Modern adaptive randomized clinical trials

10 Response–Adaptive Randomization: An Overview of DesignsLi-Xin Zhang 11 Statistical Inference Following Yanqing Yi1 and Xikui Wang2 12 Sample Size Re-Estimation in Adaptively Randomize

Modern Adaptive Randomized Clinical Trials Statistical and Practical Aspects

Shein-Chung Chow, Ph.D., Professor, Department of Biostatistics and Bioinformatics,

Duke University School of Medicine, Durham, North Carolina

Series Editors

Byron Jones, Biometrical Fellow, Statistical Methodology, Integrated Information Sciences,

Novartis Pharma AG, Basel, Switzerland

Jen-pei Liu, Professor, Division of Biometry, Department of Agronomy,

National Taiwan University, Taipei, Taiwan

Karl E Peace, Georgia Cancer Coalition, Distinguished Cancer Scholar, Senior Research Scientist

and Professor of Biostatistics, Jiann-Ping Hsu College of Public Health,

Georgia Southern University, Statesboro, Georgia

Bruce W Turnbull, Professor, School of Operations Research and Industrial Engineering,

Cornell University, Ithaca, New York

Published Titles

Adaptive Design Methods in

Clinical Trials, Second Edition

Shein-Chung Chow and Mark Chang

Adaptive Designs for Sequential

Treatment Allocation

Alessandro Baldi Antognini and

Alessandra Giovagnoli

Adaptive Design Theory and

Implementation Using SAS and R,

Applied Meta-Analysis with R

Ding-Geng (Din) Chen and Karl E Peace

Basic Statistics and Pharmaceutical

Statistical Applications, Second Edition

James E De Muth

Bayesian Adaptive Methods for

Clinical Trials

Scott M Berry, Bradley P Carlin,

J Jack Lee, and Peter Muller

Bayesian Analysis Made Simple: An Excel

GUI for WinBUGS

Ming T Tan, Guo-Liang Tian, and Kai Wang Ng

Bayesian Modeling in Bioinformatics

Dipak K Dey, Samiran Ghosh, and Bani K Mallick

Benefit-Risk Assessment in Pharmaceutical Research and Development

Andreas Sashegyi, James Felli, and Rebecca Noel

Biosimilars: Design and Analysis of Follow-on Biologics

Clinical Trial Data Analysis using R

Ding-Geng (Din) Chen and Karl E Peace

Clinical Trial Methodology

Karl E Peace and Ding-Geng (Din) Chen

Computational Methods in Biomedical

Research

Ravindra Khattree and Dayanand N Naik

Computational Pharmacokinetics

Anders Källén

Confidence Intervals for Proportions and

Related Measures of Effect Size

Shein-Chung Chow and Jen-pei Liu

Design and Analysis of Bioavailability and

Bioequivalence Studies, Third Edition

Shein-Chung Chow and Jen-pei Liu

Design and Analysis of Bridging Studies

Jen-pei Liu, Shein-Chung Chow,

and Chin-Fu Hsiao

Design and Analysis of Clinical Trials for

Predictive Medicine

Shigeyuki Matsui, Marc Buyse,

and Richard Simon

Design and Analysis of Clinical Trials with

Time-to-Event Endpoints

Karl E Peace

Design and Analysis of Non-Inferiority

Trials

Mark D Rothmann, Brian L Wiens,

and Ivan S F Chan

Difference Equations with Public Health

Applications

Lemuel A Moyé and Asha Seth Kapadia

DNA Methylation Microarrays:

Experimental Design and Statistical Analysis

Sun-Chong Wang and Arturas Petronis

DNA Microarrays and Related Genomics Techniques: Design, Analysis, and Interpretation of Experiments

David B Allison, Grier P Page,

T Mark Beasley, and Jode W Edwards

Dose Finding by the Continual Reassessment Method

Ying Kuen Cheung

Elementary Bayesian Biostatistics

Handbook of Regression and Modeling:

Applications for the Clinical and Pharmaceutical Industries

Daryl S Paulson

Inference Principles for Biostatisticians

Ian C Marschner

Interval-Censored Time-to-Event Data:

Methods and Applications

Ding-Geng (Din) Chen, Jianguo Sun, and Karl E Peace

Introductory Adaptive Trial Designs:

A Practical Guide with R

Meta-Analysis in Medicine and Health

Policy

Dalene Stangl and Donald A Berry

Mixed Effects Models for the Population

Approach: Models, Tasks, Methods and

Modern Adaptive Randomized Clinical

Trials: Statistical and Practical Aspects

Oleksandr Sverdlov

Monte Carlo Simulation for the

Pharmaceutical Industry: Concepts,

Algorithms, and Case Studies

Mark Chang

Multiple Testing Problems in

Pharmaceutical Statistics

Alex Dmitrienko, Ajit C Tamhane,

and Frank Bretz

Noninferiority Testing in Clinical Trials:

Issues and Challenges

Joseph C Cappelleri, Kelly H Zou,

Andrew G Bushmakin, Jose Ma J Alvir,

Demissie Alemayehu, and Tara Symonds

Quantitative Evaluation of Safety in Drug

Development: Design, Analysis and

Reporting

Qi Jiang and H Amy Xia

Randomized Clinical Trials of

Chul Ahn, Moonseong Heo, and Song Zhang

Sample Size Calculations in Clinical Research, Second Edition

Shein-Chung Chow, Jun Shao and Hansheng Wang

Statistical Analysis of Human Growth and Development

Yin Bun Cheung

Statistical Design and Analysis of Stability Studies

Statistical Methods for Drug Safety

Robert D Gibbons and Anup K Amatya

Statistical Methods in Drug Combination Studies

Wei Zhao and Harry Yang

Statistics in Drug Research:

Methodologies and Recent Developments

Shein-Chung Chow and Jun Shao

Statistics in the Pharmaceutical Industry, Third Edition

Ralph Buncher and Jia-Yeong Tsay

Survival Analysis in Medicine and Genetics

Jialiang Li and Shuangge Ma

Theory of Drug Development

Edited by Oleksandr Sverdlov

EMD Serono USA

Modern Adaptive

Randomized Clinical TrialsStatistical and Practical Aspects

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2 Efron’s Biased Coin Design Revisited: Statistical Properties,

Victoria Plamadeala

Alessandro Baldi Antognini1 and Maroussa Zagoraiou2

Olga M Kuznetsova1 and Yevgen Tymofyeyev2

7 Statistical Inference Following Covariate–Adaptive

D Stephen Coad

10 Response–Adaptive Randomization: An Overview of Designs

Li-Xin Zhang

11 Statistical Inference Following

Yanqing Yi1 and Xikui Wang2

12 Sample Size Re-Estimation in Adaptively Randomized

Ruitao Lin and Guosheng Yin

13 Some Caveats for Outcome Adaptive Randomization

Peter F Thall1, Patricia S Fox1 and J Kyle Wathen2

14 Efficient and Ethical Adaptive Clinical Trial Designs

Seung Won Hyun1, Tao Huang2 and Hongjian Zhu3

15 Longitudinal Covariate-Adjusted Response–Adaptive

Tao Huang1 and Hongjian Zhu2

16 Targeted Covariate-Adjusted Response–Adaptive

Antoine Chambaz1, Mark J van der Laan2 and Wenjing Zheng2,3

17 Covariate-Balanced Bayesian Adaptive Randomization:

Achieving Tradeoff between Inferential and Ethical Goals

Ying Yuan and Jing Ning

Contents ix

18 Multi-Arm Multi-Stage Designs for Clinical Trials

James Wason

Christina Yap1, Xuejing Lin2 and Ying Kuen K Cheung2

20 Accounting for Parameter Uncertainty in Two-Stage Designs

Emma McCallum1 and Bj¨orn Bornkamp2

21 A Single Pivotal Adaptive Trial in Infants with ProliferatingHemangioma: Rationale, Design Challenges, Experience

Tom Parke and Martin Kimber

23 Statistical Monitoring of Data in Response–Adaptive

Paul Gallo

of the trial results Depending on the trial objectives, one can distinguishfour major types of adaptive randomization: restricted, covariate–adaptive,response–adaptive, and covariate-adjusted response–adaptive The differentialfeature of each type is the data structure that forms the basis for designadaptations, from the simplest one, which includes a history of study patients’treatment assignments (restricted randomization), to the most complex one,which includes a history of study patients’ treatment assignments, covariates,responses, and a covariate profile of the current patient (covariate-adjustedresponse–adaptive randomization).

