The Theory of Response-Adaptive Randomization in Clinical Trials

Feifang Hu University of Virginia

Charlottesville, VA

William F. Rosenberger George Mason University

Fairfax, VA

@Z+KicjENCE A JOHN WILEY & SONS, INC., PUBLICATION

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The Theory of Response-Adaptive Randomization in Clinical Trials

This Page Intentionally Left Blank

The Theory of Response-Adaptive Randomization in Clinical Trials

Feifang Hu University of Virginia

Charlottesville, VA

William F. Rosenberger George Mason University

Fairfax, VA

@Z+KicjENCE A JOHN WILEY & SONS, INC., PUBLICATION

Copyright 0 2006 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Hu, Feifang, 1964-

F. Rosenberger p. cm.

Includes bibliographical references and index.

The theory of response-adaptive randomization in clinical trials / Feifang Hu, William

ISBN-13: 978-0-471-65396-7 (cloth) ISBN-10: 0-471-65396-9 (cloth)

1. Clinical trials. 1. Rosenberger, William F. 11. Title.

R853.C55H8 2006 6 1 0 . 7 2 ’ 4 6 ~ 2 2 2006044048

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 I

To our teachers and our students

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Contents

Dedication

Preface

1 Introduction 1.1 Randomization in clinical trials

1.1.1 Complete randomization 1.1.2 Restricted randomization procedures 1.1.3 Response-adaptive randomization procedures 1.1.4 Covariate-adaptive randomization procedures 1.1.5 Covaria fe-adjusted response-adaptive

Response-adaptive randomization in a historical context

randomization procedures 1.2

1.3 Outline of the book 1.4 References

2 Fundamental Questions of Response-Adaptive Randomization 2.1 Optimal allocation

V

xi

6

7 8 8

11 11

vii

viii CONTENTS

3

2.2

2.3 2.4 Asymptotically best procedures 2.5 References

The relationship between power andresponse-adaptive randomization The relationship for I< > 2 treatments

Likelihood-Based Inference 3. I Data structure and likelihood 3.2 Asymptotic properties of maximum likelihood

estimators 3.3 The general result for determining asymptotically best

procedures 3.4 Conclusions 3.5 References

4 Procedures Based on Urn Models 4. I Generalized Friedman 's urn

4. I . I 4.1.2 Assumptions and notation 4.1.3 Main asymptotic theorems 4. I . 4 Some examples 4.1.5 The class of ternary urn models 4.2. I Randomized Pdlya urn 4.2.2 Birth and death urn 4.2.3 Drop-the-loser rule 4.2.4 Generalized drop-the-loser rule 4.2.5

4.3 References

Historical results on asymptotic properties

Proving the main theoretical results 4.2

Asymptotic properties of the GDL rule

5 Procedures Based on Sequential Estimation 5.1 Examples 5.2

5.3 5.4 5.5 Proving the main theorems 5.6 References

Properties of procedures based on sequential estimation for K = 2 Notation and conditions for the general framework Asymptotic results and some examples

15 i8 20 21

23 23

25

27 28 28

31 31 31 34 37 41 52 56 57 58 59 61 62 64

67 68

70 75 79 85 88

CONTENTS ix

6 Sample Size Calculation 6.1 6.2 6.3 Examples

Power of a randomization procedure Three types of sample size

6.3.1 Restricted randomization 6.3.2 Response-adaptive randomization

6.4 References

7 Additional Considerations 7. 1 7.2 Continuous responses

7.3 7.4 Accommodating heterogeneity

The effects of delayed response

7.2.1 Multiple (K > 2) treatments

7.4. 1 Heterogeneity based on time trends 7.4.2 Heterogeneity based on covariates 7.4.3 Statistical inference under heterogeneity

Asymptotic variance of the four procedures

7.5 References

8 Implications for the Practice of Clinical Trials 8. 1 Standards 8.2 Binary responses 8.3 Continuous responses 8.4 8.5 Conclusions 8.6 References

