BRAR: Bayesian Response-Adaptive Randomisation

• BRAR is a family of adaptive trial procedures that update allocation probabilities using Bayesian posteriors based on accrued data.
• It enhances patient benefit by skewing allocations toward effective treatments while rigorously managing type I error and power.
• Robust computation methods, including exact recursion and Monte Carlo sampling, support its practical application in complex, multi-arm trials.

Bayesian Response-Adaptive Randomisation (BRAR) is a family of adaptive clinical trial procedures in which allocation probabilities are dynamically updated based on accrued data, so as to skew assignment toward arms with higher posterior probability of efficacy. Unlike classical fixed-ratio randomisation, BRAR systematically incorporates interim outcomes into ongoing allocation in order to increase the probability that trial participants receive effective treatments, while aiming to retain desirable inferential properties such as type I error control and high statistical power.

1. Core Structure of BRAR Designs

BRAR for binary outcomes proceeds by updating the randomisation probability after each interim or participant using Bayesian posteriors, typically with conjugate Beta-Binomial models. For $K$ arms, after observing data $D_n$ at interim $n$, each arm $j$ has a posterior

$$p_j \mid D_n \sim \mathrm{Beta}(a_{j,0} + S_{j,n},\; b_{j,0} + N_{j,n} - S_{j,n})$$

where $S_{j,n}$ is the cumulative number of successes and $N_{j,n}$ is cumulative accrual on arm $j$. In the canonical implementation, the allocation probability for the next patient to arm $j$ is set to

$$\pi_{n+1,j} = P(p_j > \max_{\ell \neq j} p_\ell \mid D_n)$$

which can be efficiently evaluated in closed form for two arms and recursively or via Monte Carlo for $K>2$ (Kaddaj et al., 2024).

The most direct BRAR algorithm is thus a variant of Thompson sampling, which randomises in exact proportion to the posterior best-arm probability. The adaptivity may be damped by a power or tuning parameter $c$, for example

$$\pi_{n+1,j} = \frac{ [P^\ast_j]^{c} }{ \sum_{k=1}^K [P^\ast_k]^{c} }$$

where $$P^\ast_j = P(p_j > \max_{\ell \neq j} p_\ell \mid D_n)$$ (Fitchett et al., 18 Nov 2025, Couturier et al., 2024, Granholm et al., 15 Jan 2025).

2. Operating Characteristics, Testing, and Error Control

A major challenge in BRAR inference is the inflation of type I error when traditional frequentist or calibrated Bayesian test statistics are combined with highly adaptive allocation. Numerical studies show, for example, that under full BRAR (no burn-in), global type I error for common test statistics can more than double nominal levels (e.g., 14% vs. 5% for $N=60$, $b=0$) (Tang et al., 25 Mar 2025). This inflation arises from discrete jumps and intrinsic dependencies in the allocation process.

Exact tests, such as unconditional Barnard-type and conditional Fisher-type procedures, provide valid type I error control under any Markovian design. The conditional exact test, which conditions on the total number of successes across both arms, achieves the highest average and minimum power for modest burn-in lengths by leveraging the maximal ancillary statistic to sharpen the critical region without sacrificing error control (Tang et al., 25 Mar 2025). Analytically, operating characteristics including type I error, power, and participant benefit can be computed by complete state enumeration of the underlying allocation Markov chain, eliminating simulation error.

Type I error and power are nonmonotonic in the burn-in; both are maximized at intermediate values rather than at the extremes of zero or full burn-in. The conditional exact test, when coupled with a small-to-moderate burn-in, yields superior power to the unconditional alternative at matched error (Tang et al., 25 Mar 2025).

3. Burn-in Periods: Role, Tuning, and Impact

BRAR deployments nearly universally incorporate an initial "burn-in" stage of equal randomisation—the allocation of a fixed number of patients per arm prior to applying the adaptive allocation rule—to mitigate the risks of early misallocation, stabilize posterior estimates, and reduce type I error inflation.

Let $N$ denote the total planned sample size, and $b$ the per-arm burn-in. Following this, subsequent patients are adaptively randomised based on the updated posterior. Empirical and analytic investigations demonstrate that

• Increasing $b$ from zero reduces, but does not eliminate, type I error inflation in non-exact testing regimes; exact tests are necessary for strict control (Tang et al., 25 Mar 2025).
• Power and participant benefit, as measured by expected proportion allocated to the superior arm (EPASA), are both nonmonotonic functions of $b$; optimal trade-offs are typically achieved for $b \in (0, N/4)$ (Tang et al., 25 Mar 2025).
• In practical guidance, $n_0 \approx N/4$ per arm is recommended for exploratory trials where stringent error control is not critical, while for confirmatory settings, the minimal $b$ satisfying the desired minimum power (across plausible parameter values) is preferred. When participant benefit is paramount, $b$ can be reduced further, subject to power constraints (Tang et al., 25 Mar 2025).

In an applied trial (ARREST), re-design with different $b$ and recalibrated thresholds demonstrates these effects directly, with power and EPASA trading off as burn-in length varies (Tang et al., 25 Mar 2025).