Adaptive randomization has been a hot topic of research in biostatisticssince the 1970s Clinical trials with adaptive randomization can frequently bemore flexible, more efficient, and more ethical than traditional fixed random-ization designs However, there is still some controversy among stakeholders

in academia, the pharmaceutical industry, and health authorities about themerits of adaptive randomization and when it is appropriate to apply such de-signs in practice The health authorities (US Food and Drug Administration,European Medicines Agency) exercise caution recommending a broad use ofadaptive randomization designs due to concerns about operational complex-ity, potentially higher vulnerability to experimental bias, and more complexstatistical inference following these designs

Over the past two decades significant progress has been made on cal and applied aspects of various adaptive randomization designs, with manypapers published in top statistical journals Valid statistical inference proce-dures following adaptive randomization have been developed Novel designsthat provide a tradeoff between competing experimental objectives have beenproposed Advances in information technology have led to the development

theoreti-of validated web-based systems to facilitate implementation theoreti-of adaptive domization in practice All these important developments signify that adap-tive randomization merits a fresh look from both statistical and regulatoryperspectives

ran-Is adaptive randomization always better than traditional fixed-schedulerandomization? Which procedures should be used and under which circum-

xii Modern Adaptive Randomized Clinical Trialsstances? What special considerations are required for adaptive randomizedtrials? What kind of statistical inference should be used to achieve valid andunbiased treatment comparisons following adaptive randomization designs?The present volume is intended to bring more insight into these questions andprovide information on recent advances in adaptive randomization.

The present volume is a collection of 23 chapters covering a wide spectrum

of topics in adaptive randomization designs in modern clinical trials The tributing authors are statisticians, clinical trialists, and subject matter ex-perts from academia and the pharmaceutical industry Some chapters provide

con-a fresh con-and criticcon-al look con-at con-alrecon-ady clcon-assiccon-al topics in con-adcon-aptive rcon-andomizcon-ation,whereas other chapters cover novel designs and very recent developments andapplications of adaptive randomization

The chapters are grouped into seven parts In Part I (Introduction), abird’s-eye view of different types of adaptive randomization is presented.Chapter 1 clarifies the taxonomy of the concept of adaptive randomizationand provides a general guidance on which designs and when they should beconsidered for use in practice

Part II is devoted to restricted randomization designs which aim at ing treatment assignments in the trial Chapter 2 discusses exact statisticalproperties, randomization-based inference, and sequential monitoring of thefamous Efron’s biased coin design Chapter 3 presents some important exten-sions of Efron’s design, the so-called adaptive biased coin designs, which can beused to achieve better treatment balance without compromising randomness

balanc-of treatment allocation Chapter 4 covers very recently developed advancedrandomization techniques such as brick tunnel randomization and wide bricktunnel randomization to achieve the pre-specified unequal allocation ratios inclinical trials

Part III deals with covariate–adaptive randomization designs which aim

at balancing treatment assignments with respect to important prognostic tors Chapter 5 discusses some stratified randomization procedures and a novelcovariate–adaptive randomization procedure called the minimal sufficient bal-ance randomization Chapter 6 discusses an important class of model-basedoptimal design covariate–adaptive randomization procedures Chapter 7 cov-ers some recent advances in statistical inference following covariate–adaptiverandomization Chapter 8 presents some novel covariate–adaptive randomiza-tion designs for studies with unequal allocation

fac-Part IV is devoted to response–adaptive randomization Chapter 9 cusses some novel optimal allocation designs for multi-arm clinical trials withcompeting requirements of treatment and inference Chapter 10 gives anoverview of various important classes of response–adaptive randomization de-signs and their asymptotic properties Chapter 11 discusses the approaches forproper statistical inference following a response–adaptive randomized clinicaltrial Chapter 12 covers an important topic of sample size re-estimation inadaptive randomized clinical trials with missing data Chapter 13 discusses

Preface xiiisome caveats inherent to response–adaptive randomization and possible com-plexities that can arise in such trials.

Part V is devoted to a novel type of adaptive randomization designs,covariate-adjusted response–adaptive (CARA) randomization, which is ap-plicable in complex clinical trials with treatment–covariate interactions wherebalanced designs may be suboptimal Chapter 14 presents efficient and eth-ical CARA randomization designs for binary outcome clinical trials Chap-ter 15 discusses longitudinal CARA randomization designs in clinical trialswith missing data Chapter 16 presents novel CARA designs based on a tar-geted maximum likelihood estimation methodology that preserves statisticalinference in a nonparametric model Chapter 17 presents covariate-balancedresponse–adaptive randomization designs which can simultaneously handlethe objectives of balancing covariate profiles between the treatment arms andassigning more patients to the empirically better treatment

Part VI is devoted to randomized designs with treatment selection ter 18 discusses multi-arm multi-stage (MAMS) designs which are new group-sequential approaches to randomized Phase II trials Chapter 19 presents se-quential elimination designs for multi-arm trials Chapter 20 presents a study

Chap-of two-stage optimal designs for phase II dose–response trials where the tive is to efficiently estimate the dose–response curve under model uncertainty.Finally, Part VII presents an application and practical aspects of adaptiverandomized clinical trials Chapter 21 presents a successful implementation of

objec-a single pivotobjec-al phobjec-ase II/III objec-adobjec-aptive triobjec-al in infobjec-ants with proliferobjec-ating hemobjec-an-gioma Chapter 22 discusses some practical aspects of phase II dose-rangingstudies Chapter 23 discusses statistical monitoring and interim analysis issues

heman-in response–adaptive randomized clheman-inical trials

I would like to extend my sincere gratitude to all the contributors andreviewers for their time and effort to make this volume appear I would like tothank John Kimmel of CRC Press for his guidance and coordination Finally, Idedicate this book to my colleagues in the scientific community and my familyand friends

Oleksandr Sverdlov PhD

Alessandro Baldi Antognini

Department of Statistical Sciences

Ying Kuen K Cheung

Mailman School of Public Health

Columbia University

New York City, New York, USA

D Stephen CoadSchool of Mathematical SciencesQueen Mary University of LondonLondon, United Kingdom

Patricia S Fox

MD Anderson Cancer CenterHouston, Texas, USAPaul Gallo

Novartis PharmaceuticalsEast Hanover, New Jersey, USAStephanie Gautier

Institut de Recherche Pierre FabreToulouse, France

Stephane HeritierMonash UniversityMelbourne, AustraliaTao Huang

Department of StatisticsShanghai University of Finance andEconomics

Shanghai, ChinaSeung Won HyunDepartment of StatisticsNorth Dakota State UniversityFargo, North Dakota, USAMartin Kimber

Tessella Ltd

Abingdon, Oxfordshire, UnitedKingdom

xvi Modern Adaptive Randomized Clinical TrialsOlga M Kuznetsova

Merck & Co., Inc

Rahway, New Jersey, USA

Ruitao Lin

Department of Statistics and

Actuarial Science

University of Hong Kong

Hong Kong, China

Sydney Medical School

The University of Sydney

MD Anderson Cancer Center

Houston, Texas, USA

Jean Jacques VoisardPierre Fabre DermatologieLavaur, France

Xikui WangDepartment of Statistics,University of ManitobaWinnipeg, Manitoba, Canada

James WasonMRC Biostatistics UnitCambridge Institute of Public HealthCambridge, United Kingdom

