BIRTH: Bayesian integration of real-world data for trials with hybrid arms

Abstract

Background

Basket trials accelerate the evaluation of targeted therapies across molecularly defined disease subtypes, but establishing reliable control groups is challenging. Historical controls risk bias, while 1:1 randomized controls are often infeasible in rare disease settings due to substantial costs and recruitment difficulties. This study aimed to develop a novel Bayesian trial design to address these challenges by efficiently integrating real-world data.

Methods

We propose the BIRTH (Bayesian Integration of Real-world data for Trials with Hybrid arms) design, a statistical framework for randomized basket trials. The methodology integrates external real-world data (RWD) using a dual-layer, dynamic borrowing architecture. It employs propensity score weighting to mitigate confounding and hierarchical Bayesian models to dynamically regulate borrowing. A key innovation is an asymmetric borrowing strategy: the control arm borrows from both RWD and internal subgroups, while the treatment arm borrows exclusively from internal trial arms, quarantining it from RWD-induced bias. The design’s operating characteristics were evaluated via extensive simulation studies.

Results

When RWD was concordant, the BIRTH design significantly reduced bias and mean squared error while substantially improving statistical power. While the design maintained robust type I error control in many scenarios, inflation was possible under extreme heterogeneity. Notably, BIRTH achieved statistical power equivalent to a 1:1 randomized trial but with a 2:1 randomization ratio, reducing the required control group sample size by 50%.

Conclusions

The BIRTH design provides a rigorous and efficient solution for designing basket trials in oncology and rare diseases. By leveraging multi-source data through a novel asymmetric borrowing strategy, it can make randomized trials more feasible where control arm recruitment is a barrier.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12874-026-02789-1.

Introduction

The advancement of precision medicine, driven by cancer genomics, is supplanting the traditional one-size-fits-all treatment paradigm, creating an urgent need for clinical trial designs that transcend the single-indication focus of conventional randomized controlled trials (RCTs) [1]. Master protocols, particularly basket trials, have emerged as a powerful framework to meet this need. By evaluating a single targeted therapy across multiple patient subgroups defined by a common biomarker, basket trials can efficiently assess efficacy and borrow information across related populations—a critical advantage in settings such as rare cancers defined by specific molecular alterations [2].

The statistical methodology for basket trials has evolved considerably. Early basket trials predominantly employed single-arm designs, relying on cross-subgroup information borrowing strategies under Bayesian frameworks. In 2023, Adetayo Kasim and Nathan Bean systematically reviewed statistical methods for information borrowing in oncology single-arm basket trials [3]. These methods have progressed from foundational Bayesian hierarchical models [4] and their calibrated versions [5] to more sophisticated frameworks. The EXNEX method, for instance, dynamically regulates information sharing through a mixture of exchangeable and non-exchangeable components [6]. Further refinements include latent variable models that achieve localized borrowing by identifying clusters of similar subgroups [7] and the Commensurate Predictive Prior (CPP) method, which offers quantitative control over the borrowing process [1]. However, the absence of a concurrent control arm leaves such trials susceptible to selection bias and confounding, compromising the reliability of their conclusions. Consequently, researchers have shifted toward randomized controlled basket trials, eliminating confounding effects through randomization and developing two primary information borrowing strategies: Treatment Effect Borrowing (TEB) models the treatment effect directly [8], while the more flexible Treatment Response Borrowing (TRB) models each arm separately, allowing for borrowing between treatment arms and control arms independently [1]. Despite their methodological robustness, randomized basket trials often face significant practical challenges, particularly in rare disease research, where patient recruitment difficulties can lead to prolonged trial durations and prohibitive costs [9]. This creates a critical need for next-generation designs that can enhance statistical power while streamlining sample size requirements.

In recent years, breakthroughs in the acquisition and application of real-world data (RWD) and real-world evidence (RWE) have extended their value from methodological exploration to clinical practice, becoming an important complement to traditional RCTs. Multi-source data integration based on real medical scenarios (such as electronic health records, insurance claims databases, and patient registry systems) not only overcomes the external validity limitations imposed by the strict inclusion criteria of traditional RCTs, but also provides more possibilities for resource-constrained research (such as rare diseases and precision oncology) through cost-effectiveness advantages. This trend is strongly supported by global regulatory bodies. The US FDA released “Considerations for the Design and Conduct of Externally Controlled Trials for Drugs and Biological Products,” encouraging the use of RWE to support regulatory decisions for medical products [10]. The European EMA issued guidelines on using registry-based RWD for new drug approvals [11]. China’s NMPA explicitly supported the use of RWE as control for single-arm trials in specific situations in its “Guidelines for the Design and Protocol Framework of Real-World Drug Research” [12]. Regulatory support has laid the institutional foundation for the clinical translation of RWD. At the methodological level, innovative clinical trial methods incorporating RWD continue to evolve. Pocock’s external data bias analysis framework, first proposed in 1976, pioneered research on synthetic control using real-world data [13]. Modern approaches frequently combine propensity score methods with Bayesian frameworks, such as meta-analytic predictive (MAP) priors [14], or employ adaptive weighting [15] and integrated testing strategies [16] to create robust synthetic or hybrid control arms. Evidently, the rational and effective utilization of RWD to assist clinical trial design has become a focal point and challenge for regulatory agencies, pharmaceutical industry, and academia worldwide. However, to date, little research has been conducted on how to design basket trials utilizing real-world data for synthetic control arms.