4. Computational Methods and Practical Implementation

Accurate and efficient computation of posterior best-arm probabilities is central to large-scale deployment of BRAR, especially for $K > 2$ arms. Several methods exist:

Method	Speed/Complexity	Accuracy / Applicability
Thompson/Miller's closed-form	Fast; $O(\min(a,b,c,d))$	Exact, but only for $K=2$ arms
General-k recursion	$O(kT^{k-1})$ or $O(C_k T)$ with memoization	Exact, feasible for $k\leq5$ under moderate $N$ (Kaddaj et al., 2024)
Gaussian Approximation (GA)	Fast per evaluation	Substantial error for small $N$ or unbalanced data [>0.1 absolute error for $N\geq 113$]
Monte Carlo (Repeated Sampling)	Parallelisable, slow unless $K$ large	Accurate (MAE $\leq 4\times10^{-3}$ for $K=10^4$)

For $K=2$ or $3$ and modest $N$, exact recursion is both feasible and recommended for both randomisation and test statistic evaluation. For large $K$ or $N$, Monte Carlo or GA may become necessary, but can lead to substantial power loss and error inflation if not carefully calibrated. Best-practice guidance is to use the exact-recursion approach wherever possible, documenting the calculation method and conducting sensitivity checks (Kaddaj et al., 2024).

5. Variants, Extensions, and Applied Contexts

BRAR is implemented in a variety of contexts, including seamless phase II/III trials with dose-finding and comparative effectiveness (e.g., Bayesian hierarchical multinomial models for multi-category outcomes) (Heath et al., 2020), platform trials with sequential multiple assignment (e.g., I-SPY2.2) (Norwood et al., 21 May 2025), and composite or mixed endpoints (e.g., mixture models for OSFD in critical care) (Xu et al., 2022).

Advanced designs embed BRAR as part of a wider adaptation framework, supporting interim arm-dropping or early stopping for efficacy or futility through Bayesian posterior probability thresholds. Performance evaluation is always simulation-based or relies on Markov chain enumeration, examining type I error, power, expected sample size, allocation proportions, and patient-benefit metrics (Granholm et al., 15 Jan 2025, Fitchett et al., 18 Nov 2025).

A broad class of information-based "uncertainty-directed" designs (BUD) have been developed as decision-theoretic extensions of BRAR, replacing the "total expected number of successes" utility (maximized by classical BRAR) with explicit information functionals (posterior variance, entropy, or inferential risk), generating allocation rules explicitly targeted at the trial's primary inferential objective (Ventz et al., 2018, Bonsaglio et al., 2021). Asymptotic analyses show that the empirical allocation proportions converge almost surely to a function of the chosen information metric and tuning parameter, with power and error rate approximations derivable from the limiting allocation vector (Bonsaglio et al., 2021).

6. Error Control, Policy Design, and Regulatory Considerations

BRAR designs must accommodate regulatory demands for strong type I error control, especially in confirmatory or licensure trials. Exact conditional or unconditional tests—computed by Markov chain enumeration—guarantee type I error control uniformly across the null parameter space for given $b$ (Tang et al., 25 Mar 2025). For multi-arm settings, familywise error rate can be controlled by adaptive testing procedures employing the conditional invariance principle, allowing strong FWER control under arbitrary adaptive allocation (Robertson et al., 2018).

Recent advances include Markov decision process formulations, in which BRAR is framed as a constrained finite-horizon CMDP: the policy is chosen to maximize a reward (e.g., patient benefit) subject to explicit constraints on type I error, power, or MSE. Efficient backward recursions and LP-based algorithms deliver optimal or near-optimal policies with guarantees on gaps to optimum, generalizing the BRAR framework to any analytic constraint set (Baas et al., 2024).

7. Software Ecosystem and Practical Guidelines

Robust open-source software for BRAR is available. Notable implementations are:

• The BATSS R package provides modular simulation of BRAR and adaptive trial designs, supporting custom allocation rules and stopping criteria, with (I)NLA-based posterior approximation and parallel execution (Couturier et al., 2024).
• The adaptr package for R implements advanced adaptive trial frameworks, including BRAR, supporting flexible outcome types, allocation restrictions (min/max), and rigorous sensitivity analyses of design performance (Granholm et al., 15 Jan 2025).
• Dedicated software supports blocked BRAR to mitigate time-trend and bias, especially in the presence of potential secular drift (Chandereng et al., 2019).
• The brar R package specifically implements point null Bayesian RAR designs, enabling shrinkage between Thompson sampling and equal randomisation (Pawel et al., 2 Oct 2025).

Best-practice recommendations from recent studies are:

• Always report the method for posterior probability computation, as approximation choice materially affects operating characteristics, patient allocation, type I error, and power (Kaddaj et al., 2024).
• For two-arm and three-arm trials with manageable $N$, use exact computation and exact test statistics calibrated to the implemented randomisation rule (Tang et al., 25 Mar 2025).
• Regularly calibrate and simulate the operating characteristics for the full design, including sensitivity to burn-in, block-size, and allocation rule parameters (Fitchett et al., 18 Nov 2025, Granholm et al., 15 Jan 2025).
• Implement allocation-probability restrictions and eligibility constraints, as required by the clinical context and to avoid excessive allocation to arms with low posterior support.
• Design adaptive stopping and arm-dropping rules using posterior probabilities tailored to realistic clinical effect sizes, with boundaries calibrated under the null hypothesis using high-throughput simulation.
• In reporting and protocol, fully specify the BRAR algorithm, calibration assumptions, approximation methods, simulation code, stopping/futility thresholds, and seed/versioning details for reproducibility.

BRAR represents a mature class of adaptive randomisation procedures for modern clinical trials, combining ethical imperatives for participant benefit with rigorous inferential design. Advances in exact computation, constrained policy optimisation, and user-friendly software environments continue to enhance its applicability and reliability for clinical development (Tang et al., 25 Mar 2025, Kaddaj et al., 2024, Fitchett et al., 18 Nov 2025, Couturier et al., 2024, Granholm et al., 15 Jan 2025, Baas et al., 2024, 2520.01734).