J Kyle WathenJanssen Research & DevelopmentTitusville, New Jersey, USA

Christina YapCancer Research UK Clinical TrialsUnit

University of BirminghamBirmingham, United Kingdom

Yanqing YiFaculty of MedicineMemorial University ofNewfoundland

Contributors xviiGuosheng Yin

Department of Statistics and

Actuarial Science

University of Hong Kong

Hong Kong, China

Ying Yuan

MD Anderson Cancer Center

Houston, Texas, USA

Zhejiang University City College

Hangzhou, Zhejiang Province, China

Wenle ZhaoDepartment of Public HealthSciences

Medical University of South CarolinaCharleston, South Carolina, USAWenjing Zheng

Division of BiostatisticsUniversity of California at BerkeleyBerkeley, California

andCenter for AIDS Prevention StudiesUniversity of California, SanFrancisco

San Francisco, California, USAHongjian Zhu

Department of BiostatisticsThe University of Texas School ofPublic Health at HoustonHouston, Texas, USA

Part I Introduction

An Overview of Adaptive Randomization Designs in Clinical Trials

Oleksandr Sverdlov

EMD Serono, Inc

CONTENTS

1.1 Introduction 4

1.2 Restricted Randomization 7

1.2.1 The Maximal Procedure 7

1.2.2 Biased Coin Designs 8

1.2.3 Randomized Urn Models to Balance Treatment Assignments 8

1.2.4 Brick Tunnel Randomization 9

1.3 Covariate–Adaptive Randomization 9

1.3.1 Minimization-Type Procedures 10

1.3.2 Model-Based Optimal Design Procedures 11

1.3.3 Covariate–Adaptive Randomization Designs That Seek Distributional Balance of Covariates 13

1.3.4 Criticism of Covariate–Adaptive Randomization Revisited 13

1.4 Response–Adaptive Randomization 14

1.4.1 Response–Adaptive Randomized Urn Models 14

1.4.2 Optimal Response–Adaptive Randomized Designs 15

1.4.3 An Example 17

1.4.4 Bayesian Adaptive Randomization 19

1.4.5 Criticism of Response–Adaptive Randomization Revisited 21

1.5 Covariate-Adjusted Response–Adaptive Randomization 22

1.5.1 Treatment Effect Mapping and Urn-Based CARA Randomization Designs 23

1.5.2 Target-Based CARA Randomization Designs 23

1.5.3 Utility-Based CARA Randomization Designs 24

1.5.4 Bayesian CARA Randomization 24

1.6 Other Designs with Elements of Adaptive Randomization 25

1.6.1 Randomized Phase I Trial Designs 25

4 Modern Adaptive Randomized Clinical Trials

1.6.2 Adaptive Optimal Dose-Finding Designs 25

1.6.3 Randomized Designs with Treatment Selection 26

1.6.4 Group Sequential Adaptive Randomization 27

1.6.5 Complex Adaptive Design Strategies 28

1.7 Concluding Remarks 29

Bibliography 30

A randomized, placebo-controlled, double-masked, equal-allocation clinical trial can be viewed as an exemplary research design to obtain generalizable re-sults on the treatment effect However, modern clinical trials are increasingly complex and often require more elaborate designs A competitive landscape of pharmaceutical research and development, an enormous number of molecules that are available as potential drugs, and limited patient resources call for clini-cal trial designs investigating the effects of multiple treatments within multiple patient subgroups Such designs should, in addition, satisfy strict regulatory requirements such as controlling the chance of making a type I error

Recognizing the challenges for research and development and recent trends for productivity decline, in 2006 the US Food and Drug Administration (FDA) released the Critical Path Initiative [72] and the Critical Path Opportunities Report [73]—two strategic documents that encourage innovation in drug de-velopment One aspect of innovation is adaptive designs—clinical trial designs that facilitate efficient learning from data in an ongoing trial and allow mod-ification of certain aspects of the study according to pre-specified criteria to achieve some pre-determined experimental objectives [74] Adaptive designs have a potential to outperform traditional parallel group fixed randomiza-tion designs by treating trial participants more efficiently, identifying promis-ing treatments more rapidly, and minimizpromis-ing unnecessary expenditures while maintaining validity and integrity of the results [61]

Dragalin’s [61] classification of adaptive designs distinguishes four major types of adaptation:

1 Adaptive allocation rule—change in the randomization procedure to mod-ify the allocation proportion or the number of treatment arms

2 Adaptive sampling rule—change in the number of study subjects or change

in study population

3 Adaptive stopping rule—early stopping due to efficacy, futility, or safety

4 Adaptive decision rule—change in the way decisions will be made about the trial (e.g., change of endpoint, change of test statistics, etc.)

An Overview of Adaptive Randomization Designs in Clinical Trials 5adaptation See Chow and Chang [58] for a treatise on various adaptive designs

in clinical trials

The current volume deals with adaptive randomization designs—the signs that fall in the first category of Dragalin’s [61] classification Rosenberger,Sverdlov and Hu [130] define adaptive randomization as a class of randomiza-tion procedures for which treatment allocation probabilities are sequentiallymodified based on accumulating data in the trial to achieve selected experi-mental objectives while protecting the study from bias and preserving infer-ential validity of the trial results Formally, let Ω = {T1, T2, , TK} denotedifferent treatment arms, which may represent K different doses of a drug or

de-K different intervention strategies, to be compared in a clinical trial ble patients are enrolled into the trial in cohorts of size c, where c is a fixedsmall positive integer Frequently, c = 1, which is referred to as sequentialenrollment Each patient will be randomized to receive one of the treatmentsfrom Ω For the jth patient, let δj ∈ Ω denote the treatment assignment,

Eligi-zj = (z1j, , zpj)0 denote a vector of important covariates (prognostic tors) observed at baseline, and Yj denote the primary outcome (response) Weassume that Yj, conditionally on δj and zj follows a statistical model

fac-E(Yj|δj, zj) = g(θ, δj, zj), (1.1)where g(.) is some regression function and θ is a vector of model parameters in-cluding the effects of treatments, covariates, and possibly treatment–covariateinteractions Note that statistical model (1.1) is used as a starting point tofacilitate the design, whereas the final inference may be based on a different,possibly nonparametric model Suppose m ≥ 1 patients have been randomizedinto the trial For j = 1, , m, let (δj, zj, yj) denote the data from the jth pa-tient (lowercase yj is used instead of Yj to emphasize the observed response).The (m + 1)st patient with covariate vector zm+1 enters the trial and must

be randomized to one of the K treatments A general adaptive randomizationprocedure is defined by specifying conditional randomization probabilities oftreatment assignments as follows:

Pm+1,k= Pr(δm+1= Tk|Dm), k = 1, , K, m ≥ 1, (1.2)where Dm is the data structure that forms the basis for design adaptations.Depending on the trial objectives, one can distinguish four types of adaptiverandomization designs [130]:

• Restricted randomization, ifDm= {δ1, , δm}, the history of previous tients’ treatment assignments The goal is to prospectively balance treat-ment numbers in the trial

pa-• Covariate–adaptive randomization, ifDm= {(δ1, z1), , (δm, zm), zm+1},the history of previous patients’ treatment assignments and covariates, andthe covariate vector of the current patient The goal is to prospectivelybalance treatment assignments overall in the trial and across selected co-variates

6 Modern Adaptive Randomized Clinical Trials

FIGURE 1.1

Classification of adaptive randomization designs

• Response–adaptive randomization, if Dm = {(δ1, y1), , (δm, ym)}, thehistory of previous patients’ treatment assignments and responses Themost common goal is to increase the chance for a patient to be assigned to

a potentially better treatment Other possible goals may include increasingestimation efficiency of the desired treatment effect or maximizing thepower of a statistical test

• Covariate-adjusted response–adaptive randomization, ifDm= {(δ1, z1, y1), , (δm, zm, ym), zm+1}, the history of previous patients’ treatment as-signments, responses and covariates, and the covariate vector of the currentpatient The most common goal is to increase the chance for a patient to

be assigned to a potentially better treatment given the patient’s covariateprofile while maintaining the power of a statistical test

