Adaptive Designs

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daptive DesignsFeifang Hu ab; Anastasia Ivanova caDepartment of Statistics, University of Virginia, Charlottesville, Virginia, U.S.A. bDivision of Biostatistics andEpidemiology, University of Virginia School of Medicine, Charlottesville, Virginia, U.S.A. cDepartment ofBiostatistics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, U.S.A.

Online Publication Date: 14 June 2004

To cite this SectionHu, Feifang and Ivanova, Anastasia(2004)'Adaptive Designs',Encyclopedia of Biopharmaceutical Statistics,1:1,1 6

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Adaptive Designs

Feifang HuUniversity of Virginia, Charlottesville, Virginia, U.S.A.

Anastasia IvanovaUniversity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, U.S.A.

INTRODUCTION

Response-adaptive designs, or response-adaptive random-ization procedures, are designs that change allocation

away from 50/50 based on responses observed so far in the

trial. The desired allocation proportion is usually moti-

vated by an ethical consideration of assigning more

patients to better treatments. This review presents several

classes of response-adaptive designs, and discusses the

advantages and disadvantages of these designs, and the

analysis of clinical trials based on adaptive designs. We

also discuss goals for the allocation proportion and the

power of treatment comparison in the trial where

response-adaptive design is used.

RESPONSE-ADAPTIVE DESIGNS

The two main goals of response-adaptive designs[1] are:

1) to maximize the individual patients personal experi-

ence in a trial under certain restrictions, and 2) to reduce

the overall number of patients (sample size) of a

randomized clinical trial. The early ideas of adaptive

designs can be traced back to Thompson[2] and Robbins.[3]

Since then, two main families of response-adaptive

designs have been proposed: 1) target-based, which is

based on an optimal allocation target, where a specific

criterion is optimized based on a population model; and 2)

design-driven, where designs are driven by intuition and

are not optimal in a formal sense.[1,4]

Urn Models

One large class of response-adaptive designs is based on

urn models (design-driven). Consider the simplest situa-

tion where two independent treatments A and B

with binary outcomes are compared in the course of a trial.

LetpAbe the probability of a success on treatment A and

let pB be the success probability of treatment B, with

qA= 1pA and qB= 1pB. In addition, let n be the totalnumber of patients in a trial, let nA be the number of

patients assigned to treatment A and let nB be the number

of patients assigned to B, so that nA+ nB= n. It is assumed

that the outcome can be observed relatively quickly.

Following the ethical imperative, one wants to changethe allocation in the course of a trial so that more

patients receive the treatment performing better thus far in

the trial.

Play-the-winner rule

One of the first response-adaptive designs, the play-the-

winner rule, was introduced by Zelen.[5] The first patient

is equally likely to receive one of the two treatments. The

subsequent patient is assigned to the same treatment

following a successful outcome, and to the opposite treat-

ment following a failure. On average, the proportion ofpatients (nA/n) assigned to treatment A is qB/(qA+ qB).[1]

Therefore this design assigned more patients to the

better treatment.

Randomized play-the-winner rule

Wei and Durham[6] and Wei[7] suggested the randomized

play-the-winner rule, a response-adaptive design that

randomizes patients. An urn contains one ball of each

type corresponding to the two treatments. When a patient

arrives, a ball is drawn from the urn and then replaced.

The patient receives corresponding treatment. If the

outcome is a success, one ball of that type is added to

the urn; otherwise, one ball of the opposite type is added

to the urn. This design has the same limiting allocation

proportion as the play-the-winner rule.

Drop-the-loser rule

According to the drop-the-loser rule,[8] the urn starts with

one ball of each treatment type and one ball of the so-

called immigration type. When a patient arrives to be

randomized to a treatment, a ball is taken from the urn. If

it is an immigration ball, the ball is replaced and two

additional balls (one of each treatment type) are added tothe urn. If it is an actual treatment ball, a patient is

assigned to corresponding treatment and response is


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observed. If response is a failure, the ball is not returned

to the urn. If response is a success, the ball is returned to

the urn; hence the urn composition remains unchanged.

On average, nqB/(qA+qB) patients are assigned to treat-

ment A.