To address this gap, we propose the BIRTH (Bayesian Integration of Real-world data for Trials with Hybrid arms) design. BIRTH is the first comprehensive framework to integrate both external RWD and internal cross-subgroup data into a randomized basket trial, offering an innovative solution that balances statistical power with resource optimization. The design’s core innovation is a dual-layer, dynamic information borrowing architecture featuring an asymmetric borrowing strategy. The control arm simultaneously borrows from both external RWD and concurrent control arms in other subgroups, whereas the treatment arm borrows only from other internal treatment arms. This novel structure effectively quarantines the treatment effect estimate from potential bias introduced by heterogeneous RWD sources. Technically, the BIRTH design achieves precise information integration through three key steps. First, it applies propensity score-based weighting to calibrate for covariate imbalances between the trial and RWD populations. Next, the framework uses hierarchical Bayesian models to dynamically regulate borrowing strength based on data heterogeneity. Finally, an enhanced TRB strategy is implemented to precisely integrate the multi-source data. As confirmed by extensive simulations, BIRTH exhibits excellent operating characteristics, providing a scientifically rigorous and efficient design for clinical research in precision oncology and rare diseases.

The remainder of this paper is organized as follows. "Method" section introduces the synthetic control arm methodology for real-world data and describes the process of designing basket trials incorporating real-world data. "Simulation" and "Simulation results" sections explore the operating characteristics of the proposed design through simulation studies, and includes sensitivity analyses of the proposed method. "Discussion" section contains discussion and conclusions.

Method

In BIRTH design, real-world data (RWD) can be collected from multiple sources, including electronic health record (EHR) databases or patient registry systems. We assume that the collected real-world studies employ the same enrollment criteria as the current clinical trial, and all real-world subjects receive the same standard treatment as control arm subjects in the current basket trial. In designs utilizing real-world data to augment clinical trials, RCTs can be conducted with any randomization ratio (e.g., 2:1), with the deficit of control subjects relative to the treatment group supplemented by external real-world data, ultimately achieving a balanced 1:1 randomization ratio for optimal statistical power [17]. Regulatory authorities typically prefer that RCT randomization ratios not fall below 2:1 to prevent external data from dominating the efficacy estimation in the current trial’s control arm. For simplicity, this design focuses on basket trials where subjects in each subgroup are randomized to treatment or control groups with a 2:1 allocation ratio.

Notations

Consider a dual-arm basket trial with subgroups(indications), where , represent the information collected from treatment arm in subgroup , where denote the treatment and control arms of the current basket trial, respectively, represents external real-world data. Let denote the number of subjects in treatment arm of subgroup in the basket trial. Let be the total number of external real-world data subjects available for subgroup of the basket trial. Typically, the number of external real-world subjects in each subgroup of the basket trial significantly exceeds the sample size of the current trial’s control arm, i.e.,. The total number of subjects from all sources can be expressed as , where. We define the nominal borrowing number from external real-world data as where , reflecting the strategy of balancing the sample sizes between treatment and control groups through borrowing from external data.

Assume that the primary endpoint of the basket trial is binary, Let represents the binary response outcome for the subject in arm of subgroup , and let denotes the total number of responders in subgroup under arm , where . Here corresponds to the experimental treatment, concurrent control, and external real-word data arms, respectively. Each subject possesses a vector of baseline covariates, denoted as . To systematically distinguish between study arms and data sources, we introduce two indicator variables, and . We define if the subject is assigned to the experimental treatment group, and if the subject belongs to a control group (encompassing both the concurrent trial control and external real-world subjects). Furthermore, we define to indicate that the subject is enrolled in the current clinical trial, while indicates that the subject is from external real-world data.

Propensity score and data adaptive weights

The propensity score was originally defined by [18] as the conditional probability of a subject being assigned to a particular group given all observable covariates. In this study, the propensity score is defined as the conditional probability that a subject belongs to the current trial rather than external real-world data , given the vector of observable baseline covariates for each subject:

1

This probability is typically estimated using a logistic regression model with subject baseline covariates as predictors.

To maximize the similarity between external and randomized standard treatment patients, we employ a propensity score range restriction strategy. First, we determine the distribution range of propensity scores for subjects in the current trial’s control arm, then retain only those external data subjects whose propensity scores fall within this interval. This screening step aims to exclude external subjects with extreme propensity score values, which often reflect potential unmeasurable variable differences that might lead to systematic bias between data sources.

Building upon this screening, the BIRTH design incorporates a data adaptive weight technique to assign individualized weights to each external subject [19]. For subgroup , we convert the propensity scores of external subjects into odds ratio form :

2

Subsequently, we obtain the final weights through normalization

3

This weighting mechanism ensures that the weights assigned to external data are proportional to their similarity with the trial population. Notably, all subjects in the current trial (whether in the treatment or control group) maintain full weights , while only external data subjects receive differential weighting.

After applying these weights, the likelihood function for external data in subgroup can be expressed as

4

Where represents the response rate in the external real-world data. Through this weighting approach, the total effective number of responders in the external data can be simplified to a weighted sum form, i.e., , laying the foundation for subsequent Bayesian inference.

Synthesis of MAP priors from real-world data

After completing the adaptive weighting of external subjects, the BIRTH design constructs a hierarchical Bayesian model to quantify and regulate the strength of information borrowing between external real-world data and the current trial. We propose a data adaptive weighted MAP prior which represents an improvement over the traditional MAP prior [6]. This enhanced methodology integrates propensity score weighting techniques with a Bayesian meta-analytic framework, thereby enabling precise borrowing from real-world data.

For the subgroup in the basket trial, we model the outcome variables for the control group and the weighted external real-world data as

5

Where represent the response rates in the current trial’s control group and the external real-world data, respectively. To facilitate Bayesian inference, we apply the logit transformation to map the response rate parameters to the real number space

6

In the hierarchical structure, we assume that these two parameters, and , arise from a common distribution,

7

where controls the information borrowed from external real-world data. We assign a non-informative prior to the hyperparameter , while the choice of should reflect the similarity between subjects in the external real-world data and the current basket trial. Theoretically, when the heterogeneity between external real-world data and the current basket trial is substantial, a prior with larger variance should be selected for , and when the between-group heterogeneity is minimal, a prior with smaller variance should be chosen for .