The class of adaptive randomization procedures can be extended further byincluding adaptive designs with treatment selection for which randomizationprobabilities for some treatment arms can be set to 0 throughout the trial.This chapter provides a bird’s-eye view of various adaptive randomiza-tion designs available in the literature The roadmap for the presentation inSections 1.2–1.6 is displayed in Figure 1.1 Section 1.7 outlines some futurework and perspectives on the use of adaptive randomization designs in con-temporary clinical trials Other key references on adaptive randomization arethe books by Rosenberger and Lachin [127], Hu and Rosenberger [88], andAtkinson and Biswas [9]

An Overview of Adaptive Randomization Designs in Clinical Trials 7

Restricted randomization is appropriate in clinical trials where balance andrandomness are two major requirements Balance is important for statisti-cally efficient treatment comparisons Randomization helps neutralize variousexperimental biases and can form the basis for valid statistical inference.Consider a two-arm clinical trial comparing an experimental treatment ver-sus control The most random procedure is the completely randomized design(CRD) for which each subject is randomized between treatments with proba-bility 1/2 and the assignments are mutually independent The CRD balancestreatment assignments asymptotically; however, it may, with high probability,result in large departures from balance in small samples To achieve balancedallocation throughout the trial, various restricted randomization procedureshave been developed The most common restricted randomization procedure

is the permuted block design (PBD) for which treatment assignments areequalized in blocks by either a random allocation rule or a truncated binomialdesign [127] A drawback of the PBD is that allocations at the tail of eachblock can be guessed with high probability, which may introduce selection bias

in the design This is particularly troublesome in single-institution unmaskedtrials where investigators can keep track of treatment assignments and mayselectively enroll patients who are thought to benefit most from a given treat-ment Several approaches were proposed to overcome a limitation of PBD andobtain less restrictive randomization procedures

1.2.1 The Maximal Procedure

In an effort to balance the conflicting requirements of treatment allocationbalance and randomness of treatment assignments, Berger, Ivanova and Knoll[33] proposed the maximal procedure, a restricted randomization design thatcontrols the maximum treatment imbalance between two treatment groupswhile providing the maximum amount of randomization An important fea-ture of the maximal procedure is that all randomization sequences in its allo-cation space (in the case of 1 : 1 randomization) are equally likely; therefore

it can be regarded as a constrained permuted block randomization where onlysequences with maximum imbalance not exceeding a pre-specified value areallowed Berger, Ivanova and Knoll [33] summarize key advantages of the max-imal procedure over the sequence of permuted blocks of small size First, themaximal procedure has fewer deterministic allocations than the sequence ofpermuted blocks This is because there is no requirement of achieving perfectbalance at intermediate steps for the maximal procedure since the maximalimbalance is controlled throughout the trial Second, as shown via simulations[33], the maximal procedure allows for less inflation of the type I error ratethan the sequence of permuted blocks with fixed or varying block sizes in case

8 Modern Adaptive Randomized Clinical Trialsbias is present The maximal procedure performs better than the sequence

of permuted blocks and is especially useful for randomizing patients withinstrata For a given sample size and a given value of the maximal toleratedimbalance, a treatment allocation sequence from the maximal procedure can

be pre-generated before any subject is enrolled into the study An efficientalgorithm for implementing the maximal procedure was proposed by Salama,Ivanova and Qaqish [133]

1.2.2 Biased Coin Designs

From a statistical perspective, slight deviations from equal allocation haveminimal impact on statistical power Efron [64] introduced a biased coin design(BCD) to achieve nearly balanced treatment assignments at any stage of thetrial For a two-arm trial, let δj = 1 if the jth subject is assigned to treatment

T1 and δj = 0 if the jth subject is assigned to treatment T2 After j ments, the treatment group sizes are N1(j) =Pj

assign-i=1δiand N2(j) = j − N1(j),respectively, and the treatment imbalance is Dj = N1(j) − N2(j) For ev-ery j ≥ 1, Dj is a random variable whose distribution is determined by therandomization procedure If Dj = 0, the treatments are balanced; if Dj < 0(> 0), the treatment T1(T2) is underrepresented Efron’s [64] BCD procedurecan be formulated as follows: if Dj = 0, the (j + 1)st subject is randomized

to either treatment with probability 1/2; if |Dj| > 0, the (j + 1)st subject

is randomized to the underrepresented treatment with some fixed probability

p > 1/2 Efron’s BCD procedure has now well-established exact properties[114] which can be used to construct randomization-based tests and facilitategroup sequential monitoring (cf Chapter 2)

Efron’s BCD was extended in a number of ways by letting the coin biasprobability depend on the magnitude of treatment imbalance The main BCDextensions include the adjustable BCD [15], the generalized BCD [141], andthe Bayesian BCD [21] See Chapter 3 for a detailed review of these designs

1.2.3 Randomized Urn Models to Balance

Treatment Assignments

Wei [163, 165] proposed a class of urn designs to sequentially balance ment assignments An urn initially contains w ≥ 0 balls of two colors, whiteand red The w/2 white balls represent treatment T1 and the w/2 red ballsrepresent treatment T2 When an eligible patient enters the trial, a ball israndomly drawn from the urn and the patient is assigned to the treatmentaccording to the color of the ball drawn The ball is then returned to the urntogether with α ≥ 0 balls of the same color and β > α balls of the oppositecolor, where α and β are predetermined numbers This design is denoted byUD(w, α, β) Let Wk(j) denote the number of color k balls after j assignments.Then Wk(j) = w/2 + αNk(j) + β(j − Nk(j)), and the probability that the

An Overview of Adaptive Randomization Designs in Clinical Trials 9(j + 1)st patient is assigned to treatment T1 is equal to

1.2.4 Brick Tunnel Randomization

In a randomized comparative trial with K ≥ 2 treatment arms, an investigatormay want to achieve some fixed treatment allocation ratio C1: C2: : CK,where Ck’s are positive integers with the greatest common divisor of 1 (Ck≡ 1for k = 1, , K corresponds to equal allocation) Let ρk = Ck/PK

j=1Cj note the target allocation proportions, where ρk ∈ (0, 1) and PK

de-k=1ρk = 1.For a trial of size n, Nk(n) subjects are randomized to the kth treatment Ingeneral, Nk(n) are random variables with PK

k=1Nk(n) = n For a suitablychosen n, a permuted block design can be cast to achieve n−1Nk(n) = ρk for

k = 1, , K However, at intermediate steps, deviations from the target cation ratio may be substantial Kuznetsova and Tymofyeyev [106] proposedthe brick tunnel randomization (BTR) which reduces the allocation space com-pared to the permuted block space, thereby providing a closer approximation

allo-to the target allocation throughout the trial An important highlight of BTR

is that it possesses an allocation ratio preserving property—at each step theunconditional probability of treatment assignment is the same as the targetallocation See Chapter 4 for more details on BTR

In many clinical trials, there are important baseline prognostic factors ates) such as age, gender, and disease severity that are known to be correlatedwith the outcome of a patient In order to have interpretable results, thesecovariates should be balanced across treatment arms Covariate–adaptive ran-domization is a class of randomization procedures which attempt to prospec-tively balance treatment assignments across selected covariates while main-taining allocation randomness McEntegart [115] gives various reasons for

10 Modern Adaptive Randomized Clinical Trialspursuing covariate-balanced designs One of the key reasons is statistical effi-ciency Under a homoscedastic linear model for the primary outcome, a designthat balances covariate profiles between treatment arms is also statisticallyefficient in the sense that it minimizes variance of the estimated treatmentdifference.

