Statistical Monitoring of Clinical Trials (eBook)

Preface 7
Contents 9
1 Introduction 14
2 A General Framework 21
2.1 Hypothesis Testing: The Null Distribution of Test Statistics Over Time 22
2.2 An Estimation Perspective 30
2.3 Connection Between Estimators, Sums, Z-Scores, and Brownian Motion 33
2.4 Maximum Likelihood Estimation 36
2.5 Other Settings Leading to E-Processes and Brownian Motion 40
2.6 The Normal Linear and Mixed Models 42
2.7 When Is Brownian Motion Not Appropriate? 48
2.8 Summary 50
2.9 Appendix 51
3 Power: Conditional, Unconditional, and Predictive 55
3.1 Unconditional Power 55
3.2 Conditional Power for Futility 57
3.3 Varied Uses of Conditional Power 65
3.4 Properties of Conditional Power 69
3.5 A Bayesian Alternative: Predictive Power 72
3.6 Summary 75
3.7 Appendix 76
4 Historical Monitoring Boundaries 79
4.1 How Bad Can the Naive Approach Be? 79
4.2 The Pocock Procedure 81
4.3 The Haybittle Procedure and Variants 81
4.4 The O’Brien-Fleming Procedure 83
4.5 A Comparison of the Pocock and O’Brien-Fleming Boundaries 84
4.6 Effect of Monitoring on Power 87
4.7 Appendix: Computation of Boundaries Using Numerical Integration 89
5 Spending Functions 92
5.1 Upper Boundaries 92
5.2 Upper and Lower Boundaries 101
5.3 Summary 103
5.4 Appendix 103
6 Practical Survival Monitoring 109
6.1 Introduction 109
6.2 Survival Trials with Staggered Entry 109
6.3 Stochastic Process Formulation and Linear Trends 111
6.4 A Real Example 112
6.5 Nonlinear Trends of the Statistics: Analogy with Monitoring a t-Test 113
6.6 Considerations for Early Termination 114
6.7 The Information Fraction with Survival Data 115
7 Inference Following a Group-Sequential Trial 122
7.1 Likelihood, Sufficiency, and (Lack of) Completeness 122
7.2 One-Tailed p-Values 125
7.3 Properties of p-Values 134
7.4 Confidence Intervals 135
7.5 Estimation 140
7.6 Summary 144
7.7 Appendix: Proof that B( t ) t Overestimates 0 in the One-Tailed Setting 144
8 Options When Brownian Motion Does Not Hold 146
8.1 Small Sample Sizes 146
8.2 Permutation Tests 152
8.3 The Bonferroni Method 158
8.4 Summary 159
8.5 Appendix 160
9 Monitoring for Safety 163
9.1 Example: Inference from a Sample Size of One 163
9.2 Example: Inference from Multiple Endpoints 164
9.3 General Considerations 165
9.4 What Safety Data Look Like 168
9.5 Looking for a Single Adverse Event 171
9.6 Looking for Multiple Adverse Events 180
9.7 Summary 181
10 Bayesian Monitoring 183
10.1 Introduction 183
10.2 The Bayesian Paradigm Applied to B-Values 184
10.3 The Need for a Skeptical Prior 185
10.4 A Comparison of Bayesian and Frequentist Boundaries 188
10.5 Example 190
10.6 Summary 192
11 Adaptive Sample Size Methods 193
11.1 Introduction 193
11.2 Methods Using Nuisance Parameter Estimates: The Continuous Outcome Case 194
11.3 Methods Using Nuisance Parameter Estimates: The Binary Outcome Case 207
11.4 Adaptive Methods Based on the Treatment Effect 211
11.5 Summary 218
12 Topics Not Covered 220
12.1 Introduction 220
12.2 Continuous Sequential Boundaries 221
12.3 Other Types of Group-Sequential Boundaries 222
12.4 Reverse Stochastic Curtailing 223
12.5 Monitoring Studies with More Than Two Arms 224
12.6 Monitoring for Equivalence and Noninferiority 225
12.7 Repeated Confidence Intervals 225
13 Appendix I: The Logrank and Related Tests 227
13.1 Hazard Functions 228
13.2 Linear Rank Statistics 231
13.3 Payment Functions and Score Functions 237
13.4 Censored Survival Data 239
13.5 The U-Statistic Approach to the Wilcoxon Statistic 240
13.6 The Logrank and Weighted Mantel-Haenszel Statistics 241
13.7 Monitoring Survival Trials 243
14 Appendix II: Group-Sequential Software 244
14.1 Introduction 244
14.2 Before the Trial Begins: Power and Sample Size 244
14.3 During the Trial: Computation of Boundaries 246
14.4 After the Trial: p-Value, Parameter Estimate, and Confidence Interval 247
14.5 Other Features of the Program 249
References 252
Index 260