The efects of delayed response

9 Incorporating Covariates 9.1 Introduction and examples

9.1. 1 Covariate-adaptive randomization procedures 9.1.2 CARA Randomization Procedures General framework and asymptotic results 9.2.1 9.2.2 Main theoretical results

Two treatments with binary responses 9.4. I Power

9.2 The procedure for K treatments

9.3 Generalized linear models 9.4

91 92 96 99 99

101 103

105 105 108 111 113 114 114 115 115 118

121 121 123 129 131 232 132

135 135 135 138 139 140 141 144 149 153

9.5 Conclusions 154

x CONTENTS

9.6 References

10 Conclusions and Open Problems 10.1 Conclusions 10.2 Open problems 10.3 References

Appendix A: Supporting Technical Material A. 1 A. 2 A. 3 A. 4

A. 5 A. 6 A. 7 A. 8

Some matrix theory Jordan decomposition Matrix recursions Mart ingales A. 4. I A.4.2 A.4.3

Cramkr- Wold device Multivariate martingales Multivariate Taylor S expansion References

Dejinition and properties of martingales The martingale central limit theorem Gaussian approximations and the law of the iterated logarithm

Appendix B: Proofs B. 1 Proofs of theorems in Chapter 4

B.l.1 Proof of Theorems 4.1-4.3 B.1.2 Proof of Theorem 4.6

B.2 Proof of theorems in Chapter 5 B.3 Proof of theorems in Chapter 7 B. 4 References

Author Index

Subject Index

155

157 157 158 159

161 161 162 163 164 164 165

167 168 168 172 172

173 173 173 189 194 205 21 4 215

217

Preface

Research in response-adaptive randomization developed as a response to a classical ethical dilemma in clinical trials. While clinical trials may provide information on new treatments that can impact countless lives in the future, the act of randomization means that volunteers in the clinical trial will receive the benefit of the new treatment only by chance. In most clinical trials, an attempt is made to balance the treatment assignments equally, but the probability that a volunteer will receive the potentially better treatment is only 1/2. Response-adaptive randomization uses accruing data to skew the allocation probabilities to favor the treatment performing better thus far in the trial, thereby mitigating the problem to some degree.

Response-adaptive allocation has a long history in the biostatistical literature, and the list of researchers who have worked (at least briefly) in the area reads like a Who’s Who of modern statistics: Anscombe, Chernoff, Colton, Cornfield, Flournoy, Greenhouse, Halperin, Louis, Robbins, Siegmund, Wei, Woodroofe, Zelen, and others. Largely because of the disastrous ECMO trial in the early 1980s, there was a general reluctance to use these procedures that has continued to this day. When the authors met in 1995, it was unclear whether these procedures were effective or could be adapted to modern clinical trials and whether certain fundamental questions could be answered. Our collaboration over the past 10 years has been an attempt to formalize the important questions regarding response-adaptive randomization in a rigorous mathematical framework and to systematically answer them. We generally had no idea that we were opening a can of worms that would require a demanding arsenal of mathematical tools. We set out to interest others in the problems, and this led to fruitful collaborations with many other investigators. This book is a result of

xi

xii PREFACE

these collaborations. It represents what we now know about the subject, and it is our attempt to form a mathematically rigorous subdiscipline of experimental design involving randomization. Two individuals were particularly influential: 2. D. Bai of Singaporc, whose collaborative work resulted in solutions to decades-old problems in urn models, largely forming the basis for Chapter 4; and L.-X. Zhang of China, whose collaborative work largely forms the basis of Chapter 5.

This book is aimed at Ph.D. students and researchers in response-adaptive ran- domization. It provides answers to some of the fundamental questions that have been asked over the years: How does response-adaptive randomization affect power? Can standard inferential tests be applied following response-adaptive randomization? What is the effect of delayed response? Which procedure is most appropriate, and how can "most appropriate" be quantified? How can heterogeneity of the patient population be incorporated? Can response-adaptive randomization be performed with more than two treatments or with continuous responses?