Once influential covariates have been identified and the decision is made

to balance treatment allocation with respect to these covariates, variouscovariate–adaptive randomization procedures can be considered The sim-plest one is stratified randomization [177] For a selected set of discrete orcategorical covariates, one forms mutually exclusive strata by taking all pos-sible combinations of covariate levels, and within each stratum randomization

is implemented using some restricted randomization procedure Stratum-levelbalance is particularly important under the homoscedastic linear model withinteractions among covariates and also when the planned analysis involvesstratified tests [100]

If the number of strata is small and the trial size is small or moderate,stratified randomization followed by stratified analysis can improve the pre-cision of estimators and the power of statistical tests [68, 78, 79] When thenumber of covariates is large, stratification may not achieve its goal (balancewithin strata) because some strata may contain only a few patients or may

be empty One can distinguish three types of CAR to achieve balance over alarge number of influential covariates without “overstratification.” These arediscussed in §1.3.1—§1.3.3 below

1.3.1 Minimization-Type Procedures

The minimization procedure was developed independently by Taves [151] andPocock and Simon [121] Unlike stratified randomization, which pursues bal-ance within strata, minimization pursues balance within covariate margins.Such marginal balance is sufficient for unbiased estimation of the treatmenteffect if responses follow a standard linear model with additive effects of thetreatment and covariates, but not their interactions [165]

The minimization procedure for a trial with treatments T1 and T2 can

be described as follows Suppose we have M discrete covariates Z1, , ZMsuch that Zi has ≥ 2 levels For a new patient with covariate profile z =(z1, , zM), compute {di1}M

i=1, the treatment imbalances within observedmargins z1, , zM, which would result if the patient is assigned to treat-ment T1 Compute G1 =PM

i=1wid2i1, the “overall imbalance” score from theassignment to treatment T1 Here {wi}M

i=1 are pre-specified positive weightsmeasuring the prognostic importance of the covariates satisfyingPM

i=1wi= 1.Similarly, for treatment T2compute G2=PM

i=1wid2 i2 If G1− G2= 0, the pa-tient is randomized to either treatment with probability 1/2 If |G1− G2| > 0,the patient is randomized to the treatment with the smaller value of Gk withsome fixed probability p > 1/2 to reduce imbalance Taves [151] suggested

An Overview of Adaptive Randomization Designs in Clinical Trials 11can be extended by considering different metrics of covariate imbalance, dif-ferent values of p (it can be chosen as a function of imbalance) and to K > 2treatment arms [121].

A comprehensive review of the minimization procedure can be found inScott et al [134] Simulation studies comparing minimization with completerandomization and stratified permuted block randomization suggest that min-imization improves balance for a large number of covariates and may improvethe power of the trial provided that covariates balanced in the design areaccounted for in the analysis [2, 3, 82, 135, 156, 158, 160, 167] Theoretical as-pects of minimization, including predictability and statistical inference, havebeen studied in the papers [28, 29, 136, 137] Recent literature reviews indicateincreased popularity of minimization in clinical trials [122, 152]

Minimization promotes balance within covariate margins but not sarily within strata To ensure balance at all three levels (stratum, covariatemargins, and trial overall), the “overall imbalance” score can be calculated

neces-as a weighted average of squared imbalances within the stratum, within thecovariate margins, and within the trial This approach was investigated by

Hu and Hu [86] and Lebowitsch et al [108] In particular, Hu and Hu [86]showed theoretically that within-stratum imbalances under this procedure arebounded in probability as sample size increases, whereas for the minimizationprocedure, these imbalances have fast-increasing variances Consequently, thenew method provides overall better balance than minimization This is alsoconfirmed by simulation under various sample sizes and covarite structures.Baldi Antognini and Zagoraiou [19] proposed an adjustable biased coin de-sign to promote balance both marginally and within strata and showed the-oretically that the design provides higher-order approximation to treatmentbalance than minimization

Several authors proposed hierarchical balancing allocation schemes whichpursue balance according to a pre-specified hierarchy of classification factors.These proposals include self-adjusting randomization plan [118], dynamic bal-ancing [84, 138], sequential balancing [44], and hybrid approaches [100, 110].Hierarchical balancing examines imbalance separately within each level in thehierarchy, starting, for example, with the stratum and proceeding through apre-determined order of the covariate margins and the overall trial If withinsome particular level the imbalance exceeds a pre-specified threshold, the newpatient is randomized to the underrepresented treatment with probabilitygreater than 1/2 to reduce imbalance This is different from minimization,which attempts to minimize, at each step, an overall imbalance score derivedfrom all covariates

1.3.2 Model-Based Optimal Design Procedures

For categorical covariates, minimization-type procedures promote balancewithin various covariate subgroups An alternative approach is to develop anallocation procedure which sequentially minimizes variance of the estimated

12 Modern Adaptive Randomized Clinical Trialstreatment difference in the presence of covariates This approach is based onoptimal design theory for linear models [1].

Suppose the following linear model defines the relationship between studyoutcome, treatment, and covariates:

E(Yj) = z0jβ + αδj, Var(Yj) = σ2 (j = 1, , n), (1.3)where Yjis the jth patient’s outcome, δj= 1(−1) if the jth patient is assigned

to treatment T1(T2), z0jis the 1×q vector of the jth patient’s covariates ing the intercept, and α is the difference between the treatment effects and β

includ-is the vector of covariate effects Define Yn= (Y1, , Yn)0, δn = (δ1, , δn)0,and let Zn denote the n × q matrix of covariate values Let Xn= (Zn, δn) and

θ =



βα

 The best linear unbiased estimator of θ is bθ = (X0nXn)−1X0nYn,with variance–covariance matrix Var(bθ) = σ2(X0nXn)−1 The variance of theestimated treatment difference is the lower diagonal element of Var(bθ):

Var(α) = σb 2{n − δ0nZn(Z0nZn)−1Z0nδn}−1 (1.4)Clearly Var(α) is minimized when the quantity Lb n = δ0nZn(Z0

nZn)−1Z0

nδn(referred to as loss [47]) is zero This is achieved when δn is orthogonal to thecolumns of Zn, i.e., δ0nZn = 0 Orthogonality is equivalent to different types

of balance (depending on the structure of Zn), including balance in treatmenttotals, balance within levels and/or within strata formed by crossing of thelevels of discrete covariates, and equal sums of continuous covariate values inthe two groups

To construct a sequential randomization procedure minimizing Var(α),bsuppose we have treatment assignments and covariates from n patients, andthe (n + 1)st patient enters the trial with covariate vector zn+1 Then thepatient’s assignment is chosen to minimize the loss Ln+1with high probability

A family of randomization procedures to achieve this goal was proposed bySmith [141]: it prescribes randomizing the (n + 1)st patient to treatment T1with probability

Pr(δn+1= 1|δn, Zn, zn+1) = φγ(z0n+1(Z0nZn)−1Z0nδn), (1.5)where φγ(x) = (1 − x)γ/{(1 − x)γ + (1 + x)γ} is non-decreasing in x and

γ ≥ 0 is a user-defined parameter controlling the degree of randomness (γ =

0 is completely randomized and γ → ∞ is almost deterministic balancedprocedure) Smith [141] makes an important observation that for designs inthis class the expected loss is approximately q(1+2γ)−1where q is the number

of columns in Zn Therefore, the loss is an increasing function of the number

of covariates in the model, and designs with larger values of γ result in lowerloss and better balance but at the expense of reduced randomness Smith[141] recommends using γ = 5 to achieve a reasonable tradeoff between lossand bias Atkinson [2–6] performed extensive simulation studies to comparevarious covariate–adaptive randomization designs in terms of loss and bias

An Overview of Adaptive Randomization Designs in Clinical Trials 13

1.3.3 Covariate–Adaptive Randomization Designs That

Seek Distributional Balance of Covariates

The ultimate goal of randomization is to achieve distributional balance ofbaseline covariates between the groups [93] A good metric of covariate im-balance should capture the difference of the whole distributions instead of thedifference in lower dimensional characteristics Endo et al [67] suggested usingKullback–Leibler divergence of two probability density functions (small value

of this metric indicates that treatment groups have similar distributions of acovariate) For a normally distributed covariate, minimizing Kullback–Leiblerdivergence implies similarity of the treatment groups in terms of means andvariances Su [142] suggested minimizing the standardized maximum absolutedifference between quartiles of covariate values in the two groups Hu and Hu[86] proposed minimizing the maximum group size difference over all possibledivisions of the covariate range Lin and Su [110] proposed the total area min-imization, which minimizes the normalized area between empirical cumulativedistribution functions for the treatment and control groups Ma and Hu [112]proposed a balancing method based on kernel densities which minimizes thedifference between probability densities of a covariate in the two groups Thelatter three methods maintain good distributional balance for both contin-uous and categorical covariates and achieve well-balanced group sizes Suchenhanced balance frequently translates into higher statistical power and bet-ter estimation precision for trials with continuous, binary, and time-to-eventoutcomes [110]