For clinical trials with K treatment (K2), animportant family of urn models is the generalized

Friedmans urn[9] (also called generalized Polyas urn in

literature), which includes the play-the-winner rule and

the randomized play-the-winner rule as special cases. The

properties of generalized Friedmans urn have been

studied extensively.[913] Another family of urn models

is based on birth-and-death urn with immigration,[14]

which includes the drop-the-loser rule as a special case.

Some special cases of birth-and-death urn have been

proposed and studied recently.[1517] Many other urn

models[1820] have been suggested for response-adaptive

designs. These urn models and their properties are

reviewed in Rosenberger.[21]

There are several advantages of designs based on urn

models: 1) urn models are intuitive and easy to implement

in clinical trials; 2) they shift the allocation probability to

better treatments; and 3) the properties of urn models have

been studied extensively. For comparing two treatments

(K= 2), the play-the-winner rule is a deterministic

allocation procedure; therefore there is a possibility of

selection bias.[1] The randomized play-the-winner rule is a

randomized design. Simulation studies[22,23] have shown

that the power of the treatment comparison might get lost

when an adaptive urn model is used for allocation. The

drop-the-loser rule is found to preserve power better thanother adaptive urn designs.[4,24,25]

There are several drawbacks of adaptive design based

on urn models: 1) it can only target a specific allocation

proportion; this allocation is usually not optimal in a

formal sense; and 2) it can only be applied to the clinical

trials with binary or multinomial responses. Modifications

are necessary to apply urn models to clinical trials with

continuous responses or survival responses.

Optimal Allocations

Response-adaptive designs were introduced following theethical goal to maximize an individual patients personal

experience in the trial. Another important goal is to

maximize the power of treatment comparison. These two

goals usually compete with each other. Optimal allocation

proportions are usually determined through some multi-

ple-objective optimality criteria.[2629] Jennison and Turn-

bull[29] described a general procedure for determining

optimal allocation.

Consider the situation when two binomials are

compared and the measure of interest is pApB. Let pAand pB be the corresponding maximum likelihood

estimators of pA and pB. The allocation that minimizes

the variance ofpApB is the Neyman allocation with:

nA

n

ffiffiffiffiffiffiffiffiffiffiffipAqA

pffiffiffiffiffiffiffiffiffiffiffipAqA

p

ffiffiffiffiffiffiffiffiffiffipBqB

p 1

Note that if, for example, bothpAand pBare less than

0.5, Neyman allocation will assign more patients to the

inferior treatment. Another goal, a compromise between

the ethical goal and efficiency, is to minimize the

expected number of failures for a fixed variance of

pApB. The optimal allocation[23] for this goal is:

nA

n

ffiffiffiffiffiffipA

pffiffiffiffiffiffipA

p ffiffiffiffiffipBp 2

For treatments with continuous outcomes, Dunnett[30]

considered an allocation rule for a set of multiplecomparisons of K1 treatments vs. a single control.Dunnett proposed the following unbalance rule with an

allocation ratio 1:1:1:. . .:1:ffiffiffiffiffiffiffiffiffiffiffiffiffiK 1p . This allocation pro-

portion is optimal when the variances of all K1treatments and control are the same. It is still unclear

how to find optimal allocations for general K treatment

clinical trials.

Target-Based Response-Adaptive Designs

Because optimal allocation usually depends on some

unknown parameters, we cannot implement it in practice,unless we estimate the unknown parameters sequentially.

In this section, we introduce two allocation procedures

that use sequential estimation of unknown parameters. We

consider the case of comparing two treatments with binary

responses. We also suppose that the optimal allocation is

the target proportion.

Sequential maximum likelihood procedure

1) To start, allocate n02 patients to both treatments Aand B, and 2) assume that we have assigned j (j 2n0)patients in the trial and their responses have been

observed. Let pA and pB be the maximum likelihood

estimates based on the j responses. Then we allocate the

(j +1)st patient to treatment A with probability:

ffiffiffiffiffiffi^pA

pffiffiffiffiffiffi^pA

p ffiffiffiffiffi^pBp 3

This design has been studied by Melfi and Page. [31]

The limiting allocation of this design is the optimal

allocation[32] ffiffiffiffiffiffipAp = ffiffiffiffiffiffipAp ffiffiffiffiffipBp , and its propertieshave also been studied in Hu and Zhang.[33]

