Efficient randomized adaptive designs for multi-arm clinical trials †

Abstract

In clinical trials, response-adaptive randomization (RAR) has gained increasing attention due to its ability to assign more patients to better-performing treatments. Consequently, several RAR methods have been proposed in recent years. Among them, the efficient response adaptive randomization design (ERADE), proposed by Hu et al. (2009), stands out as an optimal approach, with the asymptotic variance of the allocation proportion achieving the Cramér–Rao lower bound, demonstrating its statistical efficiency. However, the original ERADE is limited to trials with only two treatment arms. Given the growing prevalence of multi-arm trials in modern clinical development, the original ERADE design no longer meets all practical needs. In this paper, we extend ERADE for use in multi-arm clinical trials, proposing the multi-arm ERADE algorithm. We establish the asymptotic properties of this generalized design and demonstrate its effectiveness in finite sample settings through simulations and a real-world trial redesign.

Keywords

Efficient response adaptive randomization design biased coin design clinical trial doubly biased coin design power response-adaptive randomization

1. Introduction

The standard method of randomization in clinical trials is the randomized controlled trial (RCT), which typically assigns patients to different treatment groups with equal allocation ratios. The primary aim of this approach is to effectively determine the treatment’s efficacy with greater statistical power and reduced bias. However, one drawback is that a large number of patients may receive inferior treatments, leading to a higher number of unsuccessful outcomes within the trial. This issue becomes particularly critical in trials for severe, rare, and potentially life-threatening diseases. Therefore, a critical objective, even within clinical trials, becomes ensuring that more individuals have access to superior treatments. In response, recent advancements in clinical trial design have shifted toward adaptive designs rather than traditional methods like simple randomization. Among these, response-adaptive randomization (RAR) procedures have gained significant attention because they can allocate more patients on average to a superior treatment arm, potentially improving overall trial outcomes.

The concept of RAR was first introduced by Thompson.¹ Thompson’s approach proposed adapting patients’ allocation based on accrued responses. However, it is a non-randomized procedure, which may lead to bias.² To address this limitation, Zelen³ introduced the conceptual framework of the play-the-winner rule, which Wei and Durham⁴ later advanced into the formalized randomized play-the-winner design with specific implementation details and theoretical properties, marking the formal beginning of randomized adaptive allocation procedures. Following these foundational contributions, the literature expanded to explore the randomized play-the-winner rule and develop various modifications, all belonging to Friedman’s urn model.^5–14 These models intuitively assign the next patients in the trial to the superior treatment, as indicated by the currently available responses. Friedman’s urn models are design-driven procedures, established with intuitive motivation rather than optimality. During the subsequent 20 years, numerous randomization designs emerged with the aim of optimality.^15–19

These optimization efforts evolved to address varying priorities in clinical trial design. Different optimization frameworks emerged, each defining success according to specific goals and constraints. Some approaches prioritize minimizing the expected number of treatment failures while maintaining statistical power,¹⁶ while others focus on different balances between ethical considerations and statistical efficiency. Both frequentist and Bayesian methods contribute to this optimization landscape. Frequentist approaches typically establish a target allocation, and then the allocation rate gradually approaches this target through RAR procedures or dynamically selects the best-performing arm in a multi-arm bandit scenario to maximize overall rewards.²⁰ In contrast, Bayesian procedures continuously update prior and posterior distributions to estimate treatment effects, allocating more patients adaptively to superior treatment arms.^21–24 Each of these approaches demonstrates distinct advantages.

Most of these methodological developments initially focused on the two-arm trial setting. More recently, there has been an increasing interest in extending RAR procedures to multi-arm clinical trials. Bai et al. generalized the Friedman urn model from two-arm to multi-arm trials, establishing the foundation for urn-based approaches in multiple treatment comparisons.²⁵ Building on this work but from an optimization perspective, Hu and Zhang proposed the “optimal-driven” randomization procedure that minimizes allocation variance in multi-arm settings.¹⁸ The theoretical frameworks for RAR subsequently expanded in two major directions: (a) frequentist approaches focused on optimality properties and asymptotic guarantees, alongside alternative frameworks incorporating Bayesian and bandit perspectives^26–28 and (b) adaptations for specific outcome types.^29,30

Our approach contributes to this rich body of literature by extending the efficient randomized-adaptive design (ERADE) introduced by Hu et al.³¹ from its original two-arm setting to multi-arm scenarios. ERADE is recognized as first-order efficient because it achieves the Cramér–Rao lower bound for the asymptotic variance of allocation proportions, thus representing a highly efficient class of RAR procedures. Our generalized procedure accommodates any non-degenerate target allocation, achieves asymptotic optimality, and maintains computational simplicity for multi-arm clinical trials. This article focuses on three main goals: (a) generalizing the ERADE for K treatment ( $K \geq 2$ ), (b) studying the asymptotic properties of generalized multi-arm ERADE under some widely-satisfied conditions, and (c) examining its performance under the finite sample via numerical studies and a real-world trial redesign. The section is organized as follows: Section 2 introduces the ERADE framework for K treatments, discusses its asymptotic properties, and demonstrates the theoretical concepts with some examples. Section 3 conducts numerical studies and analyzes a migraine trial to examine the design’s performance when $K > 2$ . Section 4 concludes this section with a discussion. Finally, technical proofs and additional numerical studies are provided in the Appendix.

2. Efficient randomized-adaptive designs for multiple arms

2.1. Framework

In this section, we introduce the framework of the proposed ERADE with multiple treatments (multi-arm ERADE), followed by stating its asymptotic properties.

We consider a clinical trial with $K$ ( $K \geq 2$ ) experimental arms. $n$ patients enter the trial sequentially, and their responses are assumed to be immediate. Each patient is allocated via multi-arm ERADE to one of the $K$ treatments. Let $X_{m} = (X_{m, 1}, \dots, X_{m, K})$ be the treatment assignment of the $m$ th patient, that is, $X_{m, k} = 1$ if treatment $k$ is given to $m$ th patient and otherwise $X_{m, k} = 0$ , $k = 1, \dots, K$ . Now, suppose $ξ = (ξ_{1}, \dots, ξ_{n})^{T}$ , where $ξ_{m} = (ξ_{m, 1}, \dots, ξ_{m, K})$ is a random matrix of potential responses of the $m$ th patient on treatment $k$ and $ξ_{m, k}$ is a $d_{1}$ -dimensional random vector. In clinical trials, when the $m$ th patient is assigned to treatment $k$ , only $ξ_{m, k}$ is observed. Assume that $ξ_{m, k}, m = 1, 2, \dots$ , are independent and identically distributed (i.i.d.) with unknown parameter $θ_{k} \in R^{d_{1}}$ $(k = 1, \dots K)$ , where $d_{1}$ is the number of unknown parameter of the distribution of $ξ_{m, k}$ .