1.3.4 Criticism of Covariate–Adaptive

Randomization Revisited

In 2003, the European Committee for Proprietary Medicinal Products(CPMP) issued the guideline “Points to Consider on Adjustment for Base-line Covariates” [59] which expressed the opinion that dynamic allocationschemes such as minimization “remain highly controversial” and are “stronglydiscouraged.” This led to several follow-up discussions and commentaries[48, 49, 60, 123] In essence, the CPMP’s discouraging position concerns threeaspects: 1) predictability of covariate–adaptive randomization procedures; 2)controversy on proper statistical inference; and 3) logistical and practical com-plexity Since the issuance of the CPMP guidance in 2003, significant method-ological research has been completed on minimization and other covariate-adaptive randomization procedures Let us revisit the CPMP’s critical pointsafresh

1) Predictability In the original proposal by Taves [151], the tion procedure has no random element Inclusion of a biased coin [121]can reduce predictability at the expense of somewhat higher imbalance.Brown et al [46] and McPherson, Campbell and Elbourne [116] providerecommendations on how to judiciously select the value of the coin bias

14 Modern Adaptive Randomized Clinical TrialsAtkinson’s [3] admissibility plots can be also used to select a design that

is admissible in terms of predictability and balance

2) Proper Statistical Inference The validity and power of statistical tests lowing minimization was studied via simulation in the papers [28, 36, 75,

fol-99, 101, 132, 158] A general conclusion is that minimization achieves validand unbiased comparison and can improve power, provided that analysis

is adjusted for the covariates included in the design When an unadjustedanalysis is used, the tests are conservative Shao, Yu and Zhong [137], Shaoand Yu [136], and Ma and Hu [111] give theoretical justifications for propermodel-based statistical inference following covariate–adaptive randomiza-tion, and Hasegawa and Tango [83] and Simon and Simon [140] developedapproaches to randomization-based inference following covariate–adaptiverandomization

3) Implementation With advances in information technology, tion of trials with covariate–adaptive randomization should be straight-forward Operationally, such trials should be performed by a centralizedallocation unit using interactive voice response systems [48] Validatedinformation systems to implement covariate–adaptive randomization arediscussed in the papers [50, 51, 169, 170]

implementa-In summary, the knowledge on covariate–adaptive randomization has vanced substantially over several past decades A revised guidance on ad-justment for baseline covariates from the European Medicines Agency [66] ispending finalization

Response–Adaptive Randomization (RAR) has roots in non-randomized tive assignment procedures for selecting the best treatment, such as the play-the-winner rule by Zelen [176] Simon [139] notes that the deterministic na-ture of adaptive treatment assignment procedures is one of the major reasonsthat limits their use in practice The idea of incorporating randomization intoresponse–adaptive assignment for binary outcome trials is due to Wei andDurham [166] who introduced the randomized play-the-winner (RPW) rule.Since then, the body of knowledge on RAR has grown substantially Majorclasses of RAR designs are discussed below in §1.4.1, §1.4.2, and §1.4.4

adap-1.4.1 Response–Adaptive Randomized Urn Models

With the RPW rule of Wei and Durham [166], the treatment assignment for anew patient is determined by the color of a ball drawn from the urn, and the

An Overview of Adaptive Randomization Designs in Clinical Trials 15urn composition is sequentially updated based on accruing responses (success

or failure) from patients in the trial such that the ball representing a moresuccessful treatment is more likely to be chosen Rosenberger [125] gives anoverview of the RPW and its occasional applications in clinical trials [31, 150].The RPW rule is only one example of a broad class of randomized urn mod-els [126] Ivanova [95] introduced the drop-the-loser rule, an urn design withthe same limiting allocation as the RPW rule but with much lower variabil-ity and more desirable statistical properties Other notable RAR urn designsinclude the sequential estimated-adjusted urn [183], the generalized drop-the-loser rule [143, 181], the optimal adaptive generalized P´olya urn [174], andthe randomly reinforced urn [70] Zhang et al [185] provide a unified theory

of optimal urn designs for clinical trials

1.4.2 Optimal Response–Adaptive Randomized Designs

Hu and Rosenberger [87] proposed a mathematical template for developingoptimal RAR procedures (the term “optimal” shall be discussed momentar-ily) The template consists of three steps: 1) deriving an optimal allocation forthe selected experimental objectives; 2) constructing a RAR procedure withminimal variability to converge to the optimal allocation; and 3) assessing theoperating characteristics of the chosen design under a variety of standard toworst-case scenarios Let us examine these three steps in detail

The first step is optimal allocation Suppose that Yk, the outcome of

a patient on treatment Tk (k = 1, , K), follows some statistical modelE(Yk) = gk(θ), where θ is a vector of unknown model parameters and gk(.)are known regression functions Let ρ = (ρ1, , ρK)0 denote a design thatallocates ρk proportion of the total subjects to treatment k (0 ≤ ρk ≤ 1

k=1ρk = 1) Let M(ρ, θ) denote the Fisher information matrix for θgiven design ρ Importantly, M−1(ρ, θ) provides the lower bound on the vari-ance of an unbiased estimator of θ, and by minimizing M−1(ρ, θ) in somesense (by choice of ρ) one can achieve most accurate inference for the pa-rameters of interest Frequently, optimal allocation proportions depend on θ,i.e., ρk = ρk(θ), k = 1, , K Sverdlov and Rosenberger [144] give a compre-hensive overview of various single- and multiple-objective optimal allocationdesigns that are available in the literature Many of these designs optimizesimultaneously several criteria related to inferential efficiency and ethical con-siderations and yield unequal allocation proportions across treatment arms.The factors that may contribute to unequal allocation include nonlinearity ofregression functions, heterogeneity of the outcome variances, unequal interest

in specific treatment comparisons, and ethical and/or budgetary constraints.Baldi Antognini and Giovagnoli [16] and Baldi Antognini, Giovagnoli andZagoraiou [17] summarized allocation targets to achieve tradeoff between in-ference and ethics in binary outcome trials Azriel, Mandel and Rinott [11]and Azriel and Feigin [10] proposed some novel optimal targets for maximizingpower based on large deviations theory (cf Chapter 9)

16 Modern Adaptive Randomized Clinical TrialsThe second step is constructing a RAR design that sequentially converges

to the chosen optimal allocation ρ = (ρ1(θ), , ρK(θ))0 An initial cohort of

Km0patients (where m0is some small positive integer) is randomized to ments T1, , TK using some restricted randomization design This is done toascertain initial data for estimating θ Consider a point when m(≥ Km0)patients have been randomized into the trial Let bθm denote the maximumlikelihood estimator of θ,ρbm= (bρ1m, ,ρbKm)0 denote the estimated targetallocation (ρbkm= ρk(bθm)), and Nm/m = (N1m/m, , NKm/m)0denote thevector of treatment allocation proportions (Nkm/m is the proportion of as-signments to treatment Tk among the m patients) Then the (m + 1)st patient

treat-is randomized to treatment Tk with probability πm+1,k = πk(bρm, Nm/m),where πk’s are some appropriately chosen functions (0 ≤ πk ≤ 1 and

k=1πk = 1)

Hu and Zhang [90] extended the work of Eisele [65] and proposed thedoubly adaptive biased coin design (DBCD) for which the πk’s are defined asfollows:

random-is strongly consrandom-istent for ρ and follows an asymptotically normal drandom-istributionwith variance–covariance matrix Σγ = (1 + 2γ)−1{Σ1+ 2(1 + γ)ΣLB}, where

Σ1= diag{ρ1(θ), , ρK(θ)} − ρρ0 and ΣLB is the lower bound on the ance of a RAR procedure targeting ρ The expression for ΣLBdepends on thegradient of ρ and can be found using the methodology of Hu and Zhang [90].The third step is a comparison of various candidate RAR procedures Huand Rosenberger [87] derived a relationship between optimality, variability,and power which can be used to facilitate such a comparison In general,statistical properties of an RAR procedure are determined by the allocationtarget, the speed of convergence to the target, and the variability of treatmentallocation proportions Remarkably, the speed of convergence to the target forthe majority of RAR designs is of order n−1/2 due to the central limit the-orem for the parameter estimators employed in the designs For procedurestargeting the same allocation and for which the allocation proportions areasymptotically normal, only variability of the procedures can be compared

vari-Hu, Rosenberger and Zhang [89] introduced asymptotically best RAR dures as ones that attain a lower bound on the asymptotic variance of alloca-

An Overview of Adaptive Randomization Designs in Clinical Trials 17[176] play-the-winner rule are asymptotically best for the limiting allocationproportion of a randomized urn model, which is also the limiting allocationfor the RPW rule of Wei and Durham [166] The DBCD procedure of Hu andZhang [90] is not asymptotically best unless γ → ∞ Hu, Zhang and He [91]proposed efficient randomized adaptive designs (ERADE) for two-arm trialswhich are fully randomized, can target any allocation, and are asymptoticallybest ERADE designs for (K > 2)-arm trials are yet to be found.