Unlike traditional MAP prior, the data adaptive weighted MAP prior introduces a propensity score distribution overlap coefficient to quantify the similarity between external data and the current trial. This coefficient is expressed as

8

This coefficient measures the area of overlap between the propensity score density functions of the two data sources in subgroup . A larger indicates higher overlap, suggesting greater similarity in patient population characteristics.

Based on this similarity measure, the prior for can be specified as

Where represents the ratio of the reference overlap coefficient to the overlap coefficient of the current subgroup. is a preset reference value that can be specified by clinical experts and the research team. The prior for ensures that when the overlap is low (small ), the value increases, resulting in a more dispersed prior distribution for , thereby reducing information borrowing. Conversely, when the overlap is high, the strength of information borrowing is enhanced.

The parameter serves as a global regulatory factor to precisely control the strength of information borrowing. Crucially, there exists a strictly monotonic inverse relationship between and the resulting ESS of the data-adaptive weighted MAP prior: a larger increases the prior variance of the heterogeneity parameter , implying greater uncertainty and leading to conservative borrowing (lower ESS); conversely, a smaller concentrates the prior mass near zero, promoting stronger borrowing (higher ESS). Leveraging this property, we employ a dynamic grid search algorithm to iteratively calibrate until the prior reaches the target effective sample size (ESS) (see Supplementary Algorithm S1 for detailed update rules). The ESS is rigorously quantified by comparing the variance of the calibrated prior with that of a non-informative reference prior [22],

where is the variance of the reference non-informative prior, is the variance of the data-adaptive-weighted MAP prior, and is the sample size corresponding to the reference prior. This method allows us to quantify the information content in the data-adaptive-weighted MAP prior and ensure that its external real-world data achieves the target borrowing strength, enabling each subgroup in the basket trial to ultimately achieve a balanced 1:1 randomization ratio for optimal statistical power.

Through the above process, we obtain the key hyperparameters and , which allow us to generate a precisely calibrated, data-adaptive weighted MAP prior for each subgroup , which both retain valuable information from external data and dynamically adjust the borrowing strength to match the degree of heterogeneity between data sources. These priors will subsequently be integrated into the overall Bayesian framework of the basket trial, making the efficacy assessment of each subgroup in the basket trial more robust.

CPP priors and improved TRB information borrowing strategy

To facilitate effective information borrowing between subgroups, the BIRTH design employs the Commensurate Predictive Prior (CPP) framework and improve upon the Treatment Response Borrowing (TRB) strategy. The CPP prior was initially proposed by Zheng et al. [1] and applied to randomized basket trial designs in the work of Ouma et al. [20]. Our method introduces innovative enhancements to this foundation, fully integrating information from external real-world data.

Suppose subgroup is of interest, and consider borrowing information from subgroup , where represents our subgroup-specific parameter of interest, . We construct a normal predictive prior centered at with precision parameter ,

11

Based on this, the specific form of the CPP prior can be expressed as

12

Where represents the prior distribution of the precision parameter , and is the standard normal density function. The primary advantage of selecting the CPP prior lies in its ability to adaptively adjust the strength of information borrowing, with the value of the precision parameter determining the degree of information borrowing, i.e., when , almost no information is borrowed, while when , strong information borrowing is achieved.

To implement adaptive adjustment of the precision parameter, we use spike and slab components for the precision parameter with respective weights of and

13

When approaches 0, can reach a large value , allowing for strong information borrowing.

To quantify the degree of similarity between subgroups in the basket trial, we use the Hellinger distance as a measure of distributional difference

14

We define When , it indicates high similarity between subgroups. Conversely, when , it indicates significant differences between subgroups.

The traditional Treatment Response Borrowing (TRB) strategy, as described by [20], is predicated on the core assumption of potential similarity between the same treatment arms across different subgroups. Consequently, information borrowing is confined to these homogeneous arms. The improved TRB strategy proposed in BIRTH design extends this framework by designing a unique dual-layer information borrowing architecture for the control arm, which synergistically integrates data from both external real-world data (RWD) and the control arms of other subgroups within the current trial.

The application of this strategy is asymmetric. Specifically, for the treatment arms, the TRB model is applied conventionally, where in information is borrowed exclusively between different subgroups within the concurrent basket trial, thereby completely isolating the estimates from potential biases introduced by external RWD, as schematically shown in Fig. 1. Under this model, for any target subgroup and source subgroup , posterior distributions for their respective treatment response parameters, and are first established by updating a non-informative prior with their respective observed trial data (or). Subsequently the Hellinger distance between these posterior distributions is calculated to quantify their similarity, which in turn guides the strength of information borrowing. This design ensures that the evaluation of treatment efficacy relies solely on the internal, randomized evidence from the current trial, thereby ensuring the robustness of the estimation.

Fig. 1.

Schematic illustration of the TRB information borrowing strategy for treatment arms in basket trials

In contrast, the control arm employs a novel dual-layer extension of the TRB strategy, which integrates information from both external RWD and other internal subgroups. For a target subgroup , this sequential process commences with the first layer, where parameter estimation begins not with a non-informative prior, but with the data-adaptive-weighted MAP prior , which has already synthesized information from the external RWD cohort. Subsequently, this informative prior is updated with data from the current trial’s control arm to produce an RWD-informed posterior distribution . In the second layer, to borrow information from another source subgroup , a posterior distribution for its control arm is first obtained by updating a non-informative prior with its trial data, after which the Hellinger distance is computed between the posterior of the source subgroup and the RWD-informed posterior of the target subgroup.

This hierarchical procedure, visualized in Fig. 2, is the cornerstone of our improved TRB framework. This differentiated approach ensures that the similarity assessment for cross-subgroup borrowing in the control arm is performed after accounting for relevant external evidence, which enables the BIRTH design to construct a more robust hybrid control arm while quarantining the treatment arm from potential biases introduced by external data.

Fig. 2.

