1 Introduction 1
1.1 Reg ulatory Requirement . . . . . . . . . . . . . . . . . . . . 2
1.2 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Procedures for Sample Size Calculation . . . . . . . . . . . 13
1.4 Aims and Structure of the Book . . . . . . . . . . . . . . . 21
2 Considerations Prior to Sample Size Calculation 25
2.1 Confoundingand Interaction . . . . . . . . . . . . . . . . . 26
2.2 One-Sided Test Versus Two-Sided Test . . . . . . . . . . . . 28
2.3 Crossover Desig nVersus Parallel Desig n. . . . . . . . . . . 30
2.4 Subgroup/Interim Analyses . . . . . . . . . . . . . . . . . . 32
2.5 Data Transformation . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Comparing Means 49
3.1 One-Sample Desig n. . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Two-Sample Parallel Desig n. . . . . . . . . . . . . . . . . . 57
3.3 Two-Sample Crossover Desig n. . . . . . . . . . . . . . . . . 65
3.4 Multiple-Sample One-Way ANOVA . . . . . . . . . . . . . . 70
3.5 Multiple-Sample Williams Design . . . . . . . . . . . . . . . 74
3.6 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 78
4Larg e Sample Tests for Proportions 83
4.1 One-Sample Desig n. . . . . . . . . . . . . . . . . . . . . . . 84
4.2 Two-Sample Parallel Desig n. . . . . . . . . . . . . . . . . . 89
4.3 Two-Sample Crossover Desig n. . . . . . . . . . . . . . . . . 95
4.4 One-Way Analysis of Variance . . . . . . . . . . . . . . . . 99
4.5 Williams Design . . . . . . . . . . . . . . . . . . . . . . . . 101
4.6 Relative Risk—Parallel Desig n . . . . . . . . . . . . . . . . 104
4.7 Relative Risk—Crossover Desig n . . . . . . . . . . . . . . . 109
4.8 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 111
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5 Exact Tests for Proportions 117
5.1 Binomial Test . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Fisher’s Exact Test . . . . . . . . . . . . . . . . . . . . . . . 121
5.3 Optimal Multiple-Stage Designs for Single Arm Trials . . . 124
5.4 Flexible Desig ns forMultiple-Arm Trials . . . . . . . . . . . 141
5.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6 Tests for Goodness-of-Fit and Contingency Tables 145
6.1 Tests for Goodness-of-Fit . . . . . . . . . . . . . . . . . . . 146
6.2 Test for Independence—Sing le Stratum . . . . . . . . . . . 148
6.3 Test for Independence—Multiple Strata . . . . . . . . . . . 151
6.4 Test for Categ orical Shift . . . . . . . . . . . . . . . . . . . 153
6.5 Carry-Over Effect Test . . . . . . . . . . . . . . . . . . . . . 158
6.6 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 161
7 Comparing Time-to-Event Data 163
7.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.2 ExponentialModel . . . . . . . . . . . . . . . . . . . . . . . 166
7.3 Cox’s Proportional HazardsModel . . . . . . . . . . . . . . 174
7.4 Weig hted Log -Rank Test . . . . . . . . . . . . . . . . . . . . 179
7.5 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 185
8 Group Sequential Methods 187
8.1 Pocock’s Test . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.2 O’Brien and Fleming ’s Test . . . . . . . . . . . . . . . . . . 192
8.3 Wang and Tsiatis’ Test . . . . . . . . . . . . . . . . . . . . 193
8.4 InnerWedg eTest . . . . . . . . . . . . . . . . . . . . . . . . 197
8.5 Binary Variables . . . . . . . . . . . . . . . . . . . . . . . . 201
8.6 Time-to-Event Data . . . . . . . . . . . . . . . . . . . . . . 202
8.7 Alpha Spending Function . . . . . . . . . . . . . . . . . . . 204
8.8 Sample Size Re-Estimation . . . . . . . . . . . . . . . . . . 206
8.9 Conditional Power . . . . . . . . . . . . . . . . . . . . . . . 209
8.10 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 211
9 Comparing Variabilities 215
9.1 ComparingIn tra-Subject Variabilities . . . . . . . . . . . . 216
9.2 Comparing Intra-Subject CVs . . . . . . . . . . . . . . . . . 224
9.3 ComparingIn ter-Subject Variabilities . . . . . . . . . . . . 233
9.4 ComparingT otal Variabilities . . . . . . . . . . . . . . . . . 241
9.5 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 254
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10 Bioequivalence Testing 257
10.1 Bioequivalence Criteria . . . . . . . . . . . . . . . . . . . . . 258
10.2 Averag eBioequivalence . . . . . . . . . . . . . . . . . . . . 259
10.3 Population Bioequivalence . . . . . . . . . . . . . . . . . . . 263
10.4 Individual Bioequivalence . . . . . . . . . . . . . . . . . . . 265
10.5 In Vitro Bioequivalence . . . . . . . . . . . . . . . . . . . . 271
11 Dose Response Studies 279
11.1 Continuous Response . . . . . . . . . . . . . . . . . . . . . . 280
11.2 Binary Response . . . . . . . . . . . . . . . . . . . . . . . . 284
11.3 Time-to-Event Endpoint . . . . . . . . . . . . . . . . . . . . 285
11.4 Williams’ Test for Minimum Effective Dose (MED) . . . . . 287
11.5 Cochran-Armitag e’s Test for Trend . . . . . . . . . . . . . . 293
11.6 Dose Escalation Trials . . . . . . . . . . . . . . . . . . . . . 296
11.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 301
12 Microarray Studies 303
12.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . 304
12.2 False Discovery Rate (FDR) Control . . . . . . . . . . . . . 305
12.3 Family-wise Error Rate (FWER) Control . . . . . . . . . . 315
12.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 324
13 Bayesian Sample Size Calculation 327
13.1 Posterior Credible Interval Approach . . . . . . . . . . . . . 328
13.2 Posterior Error Approach . . . . . . . . . . . . . . . . . . . 344
13.3 The Bootstrap-Median Approach . . . . . . . . . . . . . . . 350
13.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 353
14Nonpar ametrics 355
14.1 Violation of Assumptions . . . . . . . . . . . . . . . . . . . 356
14.2 One-Sample Location Problem . . . . . . . . . . . . . . . . 357
14.3 Two-Sample Location Problem . . . . . . . . . . . . . . . . 361
14.4 Test for Independence . . . . . . . . . . . . . . . . . . . . . 365
14.5 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 369
15 Sample Size Calculation in Other Areas 373
15.1 QT/QTc Studies with Time-Dependent Replicates . . . . . 374
15.2 Propensity Analysis in Non-Randomized Studies . . . . . . 383
15.3 ANOVA with RepeatedMeasures . . . . . . . . . . . . . . . 389
15.4 Quality of Life . . . . . . . . . . . . . . . . . . . . . . . . . 394
15.5 Bridg ing Studies . . . . . . . . . . . . . . . . . . . . . . . . 399
15.6 Vaccine Clinical Trials . . . . . . . . . . . . . . . . . . . . . 408