Adaptive Designs
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Adaptive Designs
Feifang Hu ab; Anastasia Ivanova c a Department of Statistics, University of Virginia, Charlottesville, Virginia, U.S.A. b Division of Biostatistics and Epidemiology, University of Virginia School of Medicine, Charlottesville, Virginia, U.S.A. c Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, U.S.A. Online Publication Date: 14 June 2004
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Adaptive Designs
Feifang Hu
University of Virginia, Charlottesville, Virginia, U.S.A.
Anastasia Ivanova
University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, U.S.A.
INTRODUCTION Response-adaptive designs, or response-adaptive randomization 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 motivated 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.
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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 change the 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-thewinner 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 treatment following a failure. On average, the proportion of patients (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
RESPONSE-ADAPTIVE DESIGNS The two main goals of response-adaptive designs[1] are: 1) to maximize the individual patient’s personal experience 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 situation where two independent treatments ‘‘A’’ and ‘‘B’’ with binary outcomes are compared in the course of a trial. Let pA be the probability of a success on treatment A and let pB be the success probability of treatment B, with qA =1 ÀpA and qB =1 ÀpB. In addition, let n be the total number of patients in a trial, let nA be the number of patients assigned to treatment A, and let nB be the number
Encyclopedia of Biopharmaceutical Statistics DOI: 10.1081/E-EBS 120022458 Copyright D 2004 by Marcel Dekker, Inc. All rights reserved.
According to the drop-the-loser rule,[8] the urn starts with one ball of each treatment type and one ball of the socalled 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 to the 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 treatment A. For clinical trials with K treatment (K 2), an important family of urn models is the generalized Friedman’s urn[9] (also called generalized Polya’s urn in literature), which includes the play-the-winner rule and the randomized play-the-winner rule as special cases. The properties of generalized Friedman’s urn have been studied extensively.[9–13] 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.[15–17] Many other urn models[18–20] 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 than other 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 the ethical goal to maximize an individual patient’s 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 multiple-objective optimality criteria.[26–29] Jennison and Turnbull[29] described a general procedure for determining optimal allocation. Consider the situation when two binomials are ˆA compared and the measure of interest is pAÀpB. Let p ˆ B be the corresponding maximum likelihood and p
estimators of pA and pB. The allocation that minimizes ˆ AÀp ˆ B is the Neyman allocation with: the variance of p pffiffiffiffiffiffiffiffiffiffiffi pA qA nA ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pA qA þ pB qB n
ð1Þ
Note that if, for example, both pA and pB are 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 ˆ AÀ p ˆ B. The optimal allocation[23] for this goal is: p pffiffiffiffiffiffi pA nA ¼ pffiffiffiffiffiffi pffiffiffiffiffi n pA þ pB
ð2Þ
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For treatments with continuous outcomes, Dunnett[30] considered an allocation rule for a set of multiple comparisons of KÀ 1 treatments vs. a single control. Dunnett proposed the following unbalance rule with an pffiffiffiffiffiffiffiffiffiffiffiffiffi allocation ratio 1:1:1:. . .:1: K À 1. This allocation proportion is optimal when the variances of all KÀ 1 treatments 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 n0 2 patients to both treatments A and B, and 2) assume that we have assigned j ( j 2n0) patients in the trial and their responses have been ˆ A and p ˆ B be the maximum likelihood observed. Let p estimates based on the j responses. Then we allocate the ( j +1)st patient to treatment A with probability: pffiffiffiffiffiffi ^ pA pffiffiffiffiffiffi pffiffiffiffiffi ^ pB pA þ ^
ð3Þ