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Doubly adaptive biased coin design

This procedure is a generalization of the sequential maxi-

mum likelihood procedure.[31] At first, allocate n02patients to both treatments A and B. Suppose we have

assigned j ( j 2n0) patients to treatment and are about toassign patientsj +1. Let r ffiffiffiffiffiffipAp = ffiffiffiffiffiffipAp ffiffiffiffiffipBp be thedesired allocation and let r be the desired allocation,

replaced by the current estimates ^pA and ^pB. Let jA/j and

jB/jbe the current proportions of patients that have already

been allocated to A and B, respectively. Then the

probability of assigning patient j +1 to treatment A is

given by:

g jA=j;r 3where g is an allocation function. This design was first

proposed by Eisele.[34] The properties of the design have

been studied recently.[33,35]

The following family offunctions g has been proposed:[33]

gx;y yy=xg

yy=xg 1 y1 y=1 xg 4

where g0.Here we use a simple example with g=2 to illustrate

the allocation procedure. Suppose we have already

assigned j=12 patients, jA=6 to A and jB=6 to B. We

have observed a success rate ofpA=2/3 on A andpB=1/3

on B. If we are interested in optimal allocation, then we

can compute:

rffiffiffiffiffiffi^pA

p

ffiffiffiffiffiffi^pA

p ffiffiffiffiffi^pBp

ffiffiffiffiffiffiffiffi2=3

p


p


p 0:586 5

Then the probability of assigning the 13th patient to

treatment A is computed as:

g0:5; 0:586

0:5860:586= 0:52

0:5860:586 = 0:52 0:4140:414 = 0:52

0:739 6

Both sequential maximum likelihood procedures and

doubly adaptive biased coin designs can be extended to K

treatments.[33]

Treatment Effective Mapping

Another family of response-adaptive designs is called

treatment effect mapping.[36] Rosenberger[36] examined

treatment effect mappings for continuous outcomes by

using nonparametric linear rank statistic. This idea has

been applied to several statistical models.[3739]

IMPORTANT ISSUES IN ADAPTIVE DESIGN

Variability and Power ofUsing Adaptive Designs

When using response-adaptive designs in clinical trials,the number of patients assigned to each treatment is a

random variable. Therefore the power (for most testing

hypotheses and alternatives) is also a random variable.[40]

The variability of allocation has a strong effect on the

power. This has been demonstrated by some simulations

studies.[22,23] Recently, Hu and Rosenberger[24] show that

the average power of a randomization procedure is a

decreasing function of the variability of the procedure.

Furthermore, a lower bound of the asymptotic variability

of the allocation proportions for general-purpose adaptive

designs is derived.[25] Based on these theoretical results,

we can compare different response-adaptive designs.

The drop-the-loser rule has minimum variability (lower

bound) in the class of all response-adaptive procedures

targeting urn allocation qB/(qA+ qB). The doubly adaptive

biased coin design can directly target any given allocation

(may depend on unknown parameters). The parameter g

determines the variability of the procedure. At the

extreme, when g= 0, we have the sequential maximum

likelihood procedure, which has the highest variability. As

g tends to1 , we have a procedure that attains the lowerbound of any randomization procedure targeting the same

allocation. But this procedure (g=1) is entirely deter-ministic (except when jA=j

r). As g becomes smaller,

we have more randomization, but also more variability.Overall, the doubly adaptive biased coin procedure is

the most favorable design:[24,25,33] 1) it is flexible in terms

of targeting any desired allocation; 2) it can be applied to

different types of responses; and 3) one can tune the

parameter g to reflect the tradeoff between the degree of

randomization and variability, and the tradeoff between

power and expected treatment successes. In a recently

study of binary response,[4] it has been shown that the

doubly adaptive biased coin design (g=2) is always as

powerful or more powerful than complete randomization

(50/50) with fewer treatment failures.

Sample Size of Adaptive Designs

It is usually difficult to determine the required sample size

for response-adaptive designs. This is because the

allocation probabilities keep changing during a clinical

trial when using response-adaptive design. For a fixed

sample size, the number of patients assigned to each

treatment is a random variable.[40] Therefore the power is

also a random variable for a fixed sample size.

