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Sample Size Calculations in Clinical Research Third Edition

Shein-Chung Chow Jun Shao Hansheng Wang Yuliya Lokhnygina

A CHAPMAN & HALL BOOK

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

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

A CHAPMAN & HALL BOOK

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 First issued in paperback 2020 © 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 ISBN-13: 978-1-138-74098-3 (hbk) ISBN-13: 978-0-367-65776-5 (pbk) 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 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 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

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2.4 2.5 2.6

Subgroup/Interim Analyses...................................................................................... 26 2.4.1 Group Sequential Boundaries ...................................................................... 27 2.4.2 Alpha Spending Function............................................................................. 28 Data Transformation................................................................................................... 29 2.5.1 Remark ............................................................................................................ 31 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

3.6

vii

3.5.3 Test for Equivalence .......................................................................................65 3.5.4 An Example .................................................................................................... 66 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

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Contents

4.6

4.7

4.8

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 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 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

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Contents

6.4.2 6.4.3

6.5 6.6

Stuart–Maxwell Test .................................................................................... 141 Examples ....................................................................................................... 142 6.4.3.1 McNemar’s Test ............................................................................ 142 6.4.3.2 Stuart–Maxwell Test .................................................................... 142 Carryover Effect Test ................................................................................................ 143 6.5.1 Test Procedure .............................................................................................. 143 6.5.2 An Example .................................................................................................. 145 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

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Contents

8.2

O’Brien and Fleming’s Test ...................................................................................... 172 8.2.1 The Procedure .............................................................................................. 173 8.2.2 An Example .................................................................................................. 174 8.3 Wang and Tsiatis’ Test .............................................................................................. 175 8.3.1 The Procedure .............................................................................................. 175 8.3.2 An Example .................................................................................................. 175 8.4 Inner Wedge Test....................................................................................................... 177 8.4.1 The Procedure .............................................................................................. 177 8.4.2 An Example .................................................................................................. 178 8.5 Binary Variables ........................................................................................................ 180 8.5.1 The Procedure .............................................................................................. 180 8.5.2 An Example .................................................................................................. 180 8.6 Time-to-Event Data ................................................................................................... 181 8.6.1 The Procedure .............................................................................................. 181 8.6.2 An Example .................................................................................................. 182 8.7 Alpha-Spending Function ....................................................................................... 183 8.8 Sample Size Reestimation ........................................................................................ 185 8.8.1 The Procedure .............................................................................................. 185 8.8.2 An Example .................................................................................................. 186 8.9 Conditional Power .................................................................................................... 187 8.9.1 Comparing Means ....................................................................................... 187 8.9.2 Comparing Proportions .............................................................................. 188 8.10 Practical Issues .......................................................................................................... 189 9. Comparing Variabilities.................................................................................................... 191 9.1 Comparing Intrasubject Variabilities ..................................................................... 191 9.1.1 Parallel Design with Replicates ................................................................. 192 9.1.1.1 Test for Equality............................................................................ 192 9.1.1.2 Test for Noninferiority/Superiority ........................................... 193 9.1.1.3 Test for Similarity ......................................................................... 194 9.1.1.4 An Example ................................................................................... 195 9.1.2 Replicated Crossover Design ..................................................................... 195 9.1.2.1 Test for Equality............................................................................ 197 9.1.2.2 Test for Noninferiority/Superiority ........................................... 198 9.1.2.3 Test for Similarity ......................................................................... 198 9.1.2.4 An Example ................................................................................... 199 9.2 Comparing Intrasubject CVs ................................................................................... 200 9.2.1 Simple Random Effects Model ................................................................... 200 9.2.1.1 Test for Equality............................................................................ 201 9.2.1.2 Test for Noninferiority/Superiority ........................................... 202 9.2.1.3 Test for Similarity ......................................................................... 203 9.2.1.4 An Example ................................................................................... 203 9.2.2 Conditional Random Effects Model .......................................................... 204 9.2.2.1 Test for Equality............................................................................ 206 9.2.2.2 Test for Noninferiority/Superiority ........................................... 206 9.2.2.3 Test for Similarity ......................................................................... 207 9.2.2.4 An Example ................................................................................... 208 9.2.2.5 Remarks ......................................................................................... 208