Assume $N_{m} = (N_{m, 1}, N_{m, 2}, \dots, N_{m, K})$ , where $N_{m, k} = \sum_{i = 1}^{m} X_{i, k}$ denotes the number of patients in the experimental arm $k$ after the first $m$ patients have been enrolled. Upon completion of enrollment for all $n$ patients, it is possible to enumerate the number of patients allocated to each experimental arm, denoted by $N_{n, k}$ . Here, $N_{n, 1} + N_{n, 2} + \dots + N_{n, K} = n$ , ensuring that all patients are assigned to one of the experimental arms. Assume that the probability of assigning the $(m + 1)$ th patient to the treatment $k$ is $p_{m + 1, k}$ . It is noteworthy that the probability of assigning the $(m + 1)$ th patient to the arms depends on the allocations and responses of all previously enrolled $m$ patients in the trial.

Let $\hat{Θ} = ({\hat{θ}}_{1}, \dots, {\hat{θ}}_{K})$ be the estimator of $Θ = (θ_{1}, \dots, θ_{K})$ , where $θ_{k} = (θ_{1 k}, θ_{2 k}, \dots, θ_{d_{1} k})$ is the $d_{1}$ -dimensional parameter vector for treatment $k$ . The inherent dimensionality of the distribution parameters for each treatment is denoted by $d_{1}$ . For example, a Bernoulli distribution has $d_{1} = 1$ (with parameter $π_{k}$ ), while a normal distribution has $d_{1} = 2$ (with parameters $μ_{k}$ and $σ_{k}^{2}$ ). In contrast, $d$ represents the working dimension used in the allocation algorithm - the number of parameters actively utilized for decision-making. While we often assume $d = d_{1}$ for simplicity, these dimensions can differ through transformations. For instance, with normal responses, one might initially set $d = 1$ by considering only the mean. However, to incorporate variance information for optimal allocation, we can transform the parameters to work with $d = 2$ . In such cases, we use both the first moment $θ_{1 k} = μ_{k} = E (ξ_{k})$ and the second moment $θ_{2 k} = E (ξ_{k}^{2})$ to compute the variance as $σ_{k}^{2} = θ_{2 k} - θ_{1 k}^{2}$ . This transformation allows us to utilize complete distributional information and achieve more efficient allocation.

We assume that $υ = ρ (Θ)$ is the desired allocation proportion of patients assigned to each treatment, which is a function of the parameter vector $Θ$ . The objective of allocation is to have $N_{n} / n \to υ = ρ (Θ)$ , where $ρ (.) = (ρ_{1} (.), \dots, ρ_{K} (.))$ is a vector-valued function, satisfying $ρ_{1} (.) + \dots + ρ_{K} (.) = 1$ . A commonly used target allocation proportion is the RSIHR allocation, as proposed by Rosenberger et al.¹⁶ This allocation is the solution to an optimization problem that seeks to minimize the expected number of treatment failures under binary responses while keeping the variance of the test statistic fixed. Besides, several other optimal target allocations could be found in Bandyopadhyay and Biswas,³² Hu and Rosenberger,³³ and Zhang and Rosenberger.³⁴

Next, we introduce the proposed multi-arm ERADE procedure as follows:

a.
To start, allocate $K \times m_{0}$ patients to $K$ treatments via restricted randomization.
b.
Consider that $m$ ( $m \geq K m_{0}$ ) patients have been assigned, and their responses are observed. We compute ${\hat{Θ}}_{m}$ and ${\hat{ρ}}_{m} = ρ ({\hat{Θ}}_{m})$ based on previous $m$ patients’ responses and treatment assignments.
c.
Denote $p_{m + 1} = (p_{1}, \dots, p_{K})$ as the vector representing the probabilities of assigning the $(m + 1) th$ patient to each treatment arm. The probability of assigning the $(m + 1) th$ patient to the treatment $k \in {1, 2, \dots, K}$ is given by:
$\begin{aligned} p_{m + 1, k} = {\begin{cases} α {\hat{ρ}}_{m, k}, & if \frac{N_{m, k}}{m} > {\hat{ρ}}_{m, k} \\ {\hat{ρ}}_{m, k}, & if \frac{N_{m, k}}{m} = {\hat{ρ}}_{m, k} \\ \frac{(1 - α) \sum_{j \in S} {\hat{ρ}}_{m, j}}{| T |} + {\hat{ρ}}_{m, k}, & if \frac{N_{m, k}}{m} < {\hat{ρ}}_{m, k} \end{cases} \end{aligned}$
(1)
where: -
$S = {j | \frac{N_{m, j}}{m} > ρ_{m, j}}$ is the set of treatments that are over-allocated compared to their target proportions.
-
$T = {j | \frac{N_{m, j}}{m} < ρ_{m, j}}$ is the set of treatments that are under-allocated compared to their target proportions.
-
$| T |$ denotes the cardinality of set $T$ .
-
$α$ is the degree of randomization $(0 < α < 1)$ .

Remark: In this article, we consider the ERADE for $K$ treatment arms for use in multi-arm clinical trials. It reduces to the two-arm cases proposed by Hu et al.³¹ when $K = 2$ . Furthermore, we consider only three situations when comparing the actual allocation rate with the target: when the current actual allocation rate is greater than, equal to, or less than the current estimation of the target. Within each situation, the allocation probabilities are set identically. This is the form of generalization considered in this article. Although different values of $α$ could be assigned based on how much the current allocation rate deviates from the target; for example, adjusting $α$ according to the degree by which the current allocation rate exceeds the target-we do not consider this form of generalization here.
2.2. Asymptotic results

In this section, we explore the asymptotic properties of the multi-arm ERADE procedure, explicitly focusing on strong consistency and asymptotic normality. To thoroughly examine these properties, we must first establish two critical conditions:

Condition A

Assume that the response $ξ_{n} = {(ξ_{n, 1}, \dots, ξ_{n, K})}$ is a sequence of independent and identical distributed random vectors in $R^{d}$ and that $θ_{k} = E (ξ_{1, k})$ . In addition, assume that,

A1.
$E ‖ ξ_{1, k} ‖^{2} < \infty, k = 1, \dots, K$ ;
A2.
For some $ϵ > 0$ , $E ‖ ξ_{1, k} ‖^{2 + ϵ} < \infty, k = 1, \dots, K$ .