Schematic illustration of the TRB information borrowing strategy for control arms in basket trials

Specifically, as depicted in Fig. 2, the borrowing architecture is asymmetric: for the control arm, the cross-subgroup borrowing is restricted strictly to the concurrent control data of the source subgroups, explicitly excluding their RWD components. This ensures that the influence of external data remains localized to each subgroup and does not propagate across the trial.

In a basket trial containing subgroups, for a target subgroup , this borrowing process is applied to each of the remaining source subgroups. The resulting CPP priors are then aggregated with appropriate weights to obtain a comprehensive informative prior for estimating

15

Where quantifies the relative importance of information from subgroup . Assuming all parameter priors follow normal distributions, each CPP prior can be approximated as . Therefore, the comprehensive informative prior for

16

where the weights are calculated using the Hellinger distance

17

The parameter controls the influence of the Hellinger distance on the weights. The value ofcan be pre-specified before the trial begins. should not be set too large or too small to ensure it adequately reflects differences in similarity between subgroups.

In summary, the process detailed in this section culminates in the generation of a final posterior distribution for the response rate parameters in both the treatment (​) and control (​) arms for each subgroup . These posteriors are comprehensively informed, having synthesized evidence from the concurrent trial data and external RWD for the control arm. With these robust posterior distributions established, the framework can proceed to the final stage: defining the statistical rules for clinical decision-making.

Decision rules and statistical inference

In basket trials, we need to make clinical decisions for each indication subgroup separately. Define the treatment effect difference between the experimental and control arms in the subgroup as . Decisions are made across all subgroups by testing the following hypotheses:

where, represents the Minimal Clinically Important Difference (MCID), which is a threshold predetermined by clinical experts based on the therapeutic area, disease characteristics, and expected patient benefits.

In our simulation settings, we adopt a unified MCID () across all subgroups to facilitate a standardized evaluation of the design’s statistical performance. However, it is important to note that the proposed BIRTH framework is inherently flexible and can be readily adapted to real-world clinical settings where effect sizes vary by indication. In practice, the unified threshold can be replaced by subgroup-specific thresholds (i.e.,) to reflect distinct standard-of-care baselines without altering the core Bayesian inference or borrowing mechanism.

For each subgroup based on the Bayesian posterior distribution, we calculate the posterior probability that the treatment effect difference exceeds the preset threshold. When this posterior probability exceeds a pre-specified decision threshold , i.e.

18

a “Go” decision is made, recommending the subgroup to proceed to subsequent research phases. Otherwise, a “No-Go” decision is made.

The choice of decision thresholdshould balance the risks of false positive and false negative errors. It is typically set at 0.95, which corresponds to the 5% significance level in traditional frequentist methods. However, the specific threshold should be determined based on comprehensive consideration of clinical context, disease characteristics, and treatment risk-benefit profiles. In certain scenarios, such as rare diseases or areas with unmet medical needs, a lower threshold might be adopted, while for treatments with higher safety concerns, more stringent standards may be required [21].

Considering that basket trials simultaneously evaluate multiple subgroups, our Bayesian framework naturally mitigates multiplicity issues through hierarchical model structure and information borrowing mechanisms, eliminating the need for p-value adjustments required in traditional methods.

Simulation

We conducted comprehensive simulation studies to evaluate the performance of the proposed BIRTH design against four alternative approaches (NB design, MAP design, CPP design, and 1:1 trial design) across various scenarios. This simulation aimed to assess three key dimensions, including (1)treatment effect patterns across basket trial subgroups, (2)heterogeneity between subgroups, and (3)heterogeneity between real-world data and current trial data. For each scenario, we simulated 1,000 basket trials and evaluated the methods based on point estimation, interval estimation, and hypothesis testing performance.

Basic settings of basket trials

We considered a basket trial with five distinct disease subgroups. For each subgroup, we assumed that external real-world data (RWD) from 150 subjects meeting the inclusion criteria were available.

For each simulation scenario, we generated nine baseline covariates (2 binary and 7 continuous) as potential confounders. Covariate data were generated from multivariate normal distributions.

where the covariance matrices describing relationships between covariates were set to.

The covariance matrices were specified with a compound symmetric structure with parameters and . The mean vectors were determined based on different treatment effect scenarios, with the first two continuous variables converted to binary variables using a threshold of 0.

Binary outcomes for the basket trial were generated through the following logistic regression model.

Where represents covariate effects, is the random error, and represents the treatment effect in subgroup . Outcome data for external RWD were generated using the same model, but without the treatment effect term.

Simulation scenarios

We designed 10 primary scenarios (as shown in Table 1) by varying treatment effects across subgroups, encompassing three key dimensions, including.

Table 1.

Treatment effect settings for primary simulation scenarios

Scenario	Subgroupresponse rate (Treatment: Control)					Description
	1	2	3	4	5
1	0.8:0.6	0.8:0.6	0.8:0.6	0.8:0.6	0.8:0.6	All effective & homogeneous
RWD	0.6	0.6	0.6	0.6	0.6	RWD consistent with RCT
2	0.8:0.8	0.8:0.8	0.8:0.8	0.8:0.8	0.8:0.8	All null & homogeneous
RWD	0.8	0.8	0.8	0.8	0.8	RWD consistent with RCT
3	0.8:0.6	0.7:0.5	0.6:0.4	0.5:0.3	0.4:0.2	All effective & heterogeneous
RWD	0.6	0.5	0.4	0.3	0.2	RWD consistent with RCT
4	0.8:0.8	0.7:0.7	0.6:0.6	0.5:0.5	0.4:0.4	All null & heterogeneous
RWD	0.8	0.7	0.6	0.5	0.4	RWD consistent with RCT
5	0.8:0.6	0.8:0.6	0.8:0.6	0.8:0.8	0.8:0.8	Partially effective & homogeneous
RWD	0.6	0.6	0.6	0.8	0.8	RWD consistent with RCT
6	0.8:0.6	0.7:0.7	0.6:0.4	0.5:0.5	0.4:0.2	Partially effective & heterogeneous
RWD	0.6	0.7	0.4	0.5	0.2	RWD consistent with RCT
7	0.8:0.8	0.8:0.8	0.8:0.8	0.8:0.8	0.8:0.8	All null & homogeneous
RWD	0.85	0.85	0.85	0.85	0.85	RWD inconsistent with RCT
8	0.8:0.8	0.8:0.8	0.8:0.8	0.8:0.8	0.8:0.8	All null & homogeneous
RWD	0.75	0.75	0.75	0.75	0.75	RWD inconsistent with RCT
9	0.8:0.8	0.7:0.7	0.6:0.6	0.5:0.5	0.4:0.4	All null & heterogeneous
RWD	0.85	0.75	0.65	0.55	0.45	RWD inconsistent with RCT
10	0.8:0.8	0.7:0.7	0.6:0.6	0.5:0.5	0.4:0.4	All null & heterogeneous
RWD	0.75	0.65	0.55	0.45	0.35	RWD inconsistent with RCT