This design has been studied by Melfi and Page.[31] The limiting allocation of this design is the optimal pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi allocation[32]ð pA =ð pA þ pB ÞÞ, and its properties have 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 maximum likelihood procedure.[31] At first, allocate n0 2 patients to both treatments A and B. Suppose we have assigned j ( j 2n0) patients to treatment and are about to pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi assign patients j +1. Let r ¼ pA =ð pA þ pB Þ be the ^ be the desired allocation, desired allocation and let r pB . Let jA/j and replaced by the current estimates ^ pA and ^ jB/j be 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: ^Þ gð j A =j ; r
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IMPORTANT ISSUES IN ADAPTIVE DESIGN Variability and Power of Using 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 to 1 , we have a procedure that attains the lower bound of any randomization procedure targeting the same allocation. But this procedure (g = 1) is entirely deter^). As g becomes smaller, ministic (except when jA =j ¼ r 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
ð3Þ
where 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 of functions g has been proposed:[33] gð x ; y Þ ¼ y ð y =x Þ g yðy=xÞ þ ð1 À yÞðð1 À yÞ=ð1 À xÞÞg
g
ð4Þ
where g 0. 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 ˆ A = 2/3 on A and p ˆ B = 1/3 have observed a success rate of p on B. If we are interested in optimal allocation, then we can compute: pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ^ pA 2=3 pffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¼ 0:586 ð5Þ ^ ¼ pffiffiffiffiffiffi r ^ pB pA þ ^ 2=3 þ 1=3 Then the probability of assigning the 13th patient to treatment A is computed as: gð0:5; 0:586Þ ¼ 0:586ð0:586 = 0:5Þ2 0:586ð0:586 = 0:5Þ2 þ 0:414ð0:414 = 0:5Þ2
ð6Þ
¼ 0:739
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.[37–39]
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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 of randomized designs and derived an approximate distribution 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.
studied bootstrap confidence intervals for responseadaptive designs of K 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 Friedman’s 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 the treatment 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 assignments. 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 proposed. 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 treatment allocation on known covariates.[63–66] Very few responseadaptive 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.
Applying Adaptive Designs and Analyzing Data from Adaptive Designs The randomized play-the-winner rule was used in the highly controversial extracorporeal membrane oxygenation (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 complicated because the treatment assignments depend on the responses. Likelihood-based inference was studied for different response-adaptive designs,[50–53] 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]
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Adaptive Designs
Feifang Hu ab; Anastasia Ivanova c a Department of Statistics, University of Virginia, Charlottesville, Virginia, U.S.A. b Division of Biostatistics and Epidemiology, University of Virginia School of Medicine, Charlottesville, Virginia, U.S.A. c Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, U.S.A. Online Publication Date: 14 June 2004
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Adaptive Designs
Feifang Hu
University of Virginia, Charlottesville, Virginia, U.S.A.
Anastasia Ivanova
University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, U.S.A.
INTRODUCTION Response-adaptive designs, or response-adaptive randomization 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 motivated 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.
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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 change the 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-thewinner 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 treatment following a failure. On average, the proportion of patients (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
RESPONSE-ADAPTIVE DESIGNS The two main goals of response-adaptive designs[1] are: 1) to maximize the individual patient’s personal experience 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 situation where two independent treatments ‘‘A’’ and ‘‘B’’ with binary outcomes are compared in the course of a trial. Let pA be the probability of a success on treatment A and let pB be the success probability of treatment B, with qA =1 ÀpA and qB =1 ÀpB. In addition, let n be the total number of patients in a trial, let nA be the number of patients assigned to treatment A, and let nB be the number
Encyclopedia of Biopharmaceutical Statistics DOI: 10.1081/E-EBS 120022458 Copyright D 2004 by Marcel Dekker, Inc. All rights reserved.