Condition B

The proportion function $y = (y_{1}, \dots, y_{d}) = (y_{11}, \dots, y_{1 d}, \dots, y_{K 1}, \dots, y_{K d}) \to ρ (y) : R^{d K} \to (0, 1)^{K}$ is a continuous function and is twice differentiable at $Θ$ with $ρ (Θ) = υ$ .

Here, we outline the primary asymptotic properties of the multi-arm ERADE with the following notation. Let $V_{k} = V a r (ξ_{1, k}) = (cov [ξ_{1, k i}, ξ_{1, k j}]; i, j = 1, \dots d), k = 1, \dots, K$ be the variance of the responses, and

V = d i a g (\frac{1}{υ_{1}} V_{1}, \dots, \frac{1}{υ_{K}} V_{K}) and σ^{2} = (\frac{\partial ρ}{\partial y} |_{Θ})^{'} V \frac{\partial ρ}{\partial y} |_{Θ}

Theorem 1
Suppose conditions A and B are satisfied. As $n \to \infty$ we have

Strong and rates consistency
$| N_{n} - n {\hat{ρ}}_{n} | = o_{P} (\sqrt{n}), | N_{n} - n {\hat{ρ}}_{n} | = O (\sqrt{n \log \log n}) a.s.$
(2)

And
${\hat{Θ}}_{n} - Θ = O (\sqrt{n^{- 1} \log \log n}), N_{n} - n υ = O (\sqrt{n \log \log n}) a.s.$
(3)

Asymptotic normality
$\sqrt{n} ({\hat{Θ}}_{n} - Θ) \overset{D}{\to} N (0, V), \sqrt{n} (N_{n} / n - υ) \overset{D}{\to} N (0, σ^{2})$
(4)

Theorem 2
Under conditions A and B, if , $Var (ξ_{1, k}) = I_{k}^{- 1}, k = 1, \dots, K$ , then the asymptotic variance of $N_{n, 1} / n$ for the multi-arm ERADE attains the Cramér–Rao lower bound
$(\frac{\partial ρ}{\partial y} |_{Θ})^{'} diag ((υ_{1} I_{1})^{- 1}, \dots, (υ_{K} I_{K})^{- 1}) \frac{\partial ρ}{\partial y} |_{Θ}$
(5)

Remark: The theorems above are a generalization of the work by Hu et al.³¹ When $K = 2$ , our theoretical results coincide with those of Hu et al. All proofs are provided in the appendix.

In the following section, we illustrate the results of the variance calculations through some specific examples.
2.3. Some examples

In this section, we present specific examples of the proposed multi-arm ERADE methodology applied to a scenario with three treatment arms. We illustrate how the allocation probabilities for the $(m + 1)$ th patient are calculated based on the responses of the first $m$ patients under a particular target. Additionally, we specify the optimal targets for various optimization problems when $K = 3$ . Furthermore, we derive the theoretical asymptotic variance associated with trials randomized using the multi-arm ERADE approach under each target.

Example 1
(Efron’s biased Coin design) We aim for target an equal allocation, $ρ_{k} = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ for $k = 1, 2, 3$ , to ensure the experiment remains balanced among the treatment groups. The $(m + 1)$ th patient can be assigned to treatments under three different scenarios. In each scenario, we denote $p_{m + 1} = (p_{m + 1, 1}, p_{m + 1, 2}, p_{m + 1, 3})$ as the probabilities of assigning patient $m + 1$ to each treatment arm $1, 2, 3$ , which are given by: -
Scenario 1:
$p_{m + 1} = {\begin{cases} α / 3, & if N_{m, 1} / m > 1 / 3, \\ α / 3, & if N_{m, 2} / m > 1 / 3, \\ 1 - 2 α / 3, & if N_{m, 3} / m < 1 / 3, \end{cases}$

-
Scenario 2:
$p_{m + 1} = {\begin{cases} α / 3, & if N_{m, 1} / m > 1 / 3, \\ \frac{(1 - α)}{6} + 1 / 3, & if N_{m, 2} / m < 1 / 3, \\ \frac{(1 - α)}{6} + 1 / 3, & if N_{m, 3} / m < 1 / 3, \end{cases}$

-
Scenario 3:
$p_{m + 1} = {\begin{cases} α / 3, & if N_{m, 2} / m > 1 / 3, \\ 1 / 3, & if N_{m, 1} / m = 1 / 3, \\ 1 - (1 + α) / 3, & if N_{m, 3} / m < 1 / 3, \end{cases}$

where $1, 2, 3$ are the indexing for different treatment arms. $α = 2 / 3$ is a special case of Efron’s biased coin design.³⁵ Burman suggests that $α$ should be set in the range of 0.4 and 0.7 to effectively balance the treatment groups while maintaining a degree of randomness in the allocation process.³⁶

Here, since we have a constant allocation proportion, the lower bound is $0$ . Then by Theorem 1 we have $\sqrt{n} (N_{n, 1} / n - 1 / 3) \overset{p}{\to} 0$ . Hence, Efron’s biased coin design is more precise and efficient in achieving balance compared to the adaptive biased coin design⁴ and the generalized biased coin design.³⁷
Example 2
(Binary response) We examine trials with binary endpoints randomized using the multi-arm ERADE procedure under various optimization objectives. We focus on the theoretical results for optimal targets and asymptotic variances. Specifically, we consider three treatments that follow Bernoulli distributions, that is $ξ_{k} \sim$ $Bin (1, p_{k})$ , where $p_{k}$ is the success probability of treatment $k$ and $q_{k} = 1 - p_{k}$ for $k = 1, 2, 3$ . We denote $v_{l, k}$ as the $l$ th optimal target for treatment $k$ . Clearly, the binary responses and the following three common target allocations (equations (6)to (8)) satisfy conditions (A) and (B). Then, by using Theorem 1, we have
$\sqrt{n} (N_{n, k} / n - υ_{l, k}) \overset{D}{\to} N (0, σ_{l, k}^{2}), l = 1, 2, 3 and k = 1, 2, 3$
where

The generalized Polya’s urn (GPU).⁴ The target allocation proportions for each treatment $k$ $(k = 1, 2, 3)$ are
$υ_{1, 1} = \frac{1 / q_{1}}{\sum_{j = 1}^{3} 1 / q_{j}}, υ_{1, 2} = \frac{1 / q_{2}}{\sum_{j = 1}^{3} 1 / q_{j}}, υ_{1, 3} = \frac{1 / q_{3}}{\sum_{j = 1}^{3} 1 / q_{j}} .$
(6)

