1 Test for Equality .............................................................................................89 4.5.2 Test for Noninferiority/Superiority.............................................................89 4.5.3 Test for Equivalence .......................................................................................90 4.5.4 An Example ....................................................................................................91 4.5.4.1 Test for Equality ..............................................................................91 4.5.4.2 Test for Superiority .........................................................................92 4.5.4.3 Test for Equivalence .......................................................................92 viii Contents 4.6 Relative Risk—Parallel Design ..................................................................................92 4.6.1 Test for Equality .............................................................................................93 4.6.2 Test for Noninferiority/Superiority.............................................................94 4.6.3 Test for Equivalence .......................................................................................94 4.6.4 An Example ....................................................................................................95 4.6.4.1 Test for Equality ..............................................................................95 4.6.4.2 Test for Superiority .........................................................................96 4.6.4.3 Test for Equivalence .......................................................................96 4.7 Relative Risk—Crossover Design .............................................................................96 4.7.1 Test for Equality .............................................................................................97 4.7.2 Test for Noninferiority/Superiority.............................................................97 4.7.3 Test for Equivalence .......................................................................................98 4.8 Practical Issues ............................................................................................................99 4.8.1 Exact and Asymptotic Tests .........................................................................99 4.8.2 Variance Estimates .........................................................................................99 4.8.3 Stratified Analysis ........................................................................................101 4.8.4 Equivalence Test for More Than Two Proportions .................................102 5. Exact Tests for Proportions ...............................................................................................103 5.1 Binomial Test .............................................................................................................103 5.1.1 The Procedure ..............................................................................................103 5.1.2 Remarks .........................................................................................................104 5.1.3 An Example ..................................................................................................104 5.2 Negative Binomial .....................................................................................................105 5.2.1 Negative Binomial Distribution .................................................................106 5.2.2 Sample Size Requirement ...........................................................................107 5.3 Fisher’s Exact Test ......................................................................................................108 5.3.1 The Procedure ..............................................................................................109 5.3.2 Remarks .........................................................................................................109 5.3.3 An Example ..................................................................................................109 5.4 Optimal Multiple-Stage Designs for Single-Arm Trials ......................................111 5.4.1 Optimal Two-Stage Designs .......................................................................111 5.4.2 Flexible Two-Stage Designs ........................................................................113 5.4.3 Optimal Three-Stage Designs ....................................................................114 5.5 Flexible Designs for Multiple-Arm Trials ..............................................................122 5.6 Remarks ......................................................................................................................129 6. Tests for Goodness-of-Fit and Contingency Tables .....................................................131 6.1 Tests for Goodness-of-Fit .........................................................................................131 6.1.1 Pearson’s Test ................................................................................................131 6.1.2 An Example ..................................................................................................132 6.2 Test for Independence: Single Stratum ..................................................................133 6.2.1 Pearson’s Test ................................................................................................134 6.2.2 Likelihood Ratio Test ..................................................................................135 6.2.3 An Example ..................................................................................................136 6.3 Test for Independence: Multiple Strata ..................................................................136 6.3.1 Cochran–Mantel–Haenszel Test ................................................................137 6.3.2 An Example ..................................................................................................138 6.4 Test for Categorical Shift ..........................................................................................138 6.4.1 McNemar’s Test ............................................................................................139 Contents ix 6.4.2 Stuart–Maxwell Test ....................................................................................141 6.4.3 Examples .......................................................................................................142 6.4.3.1 McNemar’s Test ............................................................................142 6.4.3.2 Stuart–Maxwell Test ....................................................................142 6.5 Carryover Effect Test ................................................................................................143 6.5.1 Test Procedure ..............................................................................................143 6.5.2 An Example ..................................................................................................145 6.6 Practical Issues ..........................................................................................................145 6.6.1 Local Alternative versus Fixed Alternative .............................................145 6.6.2 Random versus Fixed Marginal Total ......................................................146 6.6.3 r × c versus p × r × c ...................................................................................146 7. Comparing Time-to-Event Data ......................................................................................147 7.1 Basic Concepts ...........................................................................................................147 7.1.1 Survival Function ........................................................................................148 7.1.2 Median Survival Time ................................................................................148 7.1.3 Hazard Function ..........................................................................................148 7.1.4 An Example ..................................................................................................149 7.2 Exponential Model ....................................................................................................150 7.2.1 Test for Equality ...........................................................................................152 7.2.2 Test for Noninferiority/Superiority...........................................................153 7.2.3 Test for Equivalence .....................................................................................154 7.2.4 An Example ..................................................................................................155 7.2.4.1 Test for Equality ............................................................................155 7.2.4.2 Test for Superiority .......................................................................156 7.2.4.3 Test for Equivalence .....................................................................156 7.2.5 Remarks .........................................................................................................156 7.2.5.1 Unconditional versus Conditional .............................................156 7.2.5.2 Losses to Follow-Up, Dropout, and Noncompliance ..............157 7.3 Cox’s Proportional Hazards Model ........................................................................158 7.3.1 Test for Equality ...........................................................................................159 7.3.2 Test for Noninferiority/Superiority...........................................................161 7.3.3 Test for Equivalence .....................................................................................162 7.3.4 An Example ..................................................................................................162 7.3.4.1 Test for Equality ............................................................................162 7.3.4.2 Test for Superiority .......................................................................163 7.3.4.3 Test for Equivalence .....................................................................163 7.4 Weighted Log-Rank Test ..........................................................................................163 7.4.1 Tarone–Ware Test .........................................................................................163 7.4.2 An Example ..................................................................................................165 7.5 Practical Issues ..........................................................................................................168 7.5.1 Binomial versus Time to Event ..................................................................168 7.5.2 Local Alternative versus Fixed Alternative .............................................168 7.5.3 One-Sample versus Historical Control .....................................................168 8. Group Sequential Methods ..............................................................................................169 8.1 Pocock’s Test ..............................................................................................................169 8.1.1 The Procedure ..............................................................................................169 8.1.2 An Example ..................................................................................................171