Some notable optimal RAR designs seeking tradeoff between statisticalefficiency and ethical criteria were developed for two-arm trials with bi-nary outcomes [76, 96, 128, 171, 172, 178], normal (Gaussian) outcomes[37, 39, 41, 80, 179], and survival outcomes [147, 180] For clinical tri-als with more than two treatment arms, Tymofyeyev, Rosenberger and Hu[161] developed a framework for finding optimal allocation designs minimiz-ing a weighted sum of group sizes subject to the minimal constraints onpower of the homogeneity test They used the DBCD procedure to sequen-tially implement optimal allocation for binary outcome trials Subsequently,more research was completed on optimal RAR designs for multi-arm trialswith possibly heteroscedastic outcomes and multiple experimental objectives[10, 42, 81, 98, 148, 149, 187]

In summary, the choice of an RAR procedure for practice should be madeafter a careful examination of operating characteristics of candidate designsfor a range of experimental scenarios Graphical approaches for visualizingdesign characteristics can be useful [69] A “good” RAR design should have

a reasonably high degree of allocation randomness, low variability and highspeed of convergence to the chosen optimal allocation

1.4.3 An Example

Consider a parallel five-arm trial comparing the effects of experimental ments T2, , T5 and the control treatment T1 with respect to a binary out-come Such trials are common in phase II of drug development (e.g., dose-ranging studies) Let pk denote the success probability for the kth treatment.Suppose the study objectives are two-fold: (i) to estimate the vector of treat-ment contrasts pc= (p2− p1, , p5− p1)0 as precisely as possible, and (ii) toassign study patients more frequently to treatments with higher success rates.The first objective is achieved by the DA-optimal allocation ρ∗= (ρ∗1, , ρ∗5)0which minimizes the log determinant of the variance–covariance matrix ofb

treat-pc= (bp2−pb1, ,pb5−pb1)0 More specifically, ρ∗ is found as a unique solution

to the following nonlinear system of equations [168]:

1

ρ∗ k

− {pk(1 − pk)}

−1

P5 j=1ρ∗j{pj(1 − pj)}−1 = 4, k = 1, , 5 (1.7)For the second objective, we take a treatment effect mapping approach [131]and define an “ethical” allocation vector ρE = (ρE1, , ρE5)0 with compo-

18 Modern Adaptive Randomized Clinical Trialsnents

ρEk= pk/(1 − pk)

P5 j=1pj/(1 − pj), k = 1, , 5. (1.8)The proportions ρEkin equation (1.8) are ordered consistently with the values

of treatment success rates: if pi ≥ pj for some i 6= j, then ρEi ≥ ρEj (withequality if and only if pi= pj) Following Sverdlov, Ryeznik and Wong [148],define a weighted optimal (WO) allocation ρα= (ρα1, , ρα5)0 as follows:

ραk = (1 − α)ρ∗k+ αρEk, k = 1, , 5 (1.9)

In equation (1.9), α ∈ [0, 1] is a pre-specified parameter that determines thetradeoff between efficiency and ethics If α = 0, we have the DA-optimalallocation; if α = 1, we have the “ethical” allocation; if 0 < α < 1, we have

an allocation providing a middle ground between efficiency and ethics Tocalibrate ρα, we consider the following characteristics:

• DA-efficiency: the ratio of the volume of the confidence ellipsoid for pcfrom the DA-optimal allocation ρ∗ to the similar quantity from WO al-location ρα For instance, the value of DA-efficiency of 0.95 means ρα is95% as efficient as ρ∗

• Power for testing the homogeneity hypothesis H0: pc= 0 vs H1: pc6= 0

We use the Wald statistic Wn=bp0cΣb−1n bpc, wherebpcis the maximum lihood estimator of pc and bΣn is a consistent estimator of Σn= Var(bpc).Given n and pc, the power could be approximated asymptotically asPr(ξ > χ24,0.95), where ξ follows a non-central chi-squared distribution with

like-4 degrees of freedom and noncentrality parameter p0cΣ−1n pc and χ2

4,0.95 isthe 95th percentile of the central chi-squared distribution with 4 degrees

of freedom

• Expected Proportion of Successes: EP S =P5

k=1ραkpk.Suppose p1= 0.2 and it is hypothesized that increases in success probabili-ties are linear with increments of 10 percentage points, i.e., pk = 0.2+0.1(k−1)for k = 2, 3, 4, 5 Under this scenario, a fixed equal allocation (EA) design with

a total sample size n = 150 (30 patients per arm) has 89% power, > 99% DAefficiency, and EP S = 0.4 (60 expected successes in a trial with 150 patients).The corresponding operating characteristics of WO allocation ρα for α from

-0 to 1 are plotted inFigure 1.2

For α = 0, the WO allocation is similar to the EA; as α increases, the

DA-efficiency and power of WO allocation are decreasing, but the EP S isincreasing The WO allocation with α = 0.5 (equal weight to efficiency andethics) has 95% DA-efficiency, 87% power, and EP S = 0.442 (66 expectedsuccesses in a trial with 150 patients)

Since the true success probabilities and the corresponding optimal

An Overview of Adaptive Randomization Designs in Clinical Trials 19

Operating Characteristics of Two Allocation Designs

we only present the results for one hypothesized experimental setup: p1= 0.2and pk = 0.2 + 0.1(k − 1) for k = 2, 3, 4, 5 Figure 1.3 shows the distribu-tions of treatment allocation proportions based on 10,000 simulation runs,andTable 1.1summarizes the key operating characteristics

Importantly, and consistent with theory, both CRD and RAR designs havenormally distributed treatment allocation proportions centered around theirtargeted values (Figure 1.3) FromTable 1.1, the RAR design has the sameaverage power (87%), slightly lower median DA-efficiency (0.963 for RAR and0.992 for CRD), and has, on average, 5.4 more successes compared to theCRD

1.4.4 Bayesian Adaptive Randomization

Bayesian adaptive randomization was originally proposed by Thompson [157].The idea is to set treatment randomization probability based on the value of

a posterior probability that one treatment is better than the other Consider

an allocation problem for a randomized trial comparing treatments T1 and

20 Modern Adaptive Randomized Clinical Trials

Treatment Arm

0.1 0.2 0.3 0.4 Completely Randomized Design

An Overview of Adaptive Randomization Designs in Clinical Trials 21

T2 with respect to binary outcomes Let p1 and p2 denote, respectively, thetreatment success probabilities which are assumed to follow independent betaprior distributions The posterior distributions of p1and p2are updated con-tinuously based on accumulating data in the trial These posteriors are alsobeta distributions with parameters updated according to the observed num-ber of successes among the patients assigned to treatments T1 and T2 Thalland Wathen [155] suggest randomizing the new patient between treatments