By the formulas in Section 2.2, we have
$σ_{1, 1}^{2} = \frac{q_{2}^{2} (p_{1} + p_{3}) + q_{3}^{2} (p_{1} + p_{2}) + 2 p_{1} q_{2} q_{3}}{q_{1}^{2} q_{2}^{2} q_{3}^{2} (\frac{1}{q_{1}} + \frac{1}{q_{2}} + \frac{1}{q_{3}})^{3}} .$

The RSIHR proportion.¹⁶ The target allocation proportions for each treatment $k$ $(k = 1, 2, 3)$ are
$υ_{2, 1} = \frac{\sqrt{p_{1}}}{\sum_{j = 1}^{3} \sqrt{p_{j}}}, υ_{2, 2} = \frac{\sqrt{p_{2}}}{\sum_{j = 1}^{3} \sqrt{p_{j}}}, υ_{2, 3} = \frac{\sqrt{p_{3}}}{\sum_{j = 1}^{3} \sqrt{p_{j}}} .$
(7)

The variance of the allocation proportions for treatment 1 is
$σ_{2, 1}^{2} = \frac{(\sqrt{p_{2}} + \sqrt{p_{3}})^{2} q_{1} \sqrt{p_{2}} \sqrt{p_{3}} + p_{1}^{3 / 2} (q_{2} \sqrt{p_{3}} + q_{3} \sqrt{p_{2}})}{4 \sqrt{p_{1} p_{2} p_{3}} (\sqrt{p_{1}} + \sqrt{p_{2}} + \sqrt{p_{3}})^{3}} .$

The Neyman proportion.³⁸ The target allocation proportions for each treatment $k$ $(k = 1, 2, 3)$ are
$υ_{3, 1} = \frac{\sqrt{p_{1} q_{1}}}{\sum_{j = 1}^{3} \sqrt{p_{j} q_{j}}}, υ_{3, 2} = \frac{\sqrt{p_{2} q_{2}}}{\sum_{j = 1}^{3} \sqrt{p_{j} q_{j}}}, υ_{3, 3} = \frac{\sqrt{p_{3} q_{3}}}{\sum_{j = 1}^{3} \sqrt{p_{j} q_{j}}} .$
(8)

The variance of the allocation proportions for treatment 1 is
$σ_{3, 1}^{2} = \frac{(1 - 2 p_{1})^{2} \sqrt{p_{2} q_{2}} \sqrt{p_{3} q_{3}} (\sqrt{p_{2} q_{2}} + \sqrt{p_{3} q_{3}})^{2} + (p_{1} q_{1})^{3 / 2} [(1 - 2 p_{2})^{2} \sqrt{p_{3} q_{3}} + (1 - 2 p_{3})^{2} \sqrt{p_{2} q_{2}}]}{4 \sqrt{p_{1} q_{1} p_{2} q_{2} p_{3} q_{3}} (\sqrt{p_{1} q_{1}} + \sqrt{p_{2} q_{2}} + \sqrt{p_{3} q_{3}})^{3}}$
.

Example 3
(Normal responses). We examine trials with continuous endpoints randomized using the multi-arm ERADE method under various optimization objectives. We assume that the responses of each treatment follow normal distributions with unknown means $μ_{1}$ , $μ_{2}$ , and $μ_{3}$ , and variances $τ_{1}^{2}, τ_{2}^{2}$ , and $τ_{3}^{2}$ . Specifically, the responses are modeled as $ξ_{1} \sim N (μ_{1}, τ_{1}^{2}), ξ_{2} \sim N (μ_{2}, τ_{2}^{2})$ , and $ξ_{3} \sim N (μ_{3}, τ_{3}^{2})$ . In this context, we define $θ_{k} = (θ_{k 1}, θ_{k 2})$ , where $θ_{k 1} = μ_{k} = E (ξ_{k})$ and $θ_{k 2} = τ_{k}^{2} =$ $E (ξ_{k}^{2}) - (E (ξ_{k}))^{2}$ for $k = 1, 2, 3$ . Then, the estimates of $τ_{1}^{2}, τ_{2}^{2}$ , and $τ_{3}^{2}$ based on $m$ patients’ responses are
${\hat{τ}}_{m, k}^{2} = \frac{\sum_{j = 1}^{m} X_{j, k} (ξ_{j, k} - {\hat{μ}}_{m, k})^{2}}{N_{m, k}} k = 1, 2, 3,$
where
${\hat{μ}}_{m, k} \frac{\sum_{j = 1}^{m} X_{j, k} ξ_{j, k}}{N_{m, k}} .$
The multi-arm ERADE could be used to target any given non-degenerate target $ρ (μ_{1}, μ_{2}, μ_{3}, τ_{1}^{2}, τ_{2}^{2}, τ_{3}^{2})$ by using their estimators. Clearly, conditions (A) and (B) are satisfied for normal response and the following allocation proportions (9),(10) and (11). Then, according to the Theorem 1, we have
$\sqrt{n} (N_{n, k} / n - υ_{l, k}) \overset{D}{\to} N (0, σ_{l, k}^{2}), l = 4, 5, 6 and k = 1, 2, 3.$
The allocation proportions with their asymptotic variance are

The Neyman allocation proportion.³⁸ The target allocation proportions for each treatment $k$ $(k = 1, 2, 3)$ are
$υ_{4, 1} = \frac{τ_{1}}{\sum_{j = 1}^{3} τ_{j}}, υ_{4, 2} = \frac{τ_{2}}{\sum_{j = 1}^{3} τ_{j}}, υ_{4, 3} = \frac{τ_{3}}{\sum_{j = 1}^{3} τ_{j}} .$
(9)

The variance of the allocation proportions for treatment 1 is
$σ_{4, 1}^{2} = \frac{τ_{1} (τ_{2} + τ_{3})}{2 (τ_{1} + τ_{2} + τ_{3})^{2}} .$

The E-optimal allocation proportion.³⁹ The target allocation proportions for each treatment $k$ $(k = 1, 2, 3)$ are
$υ_{5, 1} = \frac{τ_{1}^{2}}{\sum_{j = 1}^{3} τ_{j}^{2}}, υ_{5, 2} = \frac{τ_{2}^{2}}{\sum_{j = 1}^{3} τ_{j}^{2}}, υ_{5, 3} = \frac{τ_{3}^{2}}{\sum_{j = 1}^{3} τ_{j}^{2}} .$
(10)

By the formulas in Section 2.2, we have
$σ_{5, 1}^{2} = \frac{2 τ_{1}^{2} (τ_{2}^{2} + τ_{3}^{2})}{(τ_{1} + τ_{2} + τ_{3})^{2}} .$