1. subgroup efficacy patterns, such as all effective, all null, or partially effective;

2. inter-subgroup heterogeneity, including homogeneous or heterogeneous subgroups;

3. RWD-trial data concordance, categorized as concordant or discordant.

For each scenario, we simulated 1,000 replicate trials of five designs:

1. NB (No Borrowing) design employed a 1:1 allocation ratio (sample size ) without borrowing any external information;

2. MAP design employed a 2:1 allocation ratio (sample size ), borrowing only from RWD information;

3. CPP design employed 2:1 allocation ratio (sample size ), implementing only cross-subgroup borrowing;

4. 1:1 trial design employed 1:1 allocation ratio (sample size ), implementing only cross-subgroup borrowing;

5. Proposed BIRTH design employed 2:1 allocation ratio (sample size ), simultaneously borrowing from both RWD and cross-subgroup information.

For all methods, we specified non-informative priors for the response rate . In the CPP design, 1:1 trial design, and BIRTH design, we implemented a spike-and-slab prior for the weight parameters in the commensurate predictive prior, with hyperparameters and .The threshold parameter in the commensurate predictive prior weights was set to 0.15.

Performance evaluation metrics

To comprehensively assess the performance of the proposed BIRTH design in basket trials, we employed a multidimensional evaluation framework, encompassing point estimate accuracy, interval reliability, and decision-making precision. For each scenario and subgroup, we calculated the following metrics.

1. Bias measures the systematic deviation between estimated and true parameter values, calculated as.

   
   where represents the number of simulation iterations (set to 1,000 in this study), denotes the posterior mean of parameter in the simulation, and is the true parameter value. To facilitate comparison across methods, we computed relative bias by contrasting each method’s bias with that of the benchmark method (No Borrowing, NB).

   
2. Mean Squared Error (MSE) jointly considers estimation bias and variance, providing a more comprehensive measure of estimation precision.

   

   A smaller MSE indicates higher estimation precision.

3. Posterior Credible Interval Width evaluates interval estimation performance through the average width of 95% posterior credible intervals.

   

   Narrower interval widths indicate higher estimation precision.

4. Hypothesis Testing Performance assesses decision-making accuracy by measuring type I error rates under null scenarios (true treatment effect equals zero) and statistical power under alternative scenarios (true treatment effect differs from zero).

5. Effective Sample Size (ESS) quantifies information borrowing efficiency by calculating the posterior distribution’s effective sample size, using the same methodology as in formula (10), where is derived using the reference prior (Jeffreys prior). The ESS reflects the sample size gain achieved through information borrowing. Given the asymmetric borrowing architecture of the BIRTH design, the calculated ESS represents a cumulative metric comprising two components: a baseline ESS established by the DAW-MAP prior and a dynamic marginal ESS from cross-subgroup borrowing. Consequently, the final ESS is expected to diverge from the initial physical sample size balance, serving as an indicator of both the information gain and the specific data heterogeneity detected by the model.

Simulation results

We conducted a comprehensive evaluation of five design (NB design, MAP design, CPP design, BIRTH design, and 1:1 trial design) across 10 distinct scenarios, analyzing their performance from six dimensions, relative bias, mean squared error, credible interval width, type I error rate, statistical power, and effective sample size.

Bias

We calculated the relative bias of the five designs across all simulation scenarios. The results are illustrated in Supplementary Table S1 and Fig. 3, all designs demonstrated excellent performance with minimal relative bias in homogeneous treatment effect scenarios (Scenarios 1–2). In heterogeneous treatment effect scenarios (Scenarios 3–6), BIRTH’s adaptive borrowing mechanism effectively constrained bias, performing substantially better than the CPP design, which was prone to excessive shrinkage toward the mean. Crucially, when RWD was discordant (Scenarios 7–10), BIRTH’s asymmetric architecture and adaptive weighting reduced bias by approximately 25–30% compared to the MAP design, highlighting its ability to protect against bias from external data sources.

Fig. 3.

Comparison of relative bias across different designs under 10 simulation scenarios

Mean squared error (MSE)

We further evaluated the mean squared error (MSE) performance of the five designs across ten simulation scenarios, as presented in Supplementary Table S2 and Fig. 4. MSE simultaneously reflects both estimation bias and variance, providing a more holistic assessment of each method’s overall performance.

Fig. 4.