In the literature, sample sizes of response-adaptive

designs are calculated by ignoring the randomness of the

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allocations.[1,35] As shown in one example of Hu,[40] to

achieve a target power of 0.8, the sample size is 62 from

the formula[1] (where the randomness of the design is

ignored). If the sample size 62 is used, then, with 20%

chance, the real power is less than 0.76.

Hu[40]

studied the properties of the power function ofrandomized designs and derived an approximate distribu-

tion of the power function for comparing two treatments

when sample size is fixed. Based on the approximate

distribution, a formula of the required sample size is then

obtained. In the above example, the required sample size

should be 72, instead of 62. It also has been shown[40] that

a well-selected response-adaptive design requires a much

smaller sample size compared with the equal allocation in

certain cases.

Applying Adaptive Designs and AnalyzingData from Adaptive Designs

The randomized play-the-winner rule was used in the

highly controversial extracorporeal membrane oxygena-

tion (ECMO) study.[41] A total of 12 patients were

assigned, with only one patient assigned to conventional

therapy and 11 assigned to ECMO. The only failure

observed was in conventional therapy arm. It is reasonable

to ask about the validity of such a trial. [42,43] What went

wrong? The randomized play-the-winner rule has a high

variability of the allocation proportion (especially for

small sample size n =12). That is why only one patient

was randomized to conventional therapy arm.Adaptive designs have been implemented in several

clinical trails, including a trial of crystalloid preload in

hypotension for cesarean patients,[44] a trial of fluoxetine

vs. placebo in depression patients,[45] and a four-arm trial

in depression.[46] Adaptive designs have also been used

extensively in adaptive learning algorithms, game theory,

as well as dynamical systems.[47,48] Applications of

adaptive designs in many different disciplines can be

found in Flournoy and Rosenberger.[49] Some general

issues of implementing response-adaptive designs are

discussed in Chapter 12 of Rosenberger and Lachin.[1]

Inference for response-adaptive design is complicatedbecause the treatment assignments depend on the re-

sponses. Likelihood-based inference was studied for

different response-adaptive designs,[5053] In these papers,

it has been shown that the maximum likelihood estimators

are consistent and asymptotic normal under certain

regularity conditions. For a clinical trial (binary responses)

using the randomized play-the-winner rule,[13,54] exact

confidence intervals of the parameters pA and pB were

derived. Confidence intervals can also be constructed

following response-adaptive designs for censored survival

data and binary responses.[55] Rosenberger and Hu[56]

studied bootstrap confidence intervals for response-

adaptive designs ofK treatments.

CONCLUSION

In the above discussion, we assume that individual patient

outcome will be immediately available. For clinical trials

with delayed responses, we can update the urn when the

response becomes available.[13] The properties of urn

model (the generalized Friedmans urn) with delayed

responses are studied in some papers.[57,58] For doubly

adaptive biased coin designs, the unknown parameters are

estimated sequentially and the delayed responses effect

these estimators. The properties of doubly adaptive biased

coin design with delayed responses are unknown, and this

is a topic of future research. It is also unclear how delayed

responses will affect the birth-and-death urn and thetreatment effective mapping.

We have focused on reviewing response-adaptive

designs. In the literature, the term adaptive designs has

been also used for designs that balance treatment assign-

ments. The biased coin design of Efron[59] has been

proposed to force a sequential experiment to be balanced.

After that, adaptive biased coin designs[60,61] and

generalized adaptive biased coin designs[62] were pro-

posed. Details and comparisons of balancing properties

can be found in Rosenberger and Lachin.[1]

Covariate information is often available and usually

very important in clinical trials. In the literature,

researchers have focused on how to balance treatmentallocation on known covariates.[6366] Very few response-

adaptive designs[37,38] attempt to incorporate covariate

information. How to use covariate information in

response-adaptive design is a critical and conceptually

difficult problem. This remains a very important topic of

future research.

We have presented here our view on the use of

response-adaptive designs in clinical trials. We apologize

for not covering some important topics in adaptive

designs. We hope that the discussion will point the

readers in the right direction for a more comprehensive

discussion of the different topics introduced.

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