While the mathematics generated by these problems can sometimes be daunting, the response-adaptive randomization procedures themselves can be implemented in minutes by adding a loop to a standard randomization routine. Procedures can be simulated under various parameterizations to determine their appropriateness for use in clinical trials. Our hope is that any future objections to the use of response-adaptive randomization will not be based on logistical difficulties or the lack of theoretical justification of these procedures.

Most of the book is written at the level of graduate students in a statistics program. The technical portions of the book are mostly relegated to appendices and to brief descriptions in Chapters 4 and 5 . That material requires advanced probability and stochastic processes as well as matrix theory. Prerequisite material can be found in Appendix A for those wishing to pursue the technical details. In addition, it is recommended that readers new to the area of response-adaptive randomization begin by reading Chapters 10-12 of Rosenberger and Lachin (Randomization in Clinical Trials, Wiley, New York, 2002).

We would like to thank our colleagues 2. D. Bai, W. S. Chan, Siu Hung Cheung, Steve Durham, Nancy Flournoy, Bob Smythe, L.-X. Zhang, and Jim Zidek. In addition, we thank our current and former doctoral students Liangliang Duan and Thomas Gwise (Hu); Anastasia Ivanova, Yevgen Tymofyeyev, and Lanju Zhang (Rosenberger). Parts of this book were tested in a short course at a summer school in Torgnon, Italy, organized by Pietro Muliere of Bocconi University. We thank him and his students; in particular, the asymptotic variance in Example 5.9 was derived by the students of the course.

As we point out in Chapter 10, open problems abound in this area, and it is our sincere hope that more talented researchers will be attracted to the beauty of the complex stochastic structures encountered throughout this book. However, our greatest hope is that, by providing a firm theoretical underpinning to the concept of response-adaptive randomization in this book, clinical trialists will be motivated to apply these techniques in practice.

Finally, much of our research career has benefited from generous funding from the United States government. This included grants from the Division of Mathe-

PREFACE xiii

matical Sciences, National Science Foundation: Hu and Rosenberger 2002-2005, Rosenberger 2005-2008, Hu (Career Award) 2004-2009; and the National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health: Rosen- berger (FIRST Award) 1995-2000. These grants provided the opportunity to advance our research program, and we wish to recognize the importance of such funding for young researchers.

F. H.

Charlorresville Kqinia

W. F. R.

FairfM. KTinia

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1 Introduction

1.1 RANDOMIZATION IN CLINICAL TRIALS

We begin with a mathematical formulation of the problem. Consider a clinical trial of n patients, each of whom is to randomly receive one of K treatments. A randomization sequence is a matrix T = (TI, ..., T,)’ where T, = e,, j = 1, ..., K , i = 1, ..., n and e, is a vector of zeroes with a 1 in the j-th position. We will typically be interested in exploring important properties of the randomization sequence, which usually involves deriving asymptotic properties of the allocation proportions, given by N(n)/n, where N(n) = ( N l ( n ) , ..., N K ( ~ ) ) and N3(n) = C,”=, T,,. Necessarily IIN(n)ll = xK N3(n) = n.

Let X = (XI, ..., X,)’, where J!!$== ( X 1 l , ..., X,K), be a matrix of response variables, where X, represents the sequence of responses that would be observed if each treatment were assigned to the i-th patient independently. However, only one element of X , will be observable. Throughout the book, we will consider only probability models for X, conditional on T,. In some applications, there may be a set of covariate vectors 21, ..., 2, that are also of interest. In this case, we will consider probability models for X , conditional on T, and 2,.

Let 7, = a{T1, ..., T,,} be the sigma-algebra generated by the first n treatment assignments, let X, = a { X 1 , ..., X,} be the sigma-algebra generated by the first n responses, and let 2, = a{ 21, ..., 2,) be the sigma-algebra generated by the first n covariate vectors. Let F, = I, @ X,, @ Zn+l. A randomization procedure is defined by

1

2 INTRODUCTION

where 4n+l is F,-measurable. We can describe dn as the conditional probability of assigning treatments 1, ..., fc to the n-th patient, conditional on the previous n - 1 assignments, responses, and covariate vectors, and the current patient's covariate vector.