Comparison of MSE across different designs under 10 simulation scenarios

In terms of overall estimation accuracy, the BIRTH design consistently achieved the lowest or near-lowest MSE. In homogeneous scenarios (Scenarios 1–2), BIRTH reduced MSE by 15–20% compared to the 1:1 trial design, reflecting the efficiency gains from integrating RWD. In heterogeneous settings (Scenarios 3–6), BIRTH effectively balanced bias and variance, yielding MSE values that were superior or comparable to the 1:1 trial. When RWD was biased (Scenarios 7–10), the performance of BIRTH varied depending on the internal heterogeneity of the basket trial. In homogeneous scenarios (e.g., Scenario 8), BIRTH effectively leveraged cross-subgroup information to mitigate RWD bias, reducing MSE by approximately 12% compared to the MAP design. However, in heterogeneous scenarios (Scenarios 9–10), the MSE of BIRTH was comparable to or slightly higher than that of the MAP design. This is because, under high heterogeneity, the cross-subgroup borrowing mechanism (Layer 2) may introduce additional internal bias. For instance, in subgroups where the direction of cross-subgroup shrinkage aligns with the direction of RWD bias, the errors compound, leading to higher MSE; conversely, where they oppose, the bias may be partially offset. Overall, the BIRTH design maintains robust performance, safely balancing efficiency gains with risk control across complex scenarios (Supplementary Table S2).

95% credible interval widths

We evaluated the 95% credible interval widths across the different designs. These are visualized in Supplementary Table S3 and Fig. 5. The 1:1 trial design generated the narrowest credible intervals, while among information borrowing approaches, the BIRTH design also performed well, with 95% credible interval widths significantly smaller than both the NB and MAP designs across scenarios. In homogeneous treatment effect scenarios, the BIRTH design reduced interval widths by approximately 25% compared to the NB design. In heterogeneous scenarios, the BIRTH design maintained relatively stable interval widths through its adaptive adjustment mechanism. In scenarios incorporating external data, even in the presence of data bias, the BIRTH design sustained reasonable interval widths, providing more precise parameter estimates for clinical decision-making.

Fig. 5.

Comparison of 95%CI across different designs under 10 simulation scenarios

Type I error and statistical power

Type I error control and statistical power are core metrics for evaluating different designs performance. Supplementary Table S4 and Fig. 6 present detailed results for the five designs across ten simulation scenarios.

Fig. 6.

Comparison of power and type I error rates across different designs under 10 simulation scenarios

In scenarios with treatment effects (Scenario 1, 3, 5, and portions of Scenario 6), the BIRTH design demonstrated significant power advantages. In Scenario 1 (homogeneous effects), the BIRTH design achieved power ranging from 76.5% to 80.5%, substantially higher than the NB design (48.8%-54.1%) and MAP design (50.7%-55.4%), while approaching the 1:1 trial design (78.4%-81.2%). Notably, in Subgroup 3, the BIRTH design attained 80.5% power, nearly equivalent to the 1:1 trial (79.1%), despite using only half the control sample size. In heterogeneous effect scenarios (Scenarios 3, 5, and 6), the BIRTH design also performed excellently. For instance, in the first three subgroups of Scenario 5, the BIRTH design’s power (71.3%-74.0%) significantly exceeded that of the NB design (49.9%-53.9%) and MAP design (50.7%-55.4%), while remaining comparable to the 1:1 trial design (71.2%-74.6%). This indicates that the BIRTH design effectively identifies heterogeneity and appropriately borrows information, maintaining high power while avoiding bias from excessive borrowing.

In scenarios where the null hypothesis is true (Scenario 2, 4, 7–10, and portions of Scenarios 5 and 6), the BIRTH design exhibited varied performance in controlling the type I error rate. In homogeneous null scenarios (e.g., Scenario 2), the type I error rates for the BIRTH design ranged from 4.5% to 6.9%, closely approximating the nominal 5% level and outperforming the CPP design (7.0%-9.6%).

However, a critical observation arises in scenarios characterized by high inter-subgroup heterogeneity under the null hypothesis. Specifically, in Scenario 4, the type I error rate for subgroup 3 reached 9.0%. This inflation was more pronounced in the most challenging scenario (Scenario 10), which combined null effects, high heterogeneity, and discordant RWD, where the type I error rate for subgroup 3 rose to 12.4% (Supplementary Table S4). This indicates that while the dual dynamic borrowing mechanism is generally effective, it may not fully prevent erroneous information sharing in the presence of extreme, multi-source heterogeneity, potentially leading to a biased estimation of the control group’s response rate and an increased risk of false-positive conclusions. This finding highlights the importance of carefully considering the context of the trial before applying the BIRTH design.

In scenarios incorporating external data (Scenarios 7–10), when external real-world data showed treatment efficacy higher than the current trial control group, the BIRTH design overestimated control group efficacy, resulting in type I error rates below the nominal level. Conversely, when external real-world data showed efficacy lower than the current trial control group, the BIRTH design underestimated control group efficacy, leading to elevated type I error rates. This suggests caution when incorporating heterogeneous external real-world data to prevent erroneous efficacy estimation.

To assess global error risks, we evaluated the family-wise type I error rate (FWER) across all scenarios (Supplementary Table S5). In the global null scenario, the BIRTH design’s FWER (24.0%) was comparable to the independent no-borrowing benchmark (24.1%) and lower than the standard 1:1 trial design (28.8%). Even in highly heterogeneous settings, BIRTH consistently maintained a lower FWER than the 1:1 design, confirming that our borrowing mechanism did not introduce additional global error inflation. We further conducted a sensitivity analysis under stricter conditions (; Supplementary Table S6-S7). Results showed that the FWER for BIRTH dropped significantly to 11.2%, surpassing the control of both the no-borrowing (14.8%) and 1:1 trial (16.0%) designs. While stricter thresholds effectively suppressed false positives, they also reduced power for detecting weak signals. Given the exploratory nature of basket trials, we prioritized sensitivity for signal detection using standard settings, while demonstrating that the design retains the flexibility to enforce rigorous error control if required.

Overall, integrating power and type I error analyses, the BIRTH design demonstrated excellent balance across various scenarios, providing power comparable to the 1:1 trial design when treatment effects exist, while maintaining acceptable type I error control when the null hypothesis is true. These results further confirm the BIRTH design’s advantages in balancing information borrowing with statistical inference reliability.