The $D_{A}$ -optimal allocation proportion.⁴⁰ The target allocation proportions for each treatment $k$ $(k = 1, 2, 3)$ are
$υ_{6, 1} = \frac{τ_{1}^{4 / 3}}{\sum_{j = 1}^{3} τ_{j}^{4 / 3}}, υ_{6, 2} = \frac{τ_{2}^{4 / 3}}{\sum_{j = 1}^{3} τ_{j}^{4 / 3}}, υ_{6, 3} = \frac{τ_{3}^{4 / 3}}{\sum_{j = 1}^{3} τ_{j}^{4 / 3}} .$
(11)

The variance of the allocation proportions for treatment 1 is
$σ_{6, 1}^{2} = \frac{8 τ_{1}^{4 / 3} (τ_{2}^{4 / 3} + τ_{3}^{4 / 3})}{9 (τ_{1}^{4 / 3} + τ_{2}^{4 / 3} + τ_{3}^{4 / 3})^{2}} .$

3. Numerical study and application

3.1. Some numerical studies

In this section, we conduct numerical studies to evaluate the performance of the newly proposed multi-arm ERADE against existing methods, particularly Hu and Zhang’s doubly biased coin design (DBCD) procedures.¹⁸ Multi-arm ERADE builds upon DBCD by modifying the allocation function structure while preserving its core adaptivity principles. This methodological refinement allows ERADE to achieve lower asymptotic variability in allocation proportions—theoretically reaching the Cramér–Rao lower bound—while still maintaining DBCD’s ability to balance ethical considerations with statistical power. Our numerical studies aim to demonstrate how this theoretical advantage translates to practical improvements in allocation efficiency under various trial scenarios.

In our simulations, we compare the performance of the newly proposed multi-arm ERADE procedure with the DBCD to highlight the benefits of the new design. We examine the variability of the allocation proportion, as it is considered the main criterion for determining the efficiency (power) of a randomized design.³³ Additionally, we report the empirical power, mean of allocation proportion, and mean of number of failures throughout the trial to illustrate the algorithm’s efficiency, convergence, and ethical performance, respectively.

For our simulation setup, we examine two scenarios: trials with binary endpoints and trials with continuous endpoints, as illustrated in Examples 2 and 3. All simulations share some common settings. Initially, the first 24 patients are randomly assigned to treatments $1$ , $2$ , and $3$ using restricted randomization. The remaining patients are then allocated according to each respective design. We set the sample size to 200 patients for trials with binary endpoints and 250 patients for trials with continuous endpoints. For multi-arm ERADE, we consider $α = \frac{1}{2}$ and $α = \frac{2}{3}$ , as suggested by Huet al.³¹ For DBCD, we choose $γ = 2$ , following Rosenberger and Hu,⁴¹ to control the procedure’s variances. Each simulation is based on 10,000 replications.

Case 1: (Binary Endpoint) We evaluate the performance of the multi-arm ERADE for three treatments by considering three target allocation proportions: GPU, RSIHR, and Neyman allocation. The generalized Polya’s allocation aims to increase the probability of assigning patients to a better treatment group. The RSIHR focuses on minimizing the number of failures while ensuring sufficient statistical power, whereas Neyman allocation is designed to maximize statistical power at the fixed sample size. To evaluate the statistical power, we conducted a Chi-squared test to assess the hypothesis:

H_{0} : p_{1} = p_{2} = p_{3} versus H_{A} : p_{i} \neq p_{j} for some i \neq j

We present the results for RSIHR target in the main section, and the findings for the GPU and the Neyman targets are available in the appendix.

Table 1 shows the results for multi-arm ERADE and DBCD designs considering RSIHR allocation proportion in (7) for different values of ( $p_{1}, p_{2}, p_{3}$ ). Both designs show a reasonable match between asymptotic and simulated target allocation. However, multi-arm ERADE shows lower and more consistent variance across different scenarios, while DBCD has a slightly higher variance. The two procedures have comparable power, with multi-arm ERADE having slightly higher power in some cases.

Table 1.
Simulation results for 10,000 replications for binary responses: power, asymptotic and simulated allocation proportions $(N_{n, 1} / n)$ and their variances $n V a r (N_{n, 1} / n)$ (given in parentheses) for multi-arm ERADE and DBCD designs with $n = 200$ (RSIHR target).

Multi-arm ERADE DBCD

$α = 1 / 2$ $α = 2 / 3$ $γ = 2$

$(p_{1}, p_{2}, p_{3})$ Asymptotic Simulated Power $E$ (No.Fail)(SD) Simulated Power $E$ (No.Fail)(SD) Asymptotic Simulated Power $E$ (No.Fail)(SD)

(0.9,0.8,0.7) 0.354(0.011) 0.354(0.015) 0.790 0.196(0.027) 0.354(0.019) 0.786 0.196(0.027) 0.354(0.059) 0.354(0.058) 0.783 0.196(0.027)

(0.9,0.8,0.6) 0.362(0.013) 0.362(0.017) 0.974 0.223(0.027) 0.362(0.024) 0.972 0.223(0.027) 0.362(0.062) 0.363(0.062) 0.973 0.223(0.027)

(0.9,0.7,0.5) 0.380(0.019) 0.380(0.023) 0.999 0.280(0.028) 0.380(0.030) 0.999 0.280(0.028) 0.380(0.070) 0.381(0.072) 1.000 0.280(0.028)

(0.8,0.7,0.7) 0.348(0.018) 0.348(0.022) 0.291 0.265(0.031) 0.348(0.027) 0.288 0.265(0.031) 0.348(0.067) 0.349(0.068) 0.287 0.265(0.031)

(0.8,0.7,0.6) 0.357(0.020) 0.357(0.025) 0.646 0.295(0.031) 0.357(0.030) 0.645 0.295(0.031) 0.357(0.070) 0.357(0.071) 0.642 0.295(0.031)

(0.8,0.6,0.4) 0.389(0.033) 0.388(0.038) 0.995 0.377(0.031) 0.388(0.046) 0.997 0.378(0.031) 0.389(0.087) 0.390(0.092) 0.995 0.377(0.032)

(0.7,0.7,0.7) 0.333(0.024) 0.333(0.028) 0.055 0.300(0.032) 0.333(0.033) 0.052 0.300(0.032) 0.333(0.073) 0.333(0.073) 0.054 0.300(0.033)

(0.7,0.6,0.5) 0.361(0.033) 0.361(0.038) 0.572 0.394(0.034) 0.361(0.044) 0.575 0.394(0.034) 0.361(0.086) 0.362(0.087) 0.579 0.394(0.034)