We can describe five types of randomization procedures. We have complete randomization if

+n = E ( T n l F n - 1 ) = E(Tn) ; restricted randomization if

response-adaptive randomization if

covariate-adaptive randomization if

and covariate-adjusted response-adaptive (CARA) randomization if

This book will primarily focus on response-adaptive randomization. The latter two classes of randomization procedures, covariate-adaptive randomization and covariate- adjusted response-adaptive randomization, have been poorly studied. Chapter 9 will give an overview of the current knowledge about CARA randomization procedures. The book by Rosenberger and Lachin (2002) gives a thorough discussion of properties of complete and restricted randomization as well as a less thorough treatment of covariate-adaptive and response-adaptive randomization. We now give examples of each of these types of procedures.

1.1.1 Complete randomization

The simplest form of a randomization procedure is complete randomization. For K = 2, Ti = ( 1 , O ) or (0 , l ) according to the toss of a coin. Then 2'11, ..., Tnl are independent and identically distributed Bernoulli random variables with the prob- ability of assignment to treatment 1 given by d i1 = E(T,l) = 1/2, i = 1, ..., n. This procedure is rarely used in practice because of the nonnegligible probability of treatment imbalances in moderate samples.

1.1.2 Restricted randomization procedures

Restricted randomization procedures are the preferred method of randomization for many clinical trials because it is often desired to have equal numbers of patients assigned to each treatment, or nearly so. This is usually accomplished by changing

RANDOMIZAJION IN CLINICAL TRIALS 3

the probability of randomization to a treatment according to how many patients have already been assigned to that treatment. Rosenberger and Lachin (2002) describe four different restricted randomization procedures. The first is the random allocation rule, where, for K = 2 and n even, the probability that the i-th subject is assigned to treatment 1 is given by

The second is the truncated binomial design which, for n even and K = 2, is given by

4i1 = 1/2, if max{Nl(i - I ) ,Nz( i - I)} < n/2, = 0, if Nl(i - 1) = n/2, = 1, if Nz(i- 1) =n/2.

The third is Efon's biased coin design (Efron, 1971). From the vector N(i) , let Di = Nl(i) - Nz(i) be the imbalance between treatments 1 and 2. Define a constant a E (0.5,1]. Then the procedure is given by

q5i1 = 1/2, if Di-1 = 0, - - n, if Di-1 < 0, = 1 - a, if Di-1 > 0.

For the fourth procedure, Wei S urn design, Wei (1 978) proposed that Efron's pro- cedure be modified so that the degree of imbalance influences the randomization procedure. One technique for doing this is to establish an urn model whereby balls in the urn are labeled 1, ..., K , and each represents a treatment to be assigned. Let Y , be the urn composition, where Y,j is the number of balls labeled j in the urn after n patients have been assigned, j = 1, ..., K. One begins with an initial urn composition Y O = 1. Each randomization is accomplished as follows: a ball is drawn and replaced, its label noted, the appropriate treatment is assigned, and one ball from each of the other K - 1 labels is added to the urn. Thus the restricted randomization procedure is given by

A generalization of this procedure is found in Wei, Smythe, and Smith (1986). Let r(.) = (n l ( . ) , ..., T K ( . ) ) be a continuous function such that the following rela- tionship is satisfied: if

N j ( i - 1 ) 1 2 - 1 ' 7 7

then N ( i - 1) 1

aj ( - ) s El j = l,...,K.