Contents

9.3

9.4

9.5

xi

Comparing Intersubject Variabilities ..................................................................... 209 9.3.1 Parallel Design with Replicates ................................................................. 209 9.3.1.1 Test for Equality............................................................................ 210 9.3.1.2 Test for Noninferiority/Superiority ........................................... 211 9.3.1.3 An Example ................................................................................... 212 9.3.2 Replicated Crossover Design ..................................................................... 213 9.3.2.1 Test for Equality............................................................................ 213 9.3.2.2 Test for Noninferiority/Superiority ........................................... 215 9.3.2.3 An Example ................................................................................... 216 Comparing Total Variabilities ................................................................................. 217 9.4.1 Parallel Designs without Replicates .......................................................... 217 9.4.1.1 Test for Equality............................................................................ 218 9.4.1.2 Test for Noninferiority/Superiority ........................................... 219 9.4.1.3 Test for Similarity ......................................................................... 219 9.4.1.4 An Example ................................................................................... 220 9.4.2 Parallel Design with Replicates ................................................................. 221 9.4.2.1 Test for Equality............................................................................ 221 9.4.2.2 Test for Noninferiority/Superiority ...........................................222 9.4.2.3 An Example ...................................................................................223 9.4.3 The Standard 2 × 2 Crossover Design ...................................................... 224 9.4.3.1 Test for Equality............................................................................ 224 9.4.3.2 Test for Noninferiority/Superiority ........................................... 226 9.4.3.3 An Example ................................................................................... 227 9.4.4 Replicated 2 × 2m Crossover Design ........................................................ 227 9.4.4.1 Test for Equality............................................................................ 227 9.4.4.2 Test for Noninferiority/Superiority ........................................... 229 9.4.4.3 An Example ................................................................................... 230 Practical Issues .......................................................................................................... 231

10. Bioequivalence Testing...................................................................................................... 233 10.1 Bioequivalence Criteria ............................................................................................234 10.2 Average Bioequivalence ........................................................................................... 235 10.2.1 An Example .................................................................................................. 237 10.3 Population Bioequivalence ...................................................................................... 238 10.3.1 An Example .................................................................................................. 240 10.4 Individual Bioequivalence ....................................................................................... 242 10.4.1 An Example .................................................................................................. 246 10.5 In Vitro Bioequivalence ............................................................................................. 248 10.5.1 An Example .................................................................................................. 252 10.6 Sample Size Requirement for Analytical Similarity Assessment of Biosimilar Products .................................................................................................. 253 10.6.1 FDA’s Tiered Approach ............................................................................... 253 10.6.2 Sample Size Requirement ........................................................................... 253 11. Dose–Response Studies .................................................................................................... 257 11.1 Continuous Response ............................................................................................... 257 11.1.1 Linear Contrast Test..................................................................................... 258 11.2 Binary Response........................................................................................................ 261

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11.3 11.4 11.5 11.6 11.7

Time-to-Event Endpoint ........................................................................................... 262 Williams’ Test for Minimum Effective Dose......................................................... 264 Cochran–Armitage’s Test for Trend ....................................................................... 268 Dose Escalation Trials .............................................................................................. 271 11.6.1 A + B Escalation Design without Dose De-Escalation ........................... 272 11.6.2 A + B Escalation Design with Dose De-Escalation................................. 274 Concluding Remarks ................................................................................................ 276

12. Microarray Studies ............................................................................................................. 277 12.1 Literature Review ...................................................................................................... 277 12.2 FDR Control ............................................................................................................... 278 12.2.1 Model and Assumptions ............................................................................ 278 12.2.2 Sample Size Calculation.............................................................................. 280 12.3 FWER Control ............................................................................................................ 288 12.3.1 Multiple Testing Procedures ...................................................................... 288 12.3.2 Sample Size Calculation.............................................................................. 290 12.3.3 Leukemia Example ...................................................................................... 293 12.4 Concluding Remarks ................................................................................................ 295 13. Bayesian Sample Size Calculation .................................................................................. 297 13.1 Posterior Credible Interval Approach .................................................................... 298 13.1.1 Three Selection Criteria .............................................................................. 298 13.1.1.1 Average Coverage Criterion ........................................................ 299 13.1.1.2 Average Length Criterion............................................................ 299 13.1.1.3 Worst Outcome Criterion ............................................................300 13.1.2 One Sample ...................................................................................................300 13.1.2.1 Known Precision ..........................................................................300 13.1.2.2 Unknown Precision ..................................................................... 301 13.1.2.3 Mixed Bayesian-Likelihood ........................................................ 302 13.1.3 Two-Sample with Common Precision ......................................................305 13.1.3.1 Known Common Precision .........................................................306 13.1.3.2 Unknown Common Precision .................................................... 307 13.1.4 Two-Sample with Unequal Precisions ......................................................308 13.1.4.1 Known Precision .......................................................................... 310 13.1.4.2 Unknown Precisions .................................................................... 311 13.2 Posterior Error Approach ......................................................................................... 312 13.2.1 Posterior Error Rate ..................................................................................... 312 13.2.2 Comparing Means ....................................................................................... 314 13.3 Bootstrap-Median Approach ................................................................................... 316 13.3.1 Background................................................................................................... 317 13.3.2 Bootstrap-Median Approach ..................................................................... 318 13.4 Concluding Remarks ................................................................................................ 319 14. Nonparametrics ................................................................................................................... 321 14.1 Violation of Assumptions ........................................................................................ 321 14.2 One-Sample Location Problem ............................................................................... 323 14.2.1 Remark .......................................................................................................... 326 14.2.2 An Example .................................................................................................. 326