(0.7,0.5,0.3) 0.400(0.053) 0.400(0.062) 0.994 0.472(0.033) 0.400(0.071) 0.995 0.472(0.032) 0.400(0.112) 0.402(0.121) 0.994 0.472(0.033)

(0.6,0.5,0.4) 0.366(0.051) 0.366(0.055) 0.553 0.493(0.035) 0.366(0.063) 0.559 0.493(0.035) 0.366(0.107) 0.367(0.111) 0.558 0.493(0.035)

(0.6,0.3,0.1) 0.473(0.162) 0.475(0.187) 1.000 0.596(0.033) 0.473(0.191) 1.000 0.597(0.033) 0.473(0.245) 0.475(0.246) 1.000 0.596(0.034)

(0.5,0.4,0.3) 0.375(0.078) 0.376(0.090) 0.583 0.591(0.034) 0.374(0.101) 0.579 0.591(0.034) 0.375(0.140) 0.375(0.151) 0.575 0.592(0.034)

(0.5,0.3,0.1) 0.450(0.167) 0.452(0.198) 0.998 0.650(0.033) 0.450(0.200) 0.999 0.650(0.033) 0.450(0.250) 0.452(0.255) 0.998 0.650(0.033)

	Multi-arm ERADE	DBCD
(0.9,0.8,0.7)	0.354(0.011)	0.354(0.015)	0.790	0.196(0.027)	0.354(0.019)	0.786	0.196(0.027)	0.354(0.059)	0.354(0.058)	0.783	0.196(0.027)
(0.9,0.8,0.6)	0.362(0.013)	0.362(0.017)	0.974	0.223(0.027)	0.362(0.024)	0.972	0.223(0.027)	0.362(0.062)	0.363(0.062)	0.973	0.223(0.027)
(0.9,0.7,0.5)	0.380(0.019)	0.380(0.023)	0.999	0.280(0.028)	0.380(0.030)	0.999	0.280(0.028)	0.380(0.070)	0.381(0.072)	1.000	0.280(0.028)
(0.8,0.7,0.7)	0.348(0.018)	0.348(0.022)	0.291	0.265(0.031)	0.348(0.027)	0.288	0.265(0.031)	0.348(0.067)	0.349(0.068)	0.287	0.265(0.031)
(0.8,0.7,0.6)	0.357(0.020)	0.357(0.025)	0.646	0.295(0.031)	0.357(0.030)	0.645	0.295(0.031)	0.357(0.070)	0.357(0.071)	0.642	0.295(0.031)
(0.8,0.6,0.4)	0.389(0.033)	0.388(0.038)	0.995	0.377(0.031)	0.388(0.046)	0.997	0.378(0.031)	0.389(0.087)	0.390(0.092)	0.995	0.377(0.032)
(0.7,0.7,0.7)	0.333(0.024)	0.333(0.028)	0.055	0.300(0.032)	0.333(0.033)	0.052	0.300(0.032)	0.333(0.073)	0.333(0.073)	0.054	0.300(0.033)
(0.7,0.6,0.5)	0.361(0.033)	0.361(0.038)	0.572	0.394(0.034)	0.361(0.044)	0.575	0.394(0.034)	0.361(0.086)	0.362(0.087)	0.579	0.394(0.034)
(0.7,0.5,0.3)	0.400(0.053)	0.400(0.062)	0.994	0.472(0.033)	0.400(0.071)	0.995	0.472(0.032)	0.400(0.112)	0.402(0.121)	0.994	0.472(0.033)
(0.6,0.5,0.4)	0.366(0.051)	0.366(0.055)	0.553	0.493(0.035)	0.366(0.063)	0.559	0.493(0.035)	0.366(0.107)	0.367(0.111)	0.558	0.493(0.035)
(0.6,0.3,0.1)	0.473(0.162)	0.475(0.187)	1.000	0.596(0.033)	0.473(0.191)	1.000	0.597(0.033)	0.473(0.245)	0.475(0.246)	1.000	0.596(0.034)
(0.5,0.4,0.3)	0.375(0.078)	0.376(0.090)	0.583	0.591(0.034)	0.374(0.101)	0.579	0.591(0.034)	0.375(0.140)	0.375(0.151)	0.575	0.592(0.034)
(0.5,0.3,0.1)	0.450(0.167)	0.452(0.198)	0.998	0.650(0.033)	0.450(0.200)	0.999	0.650(0.033)	0.450(0.250)	0.452(0.255)	0.998	0.650(0.033)

Case 2: (Continuous Endpoint) In this case, we evaluate the multi-arm ERADE in trials with continuous endpoints. Similarly, we compare the performance of the multi-arm ERADE and DBCD using Neyman, E-optimal, and $D_{A}$ -optimal targets. The results using the Neyman target (equation (9)) are shown below, while the results for E-optimal and $D_{A}$ -optimal allocations are presented in the appendix. The Neyman allocation seeks to determine the optimal number of samples to allocate to each arms in a way that maximum the power for a fixed total sample size. To assess the statistical power, we performed an ANOVA test to test the following hypothesis:

H_{0} : μ_{1} = μ_{2} = μ_{3} versus H_{A} : μ_{i} \neq μ_{j} for some i \neq j

In Table 2, we show the performance of multi-arm ERADE and DBCD via expected allocation proportion and its standard deviation, empirical power, expected number of failures. After examining various sets of parameters $(μ_{1}, τ_{1}^{2}, μ_{2}, τ_{2}^{2}, μ_{3}, τ_{3}^{2})$ , the findings indicate that multi-arm ERADE and DBCD exhibit similar performance in achieving target allocation proportions. However, multi-arm ERADE consistently demonstrates slightly lower variances in allocation proportions compared to DBCD. Besides, multi-arm ERADE shows that the expected number of failures and its standard deviations are comparable to those of DBCD. In summary, while both multi-arm ERADE and DBCD perform well in terms of power and allocation proportions, multi-arm ERADE tends to achieve more consistent and stable results in terms of variance.

Table 2.

Simulation results for 10,000 replications for continuous responses: power, asymptotic, and simulated allocation proportions $(N_{n, 1} / n)$ and their variances $n v a r (N_{n, 1} / n)$ (given in parentheses) for multi-arm ERADE and DBCD designs with $n = 250$ (Neyman target).