Effective sample size (ESS)

Finally, we calculated the average effective sample size (ESS) for each method to quantify information borrowing efficiency. As shown in Table 2, in homogeneous treatment effect scenarios, the BIRTH design effectively integrated subgroup information, achieving approximately 70% equivalent sample size gains. In heterogeneous effect scenarios (Scenarios 3–6), the ESS of the BIRTH design appropriately decreased with increasing heterogeneity, reflecting its adaptive adjustment of borrowing strength. In contrast, the CPP design showed significant ESS reduction under high heterogeneity conditions, while the NB and MAP designs maintained lower but stable ESS values. When bias existed in external data, the BIRTH design appropriately adjusted its ESS to 41.7–53.7, demonstrating sensitivity to data quality while maintaining robustness.

Table 2.

Comparison of effective sample size (ESS) across different designs under ten simulation scenarios

Scenario		Average ESS
	Designs	1	2	3	4	5
Scenario 1	NB	39.2	39.3	39.2	39.3	39.3
	MAP	38.4	37.7	38.1	38.3	38.2
	CPP	44.2	44.1	44.1	43.5	44.2
	BIRTH	72.1	70.9	71.7	70.5	71.1
	1:1 trial	98.9	97.7	99.6	98.3	98.6
Scenario 2	NB	39.9	39.9	40.0	40.0	40.0
	MAP	38.0	37.9	38.0	37.5	38.1
	CPP	45.1	44.8	45.1	44.9	45.1
	BIRTH	67.2	67.2	67.2	66.5	67.6
	1:1 trial	98.6	97.2	98.3	98.7	98.6
Scenario 3	NB	39.2	39.1	39.2	39.4	40.0
	MAP	38.4	38.4	37.5	37.6	37.2
	CPP	38.1	38.4	37.6	37.2	34.1
	BIRTH	48.7	54.4	53.3	50.9	43.1
	1:1 trial	77.7	78.0	75.9	73.7	69.6
Scenario 4	NB	40.0	39.5	39.3	39.1	39.3
	MAP	38.0	38.3	38.1	37.4	37.9
	CPP	34.8	37.0	38.1	38.2	36.9
	BIRTH	44.2	51.9	53.7	53.0	47.7
	1:1 trial	70.2	74.6	74.6	76.3	75.0
Scenario 5	NB	39.1	39.2	39.3	39.9	40.0
	CPP	38.4	37.8	38.0	37.5	38.1
	MAP	40.9	41.5	40.9	37.0	37.7
	BIRTH	58.9	59.5	59.7	51.2	51.2
	1:1 trial	89.1	87.6	88.1	76.2	77.4
Scenario 6	NB	39.3	39.4	39.3	39.1	40.0
	MAP	38.4	38.3	37.5	37.4	37.1
	CPP	37.1	36.3	34.9	35.9	28.9
	BIRTH	52.4	46.3	46.3	50.3	32.7
	1:1 trial	77.6	74.4	69.3	71.7	51.2
Scenario 7	NB	39.9	39.9	40.0	40.0	40.0
	MAP	40.2	40.1	40.7	40.1	40.8
	CPP	45.1	44.8	45.1	44.9	45.1
	BIRTH	63.6	64.3	64.1	63.5	63.9
	1:1 trial	98.6	97.2	98.3	98.7	98.6
Scenario 8	NB	39.9	39.9	40.0	40.0	40.0
	MAP	36.3	36.1	37.3	35.8	36.1
	CPP	45.1	44.8	45.1	44.9	45.1
	BIRTH	65.9	65.0	65.9	65.2	65.7
	1:1 trial	98.6	97.2	98.3	98.7	98.6
Scenario 9	NB	40.0	39.5	39.3	39.1	39.3
	MAP	40.2	39.7	39.8	38.3	38.0
	CPP	34.8	37.0	38.1	38.2	36.9
	BIRTH	41.7	51.1	53.2	53.7	49.3
	1:1 trial	70.2	74.6	74.6	76.3	75.0
Scenario 10	NB	40.0	39.5	39.3	39.1	39.3
	MAP	36.2	37.3	38.2	37.7	38.2
	CPP	34.8	37.0	38.1	38.2	36.9
	BIRTH	46.1	52.1	53.5	51.7	45.1
	1:1 trial	70.2	74.6	74.6	76.3	75.0

In conclusion, the BIRTH design demonstrates excellent balance across various complex scenarios, maintaining low bias and good type I error control while providing power comparable to 1:1 trials with significant sample size savings. These results comprehensively validate the design’s potential and value for safely and effectively borrowing external data in precision medicine basket trials.

Sensitivity analysis

To evaluate the robustness of the BIRTH design, we conducted systematic sensitivity analyses covering effect heterogeneity, key parameter settings, and sample size variations.

To assess resilience against assumption violations, we extended simulations to include scenarios with heterogeneous treatment effects (HTE) and stochastic efficacy patterns (Scenarios 11–16, detailed in Supplementary Table S8). The BIRTH design demonstrated robust operating characteristics across these complex settings. While a moderate trade-off in type I error (ranging from 7% to 11%) was observed for subgroups under the null hypothesis due to information borrowing, it remained comparable to or lower than competing designs. Crucially, the design significantly improved statistical power for detecting weak signals—for instance, substantially enhancing the detection capability for subgroups with weak treatment effects in mixed scenarios compared to the no-borrowing approach—without introducing substantial estimation bias (Supplementary Table S9-S13 and Figure S1 –S4).

The hyperparameter plays a pivotal role in the borrowing mechanism by controlling the impact of the Hellinger distance on weight allocation, as explicitly defined in calculation formula (17). Mathematically, as derived in Supplementary Proof 1, the derivative of the weight with respect to distributional discrepancy is inversely proportional to :

19

Consequently, this parameter governs a fundamental bias-variance trade-off: smalle values amplify sensitivity to subgroup heterogeneity, whereas larger values promote more uniform information sharing regardless of similarity.