T1and T2 with probabilities

λ1= {Pr(p1> p2|Data)}γ

{Pr(p1> p2|Data)}γ+ {Pr(p1< p2|Data)}γ and λ2= 1 − λ1,

(1.10)where γ ≥ 0 is a tuning parameter The design with γ = 0 corresponds tocomplete randomization, whereas the design with γ = 1 is Thompson’s [157]procedure Based on empirical evidence, Thall and Wathen [155] recommendusing γ ∈ (0, 1); e.g., γ = 1/2 or γ = n/2N , where n is the current samplesize and N is the maximum sample size of the study

Several authors considered extensions of (1.10) to multi-armed ized clinical trials [153, 159, 162, 173] Trippa et al [159] and Wason andTrippa [162] proposed a Bayesian adaptive randomization procedure whichfavors more successful experimental treatments, but at the same time main-tains approximate balance between the most frequently assigned experimentaltreatment and the control arm

random-A particularly useful application of Bayesian adaptive randomization is indose-ranging studies which involve searches over a prespecified set of multipledose levels One famous example is the ASTIN trial [105], an adaptive dose-ranging study in acute ischemic stroke, where the primary objective was todetermine the dose producing 95% of the maximum treatment effect (ED95).This study involved 15 different dose levels and the placebo, which would havemade an equal allocation design prohibitively expensive Instead, a Bayesianadaptive randomization design was used where randomization probability toany dose level was set proportional to posterior probability that the given dose

is the ED95 The study was terminated early for futility (the dose–responsecurve was found to be flat); thereby substantial savings in the study cost andpatient resources were achieved

While Bayesian adaptive randomization is intuitively appealing, it also hascertain limitations (cf Chapter 13)

22 Modern Adaptive Randomized Clinical Trialsanalysis is not as easily interpretable as when fixed randomization probabilitiesare used” [74] Some important issues related to RAR include unblinding ofdata as the trial progresses, extra variability of RAR procedures and its impact

on statistical power, statistical inference following RAR designs, the issue ofestimation bias in the presence of time trends, the impact of delayed responses,and interim monitoring of RAR trials Rosenberger, Sverdlov and Hu [130] andSverdlov and Rosenberger [145] discuss how to handle these issues in practice.Chapter 11 discusses proper statistical inference following RAR

Overall, we think that RAR can be a useful research design option forcomplex clinical trials driven by multiple objectives which may include statis-tical efficiency and ethical considerations Berry [34] notes that “the adaptiverandomization light shines brightest in complicated multiarm settings,” andindeed, one important area of application of RAR is phase II dose-rangingstudies where the goal is to determine dose(s) most suitable for subsequentconfirmatory studies Another important area for application of RAR is clini-cal trials in rare diseases where the target patient population is very small and

a great number of patients with the disease will receive treatment in the trial,and trials for highly contagious diseases (such as Ebola), where it is hopedthat the disease might be eradicated by treatment or vaccine under study[32] In this case RAR can help maximize the beneficial experience of studyparticipants while achieving reliable inference on the treatment effect Ad-vances in statistical theory, information technology and guidance from healthauthorities should encourage a broader use of RAR designs in the future

RandomizationCovariate-Adjusted Response–Adaptive (CARA) randomization is applica-ble in clinical trials with heteroscedastic and nonlinear models with possi-bly treatment–covariate interactions where balanced designs may be subopti-mal As noted by Sverdlov, Rosenberger and Ryeznik [146], there are at leastthree important reasons why CARA randomization merits consideration inclinical trial practice These include: 1) ethical considerations; 2) nonlinearityand heteroscedasticity of statistical models; and 3) presence of treatment-by-covariate interactions (when the magnitude and direction of the treatmenteffect may differ for patient subgroups within a treatment) CARA randomiza-tion designs can be viewed as an important step toward personalized medicine[85] Recently, theoretical properties of a very general class of CARA proce-dures were established by Zhang et al [184] and Zhang and Hu [182] Rosen-berger and Sverdlov [129] discuss the appropriateness of CARA randomization

in clinical trials In §1.5.1–§1.5.4 we describe major types of CARA ization designs

An Overview of Adaptive Randomization Designs in Clinical Trials 23

1.5.1 Treatment Effect Mapping and Urn-Based CARA

Randomization Designs

The idea of treatment effect mapping was introduced by Rosenberger [124].This heuristic approach has an intuitive appeal of skewing randomizationprobability toward an empirically better treatment according to the currentestimate of the treatment difference (using covariate adjustments as appropri-ate) Rosenberger, Vidyashankar and Agarwal [131] proposed a CARA proce-dure based on a logistic regression model for which treatment randomizationprobabilities are set proportional to the estimated treatment odds Bandy-opadhyay and Biswas [24] proposed a CARA procedure for a linear modelusing probit mapping of a covariate-adjusted estimate of the mean treatmentdifference Unfortunately, both procedures [24, 131] are not optimal in anysense and can result in highly unbalanced treatment groups and loss in power

of statistical tests

Another class of heuristic CARA procedures is based on urn models.Covariate-adjusted extensions of the randomized play-the-winner rule of Weiand Durham [166] were proposed in the papers [22, 23, 109] Covariate-adjusted extensions of Ivanova’s [95] drop-the-loser rule were proposed in thepapers [26, 40] All these designs can assign more patients to the better treat-ment within covariate subgroups Some of these designs are more variable thanothers and more research is needed to make a definitive recommendation forpractice

1.5.2 Target-Based CARA Randomization Designs

Zhang et al [184] and Zhang and Hu [182] developed a framework for CARArandomization designs that can target a covariate-adjusted version of an opti-mal allocation derived under a framework without covariates This approachensures asymptotically the desired allocation proportions for different treat-ment groups and different covariate values Under widely satisfied condi-tions, target-based CARA designs maintain strong consistency and asymp-totic normality of both parameter estimators and treatment allocation pro-portions [184], and the designs have similar power and estimation efficiency

to balanced randomization designs [129, 146] Zhang and Hu [182] and ung et al [56] showed that this general methodology is applicable for responsesfrom a generalized linear model However, the major question is what alloca-tion to target given that the individual patient covariate values are unknown

Che-at the trial outset Target-based CARA designs with various allocChe-ation targetswere developed for linear models [18, 20, 190], logistic models [56, 129], expo-nential survival models [38, 146] and longitudinal models [43, 94] Chambazand van der Laan [54] proposed group sequential CARA designs analyzed viatargeted maximum likelihood estimation methodology (cf Chapter 16)

24 Modern Adaptive Randomized Clinical Trials

1.5.3 Utility-Based CARA Randomization Designs

Atkinson and Biswas [7, 8] proposed a class of CARA designs for which domization probabilities are determined sequentially by maximizing a utilityfunction that combines inferential and ethical criteria Let φk denote somemeasure of information from applying treatment Tk to a new eligible patientwith covariate vector z, and let πk= πk(θ, z) denote the allocation proportionfor treatment Tk for a given value of z (0 < πk < 1 andPK

ran-k=1πk = 1) suchthat πk’s are skewed in favor of superior treatment arms Then the treatmentallocation probabilities P1, , Pk (PK

k=1Pk= 1) are obtained by maximizingthe utility function

U =

KX

k=1

Pkφk− γ

KX

k=1

PklogPk

πk,

where γ ≥ 0 is the tradeoff parameter (γ = 0 is “most efficient” and γ → ∞

is “most ethical” design) The optimal randomization probabilities are

con-1.5.4 Bayesian CARA Randomization

The idea of Bayesian CARA randomization is to skew randomization bility in favor of superior treatments while adjusting for patient heterogeneityaccording to some Bayesian criterion (e.g., the posterior probability that agiven treatment is most successful for a patient’s covariate profile) UnlikeCARA designs discussed thus far, Bayesian CARA randomization proceduresare selection designs—at the end of the trial the treatment with highest pos-terior probability of the criterion is selected Some important examples ofBayesian CARA designs can be found in Thall and Wathen [154] and Che-ung et al [57] In these papers, the authors showed via extensive simulationsthat their designs allocate on average substantially more patients to superiortreatments (within patient subgroups when there are treatment–covariate in-teractions) and are similar to non-adaptive balanced randomization designs