4 INTRODUCTION

Then the restricted randomization procedure is given by

In practice, the random allocation rule and truncated binomial design are usually performed within blocks of subjects so that balance can be forced throughout the course of the clinical trial. Forcing balance within blocks of fixed or random size is called a permuted block design. Alternatively, Efron’s biased coin design and Wei’s urn design, while not forcing perfect balance, adaptively balance the treatment assignments. See Rosenberger and Lachin (2002) for details.

1.1.3 Response-adaptive randomization procedures

Response-adaptive randomization procedures change the allocation probabilities for each subject according to previous treatment assignments and responses in order to meet some objective. Two objectives that we will be principally concerned with for direct application in clinical trials are maximizing power and minimizing exposure to inferior treatments. Here we give two examples of response-adaptive randomization procedures: one a generalization of Wei’s urn design and the other a generalization of Efron’s biased coin design.

One can generalize Wei’s urn design to establish a broad family of response- adaptive randomization procedures based on a generalized Friedman’s urn model (Athreya and Karlin, 1968). For response-adaptive randomization, balls are added to the urn based not only on the treatment assigned, but also on the patient’s response. Formally, the randomization of patient i is accomplished as follows: a ball is drawn from the urn composition Y,-1 and replaced, its label noted (say j ) , and the j-th treatment is assigned. One then observes the variable Xi and D 3 k balls are added to the urn, for Ic = 1, ..., K , where D j k is a measurable function on the sample space of X i . The response-adaptive randomization procedure is then given by

Note that if D j k = 1 - d j k with probability one, where dik is the Kronecker delta, we have Wei’s urn design.

The use of the generalized Friedman’s urn model for response-adaptive random- ization procedures derives from Wei and Durham (1 978) for binary response clinical trials with K = 2. Let Xi j = 1 if patient i had a success on treatment j and X i j = 0 if patient i had a treatment failure on treatment j . With this notation, if treatment j was not assigned, then Xij is not observable. Define Y o = (a , a ) for some positive integer a and let

(1.3)

for all i . When the procedure (1.2) is used, this is called the rundotnizedphy-the- winner rule.

D j k = d j k x i j -f (1 - d j k ) ( l - x i j )

RANDOMIZATION IN CLINICAL TRIALS 5

Wei (1979) extended the randomized play-the-winner rule to K > 2 treatments as follows. Let YO = crl and

Djk = ( K - 1)djkxij 4- (1 - djk)(l - xij)

for all i and again use the procedure (1.2). Note that D = { D j k , j , k = 1, ..., K}, which we call the design matrix, is

assumed to be a homogeneous matrix over all patients i = 1, ..., n. This may not be a reasonable assumption in practice, and we expend some energy later in the book dealing with heterogeneous matrices. If we assume homogeneity, many of our asymptotic results that we develop will depend on the matrix H = E ( D ) , which we call the generating matrix.

As a second example, for K = 2, Eisele (1 994) and Eisele and Woodroofe (1 995) describe a doubly-adaptive biased corn design. Unlike the generalized Friedman’s urn, the doubly-adaptive biased coin design is based on a parametric model for the response variable. Let the probability distributions of X I , ..., X, depend on some parameter vector 8 E 0. Let p(8) E (0,1> be a target allocation, i.e., the target proportion of subjects desired to be assigned to treatment 1, which depends on the parameter vector 8. Let g be a function from [0, 112 to [0,1] such that the following four regularity conditions hold: (i) g is jointly continuous; (ii) g(r,r) = r; (iii) g(p, T ) is strictly decreasing in p and strictly increasing in r on (0,l)’; and (iv) g has bounded derivatives in both arguments. At the i-th allocation, the function g represents the closeness of Nl(i - l ) / ( i - 1) to the current estimate ofp(8) in some sense. Hu and Zhang (2004) proposed the following function g for 7 2 0:

S(0,Y) = 1; d L Y ) = 0.

This function does not satisfy Eisele’s regularity condition (iv), but it satisfies alter- native conditions of Hu and Zhang (2004), which we will describe later in the book. Then, for K = 2,

Nl(i - 1) , P ( b i - l ) ) ,

where 8i-lAis some estimator of 8 based on data from the first i - 1 subjects. When y = 0 and 8i-1 is the maximum likelihood estimator of 8, the procedure reduces to

cpil = P@i-1).