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xiii

14.3 Two-Sample Location Problem ............................................................................... 327 14.3.1 Remark .......................................................................................................... 329 14.3.2 An Example .................................................................................................. 330 14.4 Test for Independence .............................................................................................. 330 14.4.1 An Example .................................................................................................. 333 14.5 Practical Issues ..........................................................................................................334 14.5.1 Bootstrapping ...............................................................................................334 14.5.2 Comparing Variabilities ..............................................................................334 14.5.3 Multiple-Sample Location Problem...........................................................334 14.5.4 Testing Scale Parameters ............................................................................ 335 15. Sample Size Calculations for Cluster Randomized Trials......................................... 337 15.1 Unmatched Trials ...................................................................................................... 338 15.1.1 Comparison of Means ................................................................................. 339 15.1.2 Comparison of Proportions ........................................................................340 15.1.3 Comparison of Incidence Rates ................................................................. 341 15.1.4 Further Remarks ..........................................................................................342 15.2 Matched Trials ...........................................................................................................342 15.2.1 Comparison of Means .................................................................................343 15.2.2 Comparison of Proportions ........................................................................344 15.2.3 Comparison of Incidence Rates .................................................................345 15.3 Stratified Trials .......................................................................................................... 347 16. Test for Homogeneity of Two Zero-Inflated Poisson Population ............................. 349 16.1 Zero-Inflated Poisson Distribution......................................................................... 350 16.2 Testing Differences between Treatment Groups .................................................. 351 16.2.1 Testing the Difference in Both Groups of Zeros and Nonzeros ........... 352 16.2.2 Testing the Difference in the Groups of Zeros ........................................ 354 16.2.3 Testing the Difference of the Groups of Nonzeros ................................. 355 16.3 Sample Size Calculation ........................................................................................... 356 16.3.1 Testing the Difference in the Groups of Both Zeros and Nonzeros ..... 356 16.3.2 Testing the Difference in the Groups of Zeros between Treatments ..... 357 16.3.3 Testing the Difference in the Groups of Nonzeros between Treatments..................................................................................................... 357 16.3.4 An Example .................................................................................................. 361 16.4 Multivariate ZIP ........................................................................................................ 363 16.4.1 Bivariate ZIP ................................................................................................. 363 16.4.2 Comparing the Effects of Control and Test Treatment........................... 366 16.4.3 Sample Size Calculation.............................................................................. 367 16.4.4 An Example .................................................................................................. 368 16.5 Concluding Remarks ................................................................................................ 370 Appendix............................................................................................................................... 371 17. Sample Size for Clinical Trials with Extremely Low Incidence Rate...................... 373 17.1 Clinical Studies with Extremely Low Incidence Rate ......................................... 374 17.2 Classical Methods for Sample Size Determination .............................................. 374 17.2.1 Power Analysis ............................................................................................. 374 17.2.2 Precision Analysis........................................................................................ 375 17.2.3 Remarks ......................................................................................................... 376

