Sample Size Calculations in Clinical Research

511 Pages·06.964 MB·English

by  Shein-Chung Chow, Jun Shao, Hansheng Wang, Yuliya Lokhnygina

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
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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