Statistical Methods for Survival Trial Design: With Applications to Cancer Clinical Trials Using R

275 Pages·2018·10.363 MB·English

by  Jianrong Wu

Statistical Methods for
Survival Trial Design
With Applications to Cancer
Clinical Trials Using R
Chapman & Hall/CRC Biostatistics Series
Shein-Chung Chow, Duke University of Medicine
Byron Jones, Novartis Pharma AG
Jen-pei Liu, National Taiwan University
Karl E. Peace, Georgia Southern University
Bruce W. Turnbull, Cornell University
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Statistical Methods for Survival Trial Design: With Applications to Cancer Clinical
Trials Using R
Jianrong Wu
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Statistical Methods for
Survival Trial Design
With Applications to Cancer
Clinical Trials Using R
Jianrong Wu
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Wu, Jianrong, author.
Title: Statistical methods for survival trial design : with applications to cancer clinical trials using R /
Jianrong Wu.
Description: Boca Raton : Taylor & Francis, 2018. | Includes bibliographical references and index.
Identifiers: LCCN 2018008225 | ISBN 9781138033221 (hardback)
Subjects: LCSH: Clinical trials--Statistical methods. |
Cancer--Research--Methodology. | Medicine--Research--Methodology. | R
(Computer program language)
Classification: LCC R853.C55 W8 2018 | DDC 610.72/4--dc23
LC record available at https://lccn.loc.gov/2018008225
Visit the Taylor & Francis Web site at
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Contents
Preface ix
List of Figures xi
List of Tables xiii
1. Introduction to Cancer Clinical Trials 1
1.1. General Aspects of Cancer Clinical Trial Design . . . . . . . 2
1.1.1. Study Objectives . . . . . . . . . . . . . . . . . . . . . 2
1.1.2. Treatment Plan . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3. Eligibility Criteria . . . . . . . . . . . . . . . . . . . . 3
1.1.4. Statistical Considerations . . . . . . . . . . . . . . . . 3
1.2. Statistical Aspects of Cancer Survival Trial Design . . . . . . 3
1.2.1. Randomization . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2. Stratification . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3. Blinding . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.4. Sample Size Calculation . . . . . . . . . . . . . . . . . 5
2. Survival Analysis 9
2.1. Survival Distribution . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1. Exponential Distribution . . . . . . . . . . . . . . . . 11
2.1.2. Weibull Distribution . . . . . . . . . . . . . . . . . . . 14
2.1.3. Gamma Distribution . . . . . . . . . . . . . . . . . . . 14
2.1.4. Gompertz Distribution . . . . . . . . . . . . . . . . . . 16
2.1.5. Log-Normal Distribution. . . . . . . . . . . . . . . . . 19
2.1.6. Log-Logistic Distribution . . . . . . . . . . . . . . . . 19
2.2. Survival Data . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3. Fitting the Parametric Survival Distribution . . . . . . . . . 25
2.4. Kaplan-Meier Estimates . . . . . . . . . . . . . . . . . . . . . 26
2.5. Median Survival Time . . . . . . . . . . . . . . . . . . . . . . 30
2.6. Log-Rank Test . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7. Cox Regression Model . . . . . . . . . . . . . . . . . . . . . . 39
3. Counting Process and Martingale 41
3.1. Basic Convergence Concepts . . . . . . . . . . . . . . . . . . 41
3.2. Counting Process Definition . . . . . . . . . . . . . . . . . . 42
3.3. Filtration and Martingale . . . . . . . . . . . . . . . . . . . . 43
v
vi Contents
3.4. Martingale Central Limit Theorem . . . . . . . . . . . . . . . 45
3.5. Counting Process Formulation of Censored Survival Data . . 46
4. Survival Trial Design under the Parametric Model 49
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2. Weibull Model . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3. Test Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4. Distribution of the MLE test . . . . . . . . . . . . . . . . . . 53
4.5. Sample Size Formula . . . . . . . . . . . . . . . . . . . . . . 54
4.6. Sample Size Calculation . . . . . . . . . . . . . . . . . . . . . 55
4.7. Accrual Duration Calculation . . . . . . . . . . . . . . . . . . 58
4.8. Example and R code . . . . . . . . . . . . . . . . . . . . . . 58
5. Survival Trial Design under the Proportional Hazards Model 61
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2. Proportional Hazards Model . . . . . . . . . . . . . . . . . . 62
5.3. Asymptotic Distribution of the Log-Rank Test . . . . . . . . 64
5.4. Schoenfeld Formula . . . . . . . . . . . . . . . . . . . . . . . 68
5.5. Rubinstein Formula . . . . . . . . . . . . . . . . . . . . . . . 72
5.6. Freedman Formula . . . . . . . . . . . . . . . . . . . . . . . . 74
5.7. Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.8. Sample Size Calculation under Various Models . . . . . . . . 79
5.9. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.10.Optimal Properties of the Log-Rank Test . . . . . . . . . . . 88
5.10.1. Optimal Sample Size Allocation . . . . . . . . . . . . 88
5.10.2. Optimal Power . . . . . . . . . . . . . . . . . . . . . . 90
5.11.Precise Formula . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.12.Exact Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6. Survival Trial Design under the Cox Regression Model 99
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2. Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3. Asymptotic Distribution of the Score Test . . . . . . . . . . 101
6.4. Sample Size Formula . . . . . . . . . . . . . . . . . . . . . . 103
7. Complex Survival Trial Design 109
7.1. Extension of the Freedman Formula . . . . . . . . . . . . . . 111
7.1.1. Example and R code . . . . . . . . . . . . . . . . . . . 113
7.2. Lakatos Formula . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3. Markov Chain Model with Simultaneous Entry . . . . . . . . 118
7.4. Computation Formulae . . . . . . . . . . . . . . . . . . . . . 120
7.5. Markov Chain Model with Staggered Entry . . . . . . . . . . 122
7.6. Examples and R code . . . . . . . . . . . . . . . . . . . . . . 125
Contents vii
8. Survival Trial Design under the Mixture Cure Model 141
8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.2. Testing Differences in Cure Rates . . . . . . . . . . . . . . . 143
8.2.1. Mixture Cure Model . . . . . . . . . . . . . . . . . . . 143
8.2.2. Asymptotic Distribution . . . . . . . . . . . . . . . . . 145
8.2.3. Sample Size Formula . . . . . . . . . . . . . . . . . . . 147
8.2.4. Optimal Log-Rank Test . . . . . . . . . . . . . . . . . 148
8.2.5. Comparison . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2.6. Example and R code . . . . . . . . . . . . . . . . . . . 151
8.2.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 154
8.3. Testing Differences in Short- and Long-Term Survival . . . . 154
8.3.1. Hypothesis Testing . . . . . . . . . . . . . . . . . . . . 154
8.3.2. Ewell and Ibrahim Formula . . . . . . . . . . . . . . . 155
8.3.3. Simulation . . . . . . . . . . . . . . . . . . . . . . . . 158
8.3.4. Example and R code . . . . . . . . . . . . . . . . . . . 159
8.3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 164
9. A General Group Sequential Procedure 167
9.1. Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . 167
9.2. Sequential Conditional Probability Ratio Test . . . . . . . . 168
9.3. Operating Characteristics . . . . . . . . . . . . . . . . . . . . 174
9.4. Probability of Discordance . . . . . . . . . . . . . . . . . . . 175
9.5. SCPRT Design . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.Sequential Survival Trial Design 177
10.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.2.Sequential Procedure for the Parametric Model . . . . . . . . 177
10.2.1. Sequential Wald Test. . . . . . . . . . . . . . . . . . . 178
10.2.2. SCPRT for the Parametric Model . . . . . . . . . . . 180
10.3.Sequential Procedure for the Proportional Hazard Model . . 185
10.3.1. Sequential Log-Rank Test . . . . . . . . . . . . . . . . 185
10.3.2. Information Time . . . . . . . . . . . . . . . . . . . . 189