	Multi-arm ERADE							DBCD
		$α = 1 / 2$			$α = 2 / 3$				$γ = 2$
$(μ_{1}, τ_{1}^{2}, μ_{2}, τ_{2}^{2}, μ_{3}, τ_{3}^{2})$	Asymptotic	Simulated	Power	$E (Obs.Rsp)(SD)$	Simulated	Power	$E (Obs.Rsp)(SD)$	Asymptotic	Simulated	Power	$E (Obs.Rsp)(SD)$
(11,16,12,9,13,6.25)	0.421(0.122)	0.421(0.136)	0.971	11.840(0.214)	0.421(0.144)	0.971	11.843(0.217)	0.421(0.195)	0.423(0.210)	0.971	11.837(0.216)
(11,6.25,12,16,13,9)	0.263(0.097)	0.263(0.107)	0.972	12.051(0.213)	0.263(0.117)	0.972	12.052(0.216)	0.263(0.155)	0.262(0.170)	0.970	12.051(0.218)
(11,16,11,9,11,6.25)	0.421(0.122)	0.421(0.136)	0.042	10.998(0.211)	0.421(0.144)	0.044	11.000(0.213)	0.421(0.195)	0.423(0.210)	0.045	10.998(0.210)
(13,16,14,9,15,6.25)	0.421(0.122)	0.421(0.136)	0.971	13.840(0.214)	0.421(0.144)	0.971	13.843(0.217)	0.421(0.195)	0.423(0.210)	0.971	13.837(0.216)
(13,6.25,14,9,15,16)	0.263(0.097)	0.263(0.109)	0.971	14.159(0.212)	0.264(0.120)	0.970	14.155(0.216)	0.263(0.155)	0.262(0.170)	0.971	14.161(0.216)
(15,16,16,9,17,6.25)	0.421(0.122)	0.421(0.136)	0.971	15.840(0.214)	0.421(0.144)	0.971	15.843(0.217)	0.421(0.195)	0.423(0.10)	0.971	15.837(0.216)
(15,6.25,16,9,17,16)	0.263(0.097)	0.263(0.109)	0.971	16.159(0.212)	0.264(0.120)	0.970	16.155(0.216)	0.263(0.155)	0.262(0.170)	0.971	16.161(0.216)
(15,6.25,15,9,15,16)	0.263(0.097)	0.262(0.109)	0.043	15.001(0.209)	0.264(0.120)	0.045	14.997(0.212)	0.263(0.155)	0.262(0.170)	0.044	15.000(0.211)
(17,16,18,9,19.5,6.25)	0.421(0.122)	0.421(0.136)	0.995	17.972(0.216)	0.421(0.144)	0.995	17.974(0.219)	0.421(0.195)	0.423(0.210)	0.994	17.968(0.219)
(17,6.25,18,9,19.5,16)	0.263(0.097)	0.262(0.109)	0.995	18.370(0.214)	0.264(0.120)	0.994	18.365(0.218)	0.263(0.117)	0.262(0.170)	0.994	18.372(0.220)

3.2. Analysis of migraine trial

To illustrate the practical utility of our proposed method, we apply it to a real-world clinical trial. Specifically, we consider the Migraine Prevention (STRIVE) study conducted by Goadsby et al.⁴² This is a phase 3 randomized, placebo-controlled trial that evaluated the efficacy of erenumab, a monoclonal antibody targeting the calcitonin gene-related peptide receptor, for preventing episodic migraine. The secondary endpoint of the trial was achieving at least a 50% reduction in the mean number of migraine days per month. The study enrolled 955 patients who were randomly assigned in a 1:1:1 ratio to three treatment groups: 70-mg erenumab (317 patients), 140-mg erenumab (319 patients), or placebo (319 patients).

In our redesign of this trial, patient responses are simulated based on the original success rates $p_{1} = 0.5$ , $p_{2} = 0.433$ , and $p_{3} = 0.266$ , which are assumed to be unknown during the trial. We reduce the sample size to $n = 400$ from the original $n = 955$ to examine the performance of our proposed method under more constrained conditions. Specifically, we allocate the first 24 patients using restricted randomization to generate initial parameter estimates for each treatment arm. Subsequently, the remaining patients are adaptively assigned to treatment arms using each distinct allocation methods: complete randomization (CR), DBCD, and our proposed multi-arm ERADE procedures. Additionally, to illustrate the benefits of allocation design based on explicit optimization criteria, we implement different optimal allocation targets: GPU, RSIHR, and Neyman allocation. The comparative results from this redesigned analysis are summarized in Table 3.

Table 3.
The migraine clinical trial with reduced sample size (n $=$ 400).

Method $ρ$ $E (\frac{N_{1}}{n}) (S D)$ Power Patient in 140-mg erenumab Total Failure

CR 0.334(0.024) 0.967 133.33(9.27) 240.22(9.75)

multi-arm ERADE GPU 0.390(0.022) 0.963 155.65(8.96) 234.32(10.13)

DBCD GPU 0.389(0.027) 0.962 155.62(10.68) 234.25(10.18)

multi-arm ERADE RSIHR 0.376(0.015) 0.965 150.39(5.85) 234.98(9.58)

DBCD RSIHR 0.376(0.019) 0.963 150.43(7.69) 234.92(9.58)

multi-arm ERADE Neyman 0.348(0.006) 0.968 139.18(2.39) 237.99(9.20)

DBCD Neyman 0.348(0.012) 0.966 139.25(5.00) 237.98(9.30)

Method	$ρ$	$E (\frac{N_{1}}{n}) (S D)$	Power	Patient in 140-mg erenumab	Total Failure
CR		0.334(0.024)	0.967	133.33(9.27)	240.22(9.75)
multi-arm ERADE	GPU	0.390(0.022)	0.963	155.65(8.96)	234.32(10.13)
DBCD	GPU	0.389(0.027)	0.962	155.62(10.68)	234.25(10.18)
multi-arm ERADE	RSIHR	0.376(0.015)	0.965	150.39(5.85)	234.98(9.58)
DBCD	RSIHR	0.376(0.019)	0.963	150.43(7.69)	234.92(9.58)
multi-arm ERADE	Neyman	0.348(0.006)	0.968	139.18(2.39)	237.99(9.20)
DBCD	Neyman	0.348(0.012)	0.966	139.25(5.00)	237.98(9.30)

Table 3 presents results based on 10,000 simulated replications, comparing multi-arm ERADE with CR and DBCD. Since both multi-arm ERADE and DBCD allow for multiple optimal allocation targets, GPU, RSIHR, and Neyman allocation are selected for detailed comparison.

From a within-trial patient benefit perspective, allocating more patients to the most effective treatment (140-mg erenumab) reduces treatment failures. GPU-targeted multi-arm ERADE and DBCD maximize this benefit with the highest patient allocation to 140-mg erenumab (allocation proportions of approximately 0.390 and 0.389 for the best treatment arm, respectively). RSIHR-based implementations also increase allocation (approximately 0.376) compared to CR (approximately 0.334). Neyman allocation, while allocating more patients to 140-mg erenumab (approximately 0.348) than CR, provides less patient benefit than GPU and RSIHR approaches.