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Contents

17.3 Chow and Chiu’s Procedure for Sample Size Estimation ................................... 378 17.3.1 Basic Idea of Chow and Chiu’s Procedure ............................................... 378 17.3.2 Sensitivity Analysis ..................................................................................... 379 17.3.3 An Example .................................................................................................. 380 17.4 Data Safety Monitoring Procedure......................................................................... 380 17.5 Concluding Remarks ................................................................................................ 383 18. Sample Size Calculation for Two-Stage Adaptive Trial Design ............................... 389 18.1 Types of Two-Stage Adaptive Designs................................................................... 390 18.2 Analysis and Sample Size for Category SS Adaptive Designs ........................... 391 18.2.1 Theoretical Framework ............................................................................... 392 18.2.2 Two-Stage Design ........................................................................................ 394 18.2.3 Conditional Power ....................................................................................... 397 18.3 Analysis and Sample Size for Category II SD Adaptive Designs ...................... 398 18.3.1 Continuous Endpoints ................................................................................ 398 18.3.2 Binary Responses ......................................................................................... 402 18.3.3 Time-to-Event Endpoints ............................................................................405 18.4 Analysis and Sample Size for Category III DS and IV DD Two-Stage Adaptive Designs ...................................................................................................... 414 18.4.1 Nonadaptive Version ................................................................................... 415 18.4.2 Adaptive Version.......................................................................................... 416 18.4.3 A Case Study of Hepatitis C Virus Infection ........................................... 417 18.5 Concluding Remarks ................................................................................................ 419 19. Simulation-Based Sample Size and Power Analysis .................................................. 421 19.1 Example: Survival Study with Nonconstant Treatment Effect ..........................422 19.2 Example: Cluster Randomized Study with Stepped Wedge Design................. 424 20. Sample Size Calculation in Other Areas ....................................................................... 427 20.1 QT/QTc Studies with Time-Dependent Replicates .............................................. 427 20.1.1 Study Designs and Models ......................................................................... 428 20.1.2 Power and Sample Size Calculation .......................................................... 429 20.1.3 Extension ....................................................................................................... 433 20.1.4 Remarks .........................................................................................................434 20.2 Propensity Analysis in Nonrandomized Studies ................................................ 435 20.2.1 Weighted Mantel–Haenszel Test ............................................................... 435 20.2.2 Power and Sample Size ............................................................................... 436 20.2.3 Simulations ................................................................................................... 439 20.2.4 Concluding Remarks ...................................................................................440 20.3 ANOVA with Repeated Measures ..........................................................................440 20.3.1 Statistical Model ...........................................................................................440 20.3.2 Hypotheses Testing .....................................................................................442 20.3.3 Sample Size Calculation..............................................................................443 20.3.4 An Example ..................................................................................................443 20.4 Quality of Life............................................................................................................445 20.4.1 Time Series Model .......................................................................................446 20.4.2 Sample Size Calculation..............................................................................448 20.4.3 An Example ..................................................................................................448

Contents

xv

20.5 Bridging Studies ........................................................................................................ 449 20.5.1 Sensitivity Index .......................................................................................... 449 20.5.2 Assessment of Similarity ............................................................................ 452 20.5.3 Remarks ......................................................................................................... 457 20.6 Vaccine Clinical Trials .............................................................................................. 457 20.6.1 Reduction in Disease Incidence ................................................................. 458 20.6.2 Evaluation of Vaccine Efficacy with Extremely Low Disease Incidence......................................................................................... 459 20.6.3 Relative Vaccine Efficacy............................................................................. 461 20.6.4 Composite Efficacy Measure ...................................................................... 461 20.6.5 Remarks .........................................................................................................463 Bibliography................................................................................................................................ 465 Index ............................................................................................................................................. 481

Statistics

Praise for the Second Edition “… this is a useful, comprehensive compendium of almost every possible sample size formula. The strong organization and carefully defined formulae will aid any researcher designing a study.” —Biometrics “This impressive book contains formulae for computing sample size in a wide range of settings. One-sample studies and two-sample comparisons for quantitative, binary, and time-to-event outcomes are covered comprehensively, with separate sample size formulae for testing equality, non-inferiority, and equivalence. Many less familiar topics are also covered …” —Journal of the Royal Statistical Society Sample Size Calculations in Clinical Research, Third Edition presents statistical procedures for performing sample size calculations during various phases of clinical research and development. A comprehensive and unified presentation of statistical concepts and practical applications, this book includes a well-balanced summary of current and emerging clinical issues, regulatory requirements, and recently developed statistical methodologies for sample size calculation. Features • Compares the relative merits and disadvantages of statistical methods for sample size calculations • Explains how the formulae and procedures for sample size calculations can be used in a variety of clinical research and development stages • Presents real-world examples from several therapeutic areas, including cardiovascular medicine, the central nervous system, anti-infective medicine, oncology, and women’s health • Provides sample size calculations for dose response studies, microarray studies, and Bayesian approaches This new edition is updated throughout, includes many new sections, and five new chapters on emerging topics: two stage seamless adaptive designs, cluster randomized trial design, zero-inflated Poisson distribution, clinical trials with extremely low incidence rates, and clinical trial simulation.

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