Our simulation studies, summarized in Supplementary Table S14 and Fig. 7, underscore the necessity of selecting a moderate to balance these competing risks. Specifically, extremely low values () rendered the model overly sensitive to random fluctuations, resulting in minor type I error inflation (reaching 6.9%–8.9% in Scenario 4). Conversely, excessive values () led to indiscriminate borrowing that diluted subgroup-specific signals, resulting in excessive conservatism with type I error rates falling to 2.3%–3.6%. The interval of achieved an optimal equilibrium, maintaining sufficient statistical power while keeping type I error rates close to the nominal 5% level.

Fig. 7.

Impact of different on type I error rate and power of the BIRTH design

To quantify this responsiveness, we established linear regression models for each subgroup, explicitly modeling bias as a function of the logarithm of (i.e., ). In this framework, the slope serves as the sensitivity coefficient, capturing the rate of change in bias relative to logarithmic variations in . Paired-sample t-tests on the mean coefficients revealed that the BIRTH design exhibits significantly higher sensitivity compared to the 1:1 design (Fig. 8). This heightened responsiveness confirms the design’s adaptive capacity to dynamically integrate multi-source data, in contrast to the static nature of standard designs. Accordingly, we recommend setting for practical applications to ensure robust performance across varying degrees of data heterogeneity.

Fig. 8.

Parameter sensitivity of BIRTH versus 1:1 trial designs across simulation scenarios

Finally, we investigated the impact of sample size variations () to evaluate the design’s performance in resource-constrained settings. The design maintained stable type I error control across all sample sizes. As the control group sample size increased, the statistical power of the BIRTH design progressively converged with that of the 1:1 benchmark, underscoring its efficiency in small-sample settings typical of rare diseases (Supplementary Table S15 and Supplementary Figures S5–S6).

Discussion

In this paper, we proposed the BIRTH design, a novel Bayesian framework for hybrid control basket trials that synergistically integrates external RWD and internal cross-subgroup information. Our simulation studies demonstrate that BIRTH offers substantial gains in statistical efficiency, achieving statistical power comparable to a 1:1 randomized trial while reducing the control group sample size by 50%. This is achieved through a dual-layer, dynamic borrowing architecture, with the core innovation being an asymmetric borrowing strategy that protects the treatment arm from RWD-induced bias. These features make BIRTH a scientifically rigorous and resource-efficient solution for trials in precision oncology and rare diseases.

A critical finding from our simulations is the potential for type I error rate inflation under conditions of extreme, multi-source heterogeneity (Scenarios 4 and 10). This phenomenon highlights a fundamental trade-off: the adaptive mechanisms that enable powerful information borrowing in concordant settings can become overly aggressive when faced with substantial conflict across data sources. This does not invalidate the design, but rather defines its ‘safe operating space’. We therefore recommend BIRTH for clinical settings where there is a reasonable prior belief—grounded in biological plausibility or historical data—of at least partial homogeneity across subgroups. For purely exploratory trials with high uncertainty, a more cautious approach is warranted. This could involve pre-specifying more conservative borrowing parameters or, more formally, integrating an adaptive rule into the protocol that defaults to a no-borrowing model if an interim analysis indicates heterogeneity exceeding a pre-defined threshold. Transparently acknowledging and proactively managing this limitation is paramount for the design’s responsible implementation.

The validity of our approach, like all RWD-based methods, is contingent upon the correct specification of the propensity score model to adjust for observed confounders. While we employed best practices, the risk of unobserved confounding remains. Future applications could incorporate sensitivity analyses, such as calculating the E-value, to quantify the potential impact of unobserved confounders on study conclusions. Furthermore, it is critical to distinguish between the design-stage target and the posterior estimation outcome regarding sample size balance. Our choice to pre-specify a nominal borrowing number from RWD (via the target ESS) was a deliberate design feature to enhance regulatory transparency and prevent RWD from unduly dominating the analysis. However, the final Effective Sample Size (ESS) is a cumulative metric that also incorporates the marginal information gain from cross-subgroup borrowing. Given the asymmetric borrowing architecture—where the treatment arm borrows from internal pools of size while the control arm borrows strictly from concurrent pools of size (excluding RWD)—the final ESS naturally diverges from the initial 1:1 target. This variation is not a theoretical inconsistency but a demonstration of the model’s robustness: it maximizes power when safe while strictly limiting bias when data conflict.

Our study has several strengths. First, the simulation design and data generation process were constructed to reflect real-world stochastic variability rather than artificially forcing response rates to match targets. Second, regarding practical implementation, while this study employed a fixed clinical threshold () for benchmarking purposes, we recognize that basket trials often span indications with varying prognostic profiles. A key advantage of the BIRTH design is its adaptability; trial sponsors can readily specify subgroup-specific thresholds () to align with distinct clinical variations across histological subtypes. This flexibility ensures that the design remains scientifically robust even when decision criteria are heterogeneous across baskets. Future research should focus on extending the BIRTH framework to non-binary and time-to-event endpoints and developing adaptive sample size adjustment strategies. As regulatory acceptance of RWE continues to grow, real-world case studies will be crucial to validate BIRTH’s performance in multi-center clinical trials.

In conclusion, the BIRTH design marks a significant methodological advancement by providing a standardized, efficient, and robust framework for leveraging multi-source data in basket trials. It has the potential to become a key design paradigm for accelerating the development of innovative therapies in rare diseases and targeted oncology.

Authors’ contributions

WW contributed to the conception and design of the work. WW finished the code and analyses. WW, XC, LB, LJ and FY interpreted the results. WW drafted the manuscript, and all authors read, edited, and approved the final manuscript.

Funding

This work was supported by the National Natural Science Foundation of China [No. 82273735; 82473737; 82304252].

Data availability

The R code used to implement the proposed BIRTH methodology, perform the simulations, and generate the results reported in this manuscript is openly available in the GitHub repository: https://github.com/HM-WWX/BIRTH-design.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.