According to the drop-the-loser rule,[8] the urn starts with one ball of each treatment type and one ball of the socalled 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 to the 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 treatment A. For clinical trials with K treatment (K 2), an important family of urn models is the generalized Friedman’s urn[9] (also called generalized Polya’s urn in literature), which includes the play-the-winner rule and the randomized play-the-winner rule as special cases. The properties of generalized Friedman’s urn have been studied extensively.[9–13] 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.[15–17] Many other urn models[18–20] 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 than other 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 the ethical goal to maximize an individual patient’s 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 multiple-objective optimality criteria.[26–29] Jennison and Turnbull[29] described a general procedure for determining optimal allocation. Consider the situation when two binomials are ˆA compared and the measure of interest is pAÀpB. Let p ˆ B be the corresponding maximum likelihood and p
estimators of pA and pB. The allocation that minimizes ˆ AÀp ˆ B is the Neyman allocation with: the variance of p pffiffiffiffiffiffiffiffiffiffiffi pA qA nA ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pA qA þ pB qB n
ð1Þ
Note that if, for example, both pA and pB are 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 ˆ AÀ p ˆ B. The optimal allocation[23] for this goal is: p pffiffiffiffiffiffi pA nA ¼ pffiffiffiffiffiffi pffiffiffiffiffi n pA þ pB
ð2Þ
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For treatments with continuous outcomes, Dunnett[30] considered an allocation rule for a set of multiple comparisons of KÀ 1 treatments vs. a single control. Dunnett proposed the following unbalance rule with an pffiffiffiffiffiffiffiffiffiffiffiffiffi allocation ratio 1:1:1:. . .:1: K À 1. This allocation proportion is optimal when the variances of all KÀ 1 treatments 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 n0 2 patients to both treatments A and B, and 2) assume that we have assigned j ( j 2n0) patients in the trial and their responses have been ˆ A and p ˆ B be the maximum likelihood observed. Let p estimates based on the j responses. Then we allocate the ( j +1)st patient to treatment A with probability: pffiffiffiffiffiffi ^ pA pffiffiffiffiffiffi pffiffiffiffiffi ^ pB pA þ ^
ð3Þ
This design has been studied by Melfi and Page.[31] The limiting allocation of this design is the optimal pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi allocation[32]ð pA =ð pA þ pB ÞÞ, and its properties have 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 maximum likelihood procedure.[31] At first, allocate n0 2 patients to both treatments A and B. Suppose we have assigned j ( j 2n0) patients to treatment and are about to pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi assign patients j +1. Let r ¼ pA =ð pA þ pB Þ be the ^ be the desired allocation, desired allocation and let r pB . Let jA/j and replaced by the current estimates ^ pA and ^ jB/j be 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: ^Þ gð j A =j ; r
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IMPORTANT ISSUES IN ADAPTIVE DESIGN Variability and Power of Using 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 to 1 , we have a procedure that attains the lower bound of any randomization procedure targeting the same allocation. But this procedure (g = 1) is entirely deter^). As g becomes smaller, ministic (except when jA =j ¼ r 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
ð3Þ
where 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 of functions g has been proposed:[33] gð x ; y Þ ¼ y ð y =x Þ g yðy=xÞ þ ð1 À yÞðð1 À yÞ=ð1 À xÞÞg
g
ð4Þ
where g 0. 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 ˆ A = 2/3 on A and p ˆ B = 1/3 have observed a success rate of p on B. If we are interested in optimal allocation, then we can compute: pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ^ pA 2=3 pffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¼ 0:586 ð5Þ ^ ¼ pffiffiffiffiffiffi r ^ pB pA þ ^ 2=3 þ 1=3 Then the probability of assigning the 13th patient to treatment A is computed as: gð0:5; 0:586Þ ¼ 0:586ð0:586 = 0:5Þ2 0:586ð0:586 = 0:5Þ2 þ 0:414ð0:414 = 0:5Þ2
ð6Þ
¼ 0:739
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.[37–39]
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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 of randomized designs and derived an approximate distribution 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.
studied bootstrap confidence intervals for responseadaptive designs of K 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 Friedman’s 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 the treatment 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 assignments. 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 proposed. 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 treatment allocation on known covariates.[63–66] Very few responseadaptive 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.
Applying Adaptive Designs and Analyzing Data from Adaptive Designs The randomized play-the-winner rule was used in the highly controversial extracorporeal membrane oxygenation (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 complicated because the treatment assignments depend on the responses. Likelihood-based inference was studied for different response-adaptive designs,[50–53] 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]
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