Considering both stability and efficiency, Neyman allocation yields the most stable assignments in both multi-arm ERADE and DBCD, with the lowest standard deviations (0.006–0.012), as it directly minimizes variance. It also achieves high power (0.968 for ERADE, 0.966 for DBCD), though not the highest across all methods. RSIHR allocation offers moderate stability (SD: 0.015–0.019) and strong power (0.965 for ERADE, 0.963 for DBCD), balancing efficiency and ethical considerations by reducing treatment failures. GPU, while maximizing patient benefit, incurs higher variability (0.022–0.027) and slightly reduced power (0.963 and 0.962). CR, with SD of 0.024 and power of 0.967, lacks both the stability and ethical advantages of targeted allocations.

Empirical type I error rates were close to the nominal 5% level: CR (0.050), GPU (0.055–0.056), RSIHR (0.051–0.052), and Neyman (0.052–0.055). These small deviations are reasonable given the finite-sample properties of adaptive test statistics.

Overall, multi-arm ERADE with RSIHR targeting offers the most favorable balance between ethics and statistical efficiency. It adaptively favors better treatments, maintains stability, controls type I error, and avoids the narrow focus of GPU, or Neyman targets—making it a balanced and practical choice for this trial.

4. Discussion

In many clinical trials, researchers aim to compare the effectiveness of multiple treatments simultaneously. Recently, there has been a growing interest in evaluating the efficacy of multiple active compounds or varies doses of a single intervention compared with a standard treatment. However, most previously developed advantageous procedures were designed specifically for two-arm trials (consisting of one control and one treatment), restricting their applicability in modern multi-arm studies. To address this gap, we propose a multi-arm ERADE procedure, extending the two-arm ERADE framework introduced by Hu et al.,³¹ and derive its asymptotic properties, including strong consistency and asymptotic normality. Our theoretical analysis confirms that the proposed multi-arm ERADE attains the Cramér-Rao lower bound for allocation variance under suitable conditions, highlighting its optimal efficiency. Numerical studies and a redesign of a real-world trial further validate these theoretical findings, demonstrating the efficiency and practical utility of the proposed method. Compared to the original two-arm ERADE approach, the multi-arm extension offers greater flexibility and broader applicability in contemporary clinical trial settings.

Currently, a growing body of research focuses on multi-arm randomization procedures, aligning with the evolving needs of contemporary trials. Bai et al.²⁵ generalized the Friedman urn model from two-arm to multi-arm trials, followed by Hu and Zhang,¹⁸ who introduced an “optimal-driven” randomization procedure suitable for multiple arms. Subsequent studies have further explored multi-arm RAR procedures, considering various responses and proposing modified versions.^29,43,44 Our multi-arm ERADE design builds upon these works by extending the two-arm ERADE and demonstrating its asymptotic efficiency. Additionally, multi-arm RAR procedures based on Bayesian and bandit methodologies also have been extensively researched.^26,27,30 Unlike these Bayesian and bandit-based methods, our multi-arm ERADE offers a frequentist alternative characterized by strong consistency and asymptotic normality, achieving optimal allocation variance efficiency. Thus, our contribution provides a robust and versatile randomization approach for modern clinical trials.

This section introduces a multi-arm ERADE procedure within the frequentist RAR framework, laying an important methodological foundation for many future studies. For example, with the growing emphasis on precision medicine, incorporating covariate balance and subgroup considerations into randomization designs has become increasingly important. While several recent studies have explored how to account for patients’ prognostic factors within RAR designs to improve treatment effect estimation,^45–50 simultaneous consideration of accrued responses and covariate balancing in multi-arm trials remains largely unexplored. Our proposed multi-arm ERADE provides a robust theoretical and methodological framework, opening numerous possibilities for future research. Extending ERADE to incorporate covariate balancing and subgroup analyses will likely lead to substantial advancements, both theoretically and practically, in the design of modern clinical trials.

Supplemental Material

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Supplemental material, sj-pdf-1-smm-10.1177_09622802251362644 for Efficient randomized adaptive designs for multi-arm clinical trials by Norah Alkhnefr, Feifang Hu and Guannan Zhai in Statistical Methods in Medical Research

Supplemental Material

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Supplemental material, sj-rmd-2-smm-10.1177_09622802251362644 for Efficient randomized adaptive designs for multi-arm clinical trials by Norah Alkhnefr, Feifang Hu and Guannan Zhai in Statistical Methods in Medical Research

Supplemental Material

sj-rmd-3-smm-10.1177_09622802251362644 - Supplemental material for Efficient randomized adaptive designs for multi-arm clinical trials

Supplemental material, sj-rmd-3-smm-10.1177_09622802251362644 for Efficient randomized adaptive designs for multi-arm clinical trials by Norah Alkhnefr, Feifang Hu and Guannan Zhai in Statistical Methods in Medical Research

Footnotes

Declaration of conflicting interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Feifang Hu

Guannan Zhai

Supplemental material

Supplemental material for this article is available online.

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Supplementary Material

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Efficient randomized adaptive designs for multi-arm clinical trials †

Abstract

Keywords

1. Introduction

2. Efficient randomized-adaptive designs for multiple arms

2.1. Framework

Condition A

A1. E ‖ ξ 1 , k ‖ 2 < ∞ , k = 1 , … , K ; A2. For some ϵ > 0 , E ‖ ξ 1 , k ‖ 2 + ϵ < ∞ , k = 1 , … , K . Condition B

3.1. Some numerical studies

Supplemental Material

sj-pdf-1-smm-10.1177_09622802251362644 - Supplemental material for Efficient randomized adaptive designs for multi-arm clinical trials

Supplemental Material

sj-rmd-2-smm-10.1177_09622802251362644 - Supplemental material for Efficient randomized adaptive designs for multi-arm clinical trials

Supplemental Material

sj-rmd-3-smm-10.1177_09622802251362644 - Supplemental material for Efficient randomized adaptive designs for multi-arm clinical trials

Footnotes

Declaration of conflicting interest

Funding

ORCID iDs

Supplemental material

References

Supplementary Material

A1.
$E ‖ ξ_{1, k} ‖^{2} < \infty, k = 1, \dots, K$ ;
A2.
For some $ϵ > 0$ , $E ‖ ξ_{1, k} ‖^{2 + ϵ} < \infty, k = 1, \dots, K$ .

Condition B