This is called the sequential maximum likelihoodprocedure, which has been studied by Melfi and Page (2000) and Melfi, Page, and Geraldes (2001).

These two examples, one based on a class of urn models and the other on a class of adaptive biased coin designs, lead to an important distinction motivating the approach to response-adaptive randomization. The first approach is completely nonparametric

6 INTRODUCTION

and is not designed to target some specific allocation based on unknown parameters. The second approach begins with a parametric response model and a target allocation based on unknown parameters of that model and sequentially substitutes updated estimates of those parameters. This book will explore both approaches: the first approach in the context of various urn models and the second approach in the context of the doubly-adaptive biased coin design. In fhture chapters we will refer to the first approach as procedures based on urn models and the second approach as procedures based on sequential estimation.

1.1.4 Covariate-adaptive randomization procedures

Clinical trialists are often concerned that treatment arms will be unbalanced with respect to key covariates of interest. To prevent this, covariate-adaptive randomization is sometimes employed. As one example, the Pocock-Simon procedure (Pocock and Simon, 1975) is perhaps the most widely used covariate-adaptive randomization procedure. Let 21, ..., 2, be the covariate vector of patients 1, ..., n. Further, we assume that there are S covariates of interest (continuous or otherwise) and they are divided into n,, s = 1, ..., S, different levels. Define Nsik(n), s = 1, ..., S, i = 1, ..., n,, k = 1 , 2 to be the number ofpatients in the i-th level ofthe s-th covariate on treatment k. Let patient n + 1 have covariate vector Zn+l = ( T I , ..., T S ) . Define a metric Ds(n) = N,,, ~ ( n ) - Nsr,2(n), which is the difference between the numbers ofpatients on treatments 1 and 2 for members of level T, ofcovariate s. Let w1, ..., ws be a set of weights and take the weighted aggregate D(n) = xfZl w,D,(n). Establish a probability 7r E (1/2,1]. Then the procedure allocates to treatment 1 according to

46il = E(TillZ-1,Zi) = 1/2, if D ( i - 1) = 0, T, if D(i - 1) < 0,

= 1 -7r , if D(i - 1) > 0.

- -

1.1.5 Covariate-adjusted response-adaptive randomization procedures

A relatively new concept in response-adaptive randomization is CARA randomiza- tion, in which the previous patient responses and covariate vectors and the current patient’s covariate vector are used to obtain a covariate-adjusted probability of as- signment to treatment. One approach to this, for binary response (Xi j = 0 or 1, i = 1, ..., n, j = 1, ..., K ) using a logistic regression model, was given by Rosen- berger, Vidyashankar, and Agarwal(200 I). Consider the covariate-adjusted logistic regression model, given by

where a is the intercept, p is the treatment main effect, -y is an S-dimensional vector of covariate main effects, and 6 is an S-dimensional vector of treatment-by-covariate interactioqs. The fitted model acer n patients yields maximum likelihood estimators bn of a, 0, of P, +, of 7, and 6, of 6. The covariate-adjusted odds ratio based on

RESPONSE-ADAPTIVE RANDOMIZATION IN A HISTORICAL CONTEXT 7

the data for n patients and the covariate vector of the (n + 1)-th patient is given by

Then we define the randomization procedure by

1.2 RESPONSE-ADAPTIVE RANDOMIZATION IN A HISTORICAL CONTEXT

In this book, the response-adaptive randomization procedures have three defining characteristics: (1) they are myopic; (2) they are fully randomized; (3) they require a fixed sample size. Let us examine these characteristics in a historical context.