10.3.3. SCPRT for the PH Model . . . . . . . . . . . . . . . . 191
11.Sequential Survival Trial Design Using Historical Controls 199
11.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11.2.Sequential Log-Rank Test with Historical Controls . . . . . . 200
11.2.1. Sample Size Calculation . . . . . . . . . . . . . . . . . 201
11.2.2. Information Time . . . . . . . . . . . . . . . . . . . . 202
11.2.3. Group Sequential Procedure . . . . . . . . . . . . . . . 204
11.3.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
viii Contents
12.Some Practical Issues in Survival Trial Design 211
12.1.Parametric vs. Nonparametric Model . . . . . . . . . . . . . 211
12.2.Nonproportional Hazards Model . . . . . . . . . . . . . . . . 212
12.3.Accrual Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 212
12.4.Mixed Populations . . . . . . . . . . . . . . . . . . . . . . . . 213
12.5.Loss to Follow-Up . . . . . . . . . . . . . . . . . . . . . . . . 213
12.6.Noncompliance and Drop-In . . . . . . . . . . . . . . . . . . 214
12.7.Competing Risk . . . . . . . . . . . . . . . . . . . . . . . . . 215
A. Likelihood Function for the Censored Data 219
B. Probability of Failure under Uniform Accrual 221
C. Verification of the Minimum Sample Size Conditions 223
D.R Codes for the Sample Size Calculations 225
E. Derivation of the Asymptotic Distribution 231
F. Derivation of Equations for Chapter 8 235
Bibliography 237
Index 247
Preface
The clinical trial has become a major research tool for developing advanced
cancer treatments. In recent years, the pharmaceutical industry, hospitals,
and other research centers have been increasingly active in conducting cancer
clinical trials with time-to-event endpoints. Statistical methods for designing
and monitoring such trials have been available for many years, and a number
of books have addressed this topic, including those by Chow et al. (2003),
Julious (2010), Cook and DeMets (2008), Jennison and Turnbull (2000), and
Proschanetal.(2006).However,asnoneofthesebookswaswrittenspecifically
to address the design and monitoring of survival clinical trials, they contain
onlylimitedmaterialonthattopic.Forexample,inthebookbyJulious,only
onechapterdiscussessamplesizecalculationundertheexponentialmodeland
proportional hazards model, and some of the presented approaches to sample
sizecalculationandadjustingforlosstofollow-upareinappropriate.Thebook
by Chow et al. is also limited with regard to its discussion of the exponential
model, and its presentation of the method of Lakatos for complex survival
trial design not only lacks sufficient detail but also contains errors. The book
by Jennison and Turnbull is an excellent reference for group sequential trial
design, but its coverage of survival trial design is limited to a single chapter.
Cancertreatmenthasprogresseddramaticallyinrecentdecades,suchthat
itisnowcommontoseeacureorlong-termsurvivalinasignificantproportion
ofpatientswithvarioustypesofcancer,e.g.,breastcancer,non-Hodgkinlym-
phoma,leukemia,prostatecancer,melanoma,orheadandneckcancer(Ewell
and Ibrahim, 1997). Until now, however, the principles of designing survival
trials in which a proportion of the participants are expected to be cured have
notbeenpublishedinanybookorimplementedinanycommerciallyavailable
software.
Thisbookisintendedtoprovideacomprehensiveintroductiontothemost
commonlyusedmethodsforsurvivaltrialdesignandmonitoringandtohigh-
light some recent developments in the area. I hope this book will serve as a
referencebookforresearchersconductingclinicaltrialswithsurvivalendpoints
and for statisticians designing survival clinical trials for the pharmaceutical
industry, hospitals, or other cancer research centers. It may also serve as a
textbook for graduate students in the biostatistics field who wish to learn
survival analysis and/or acquire basic knowledge of clinical trial design and
sample size calculation.
The main focus of this book is on the methodology of sample size calcula-
tion, and clinical trial design and monitoring are illustrated using data from
ix