The response-adaptive randomization procedures we examine are myopic proce- dures in that they incorporate current data on treatment assignments and responses into decisions about treatment assignments for the next subject. After each subject, the data are updated, and a decision is made only about the next subject. This differs from the approach in multi-armed banditproblems (see Berry and Fristedt, 1986, for example) in which all possible sequences of treatment assignments and responses are enumerated, and the sequence that optimizes some objective is selected. The latter approach is computationally intensive, so myopic procedures are often preferable from a logistical consideration, although there is no guarantee that such procedures are globally optimal. Most of the work on multi-armed bandit problems has been in the context of nonrandomized studies, although recently there has been some work on randomized bandits by Yang and Zhu (2002).

In addition to being myopic, the response-adaptive randomization procedures described are fully randomized in that each subject is assigned a treatment by random chance, the cornerstone of the modern clinical trial. Randomization is necessary to prevent selection bias, covariate imbalances, and to provide a basis for inference (Rosenberger and Lachin, 2002). Historically, response-adaptive treatment allocation designs were developed for the purpose of assigning patients to the better treatment with probability 1 . The preliminary ideas can be traced back to Thompson (1933) and Robbins (1952) and then to a series of papers in the 1960s by Anscombe (1963), Colton (1963), Zelen (l969), and Cornfield, Halperin, and Greenhouse (1969). The idea of incorporating randomization in the context of response-adaptive treatment allocation designs stems from Wei and Durham (1978).

Finally, the response-adaptive procedures in this book assume a fixed sample size. Historically, response-adaptive treatment allocation designs were often viewed in the context of sequential analysis, where a random number of patients N would be determined according to an appropriate stopping boundary. Early papers in this context were by Chernoff and Roy (1 965), Flehinger and Louis (1 97 l), Robbins and Siegmund (1 974), and Hayre ( 1 979). However, most modem clinical trials in the

8 INTRODUCTION

United States are conducted with a fixed, predetermined sample size set according to power and budgetary considerations. Interim monitoring plans are instituted to potentially stop the clinical trial early for safety or efficacy reasons. The imposition of an early stopping boundary is flexible, and in fact, many clinical trials that cross an early stopping boundary are not terminated early, usually for reasons unrelated to the primary outcome of the study.

Thus we restrict our attention to myopic, hlly randomized, fixed sample size procedures, which reflect very well the modem conduct of clinical trials. In fact, to implement these procedures in an actual clinical trial requires nothing more than a simple modification of the randomization sequence generator. (Of course, there are many interesting and important statistical questions that arise in the context of response-adaptive randomization, which will be the subject of this book.) The methodology for such response-adaptive randomization procedures thus begins in the late 1970s, and much of our current understanding is very recent. Many open problems still abound, and this is a fruitful area for hture research.

1.3 OUTLINE OF THE BOOK

In Chapter 2, we identify fundamental questions of response-adaptive randomization, namely, what allocation should we target to achieve requisite power while resulting in fewer treatment failures, and how do we employ randomization to attain that target allocation? In Chapter 3, we describe the impact on likelihood-based inference resulting from response-adaptive randomization procedures and derive a benchmark for measuring the efficiency of a response-adaptive randomization procedure. We then detail the two approaches to response-adaptive randomization: in Chapter 4, we describe procedures based on urn models and derive their asymptotic properties; in Chapter 5 , we describe procedures based on sequential estimation and derive their properties. In Chapter 6, we deal with the subtle concept of sample size computation in the context of random sample fractions. Chapter 7 discusses additional technical and practical considerations for response-adaptive randomization: delayed responses, heterogeneity, and more general response types and multiple treatments. Chapter 8 describes some practical considerations in selecting response-adaptive randomization procedures. Chapter 9 gives an overview of our current knowledge about C A M procedures. Two appendices describe certain mathematical background that forms a prerequisite for the more technical parts of the book and gives exhaustive proofs of some of the most important theorems in Chapters 4 and 5 .

1.4 REFERENCES

ANSCOMBE, F. J. (1963). Sequential medical trials. Journal ofthe American Sta- tistical Association 58 365-384.

ATHREYA, K. B. AND KARLIN, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems.