Bayesian fusion forests for heterogeneous treatment effects on survival from randomised and real-world data

Technical Summary: Bayesian Fusion Forests for Heterogeneous Treatment Effects on Survival

Problem Statement
The paper addresses the challenge of estimating heterogeneous treatment effects (HTE) on survival outcomes by combining data from a Randomised Controlled Trial (RCT) and Real-World Data (RWD). While RCTs provide unbiased causal estimates, they are often underpowered for subgroup analysis and have limited follow-up. Conversely, RWD offers larger sample sizes and longer follow-up but suffers from unmeasured confounding.

Three specific challenges arise when fusing these sources for survival data:

1. Unmeasured Confounding: In RWD, treatment assignment is not random and may depend on unobserved factors, biasing treatment effect estimates.
2. Between-Source Heterogeneity: The baseline prognosis (survival without treatment) and error distributions often differ between the trial and observational populations due to differences in eligibility, protocols, and case mix. Simple pooling confounds these baseline differences with the treatment effect.
3. Censoring: Survival data often involves right-censoring (event time exceeds follow-up) and interval-censoring (event occurs between periodic visits), which complicates likelihood construction and estimation.

Existing methods typically address only one or two of these challenges, often failing to accommodate interval censoring or unmeasured confounding simultaneously while estimating HTE.

Methodology: The Bayesian Fusion Forest
The authors propose a nonparametric framework called the Bayesian Fusion Forest, which models the log-survival time using an Accelerated Failure Time (AFT) decomposition. The model jointly analyzes data from both sources ($S \in \{0, 1\}$) and decomposes the conditional log-survival time as follows:

$$\log T = m_0(X, S) + \tau(X)A + (1-S)A c(X) + \varepsilon$$

Where:

• $A$ is the binary treatment indicator.
• $X$ represents pre-treatment covariates.
• $m_0(X, S)$ (Baseline Prognosis): Modeled as a shared component $m_0^{sh}(X)$ plus a source-specific deviation $(1-S)d(X)$. This structure allows the model to borrow strength across sources while accommodating differences in baseline prognosis.
• $\tau(X)$ (Treatment Effect): The Conditional Average Treatment Effect (CATE), defined as $E[\log T(1) - \log T(0) | X]$. This is the primary estimand, interpreted as an acceleration factor on the survival time scale.
• $c(X)$ (Confounding Function): A term active only in the RWD ($S=0$) that absorbs the bias caused by unmeasured confounding. It is defined as the difference between the observed RWD treatment contrast and the true causal effect.
• $\varepsilon$ (Error Term): Modeled using a Hierarchical Dirichlet Process Mixture (HDPM) of normals. This allows the error distribution to differ in shape and scale between the two sources while sharing mixture components, capturing non-Gaussianity and heterogeneity without assuming a single parametric form.

Priors and Regularization
Each functional component ($m_0^{sh}, d, \tau, c$) is assigned a Bayesian Additive Regression Tree (BART) prior. The authors implement a "ladder" of regularization to reflect the information available for each component:

• Baseline ($m_0^{sh}$): The most flexible prior (200 trees) to capture complex disease-intrinsic structures.
• Deviation ($d$): Moderate flexibility (50 trees) to capture residual between-source differences.
• Treatment Effect ($\tau$): Stronger regularization (100 trees, depth penalty) to shrink towards homogeneity, reducing spurious heterogeneity.
• Confounding Function ($c$): The strongest regularization (50 trees, low root-split probability, depth penalty) to shrink towards a uniform bias, reflecting the assumption that unmeasured confounding varies little with covariates.

Inference
Posterior inference is performed via a blocked Gibbs sampler. The sampler updates the BART forests against partial residuals and handles censored observations (both right and interval) through data augmentation (Tanner and Wong, 1987). The HDPM error block is updated via a separate Gibbs step over cluster assignments, atoms, and concentration parameters.

Key Results

• Simulation Study:

  • Robustness to Confounding: The Bayesian Fusion Forest remains essentially unbiased across varying levels of unmeasured confounding ($\lambda_u$), whereas RWD-only estimates become increasingly biased.
  • Robustness to Heterogeneity: The method remains unbiased even as between-source heterogeneity in baseline prognosis increases ($\lambda_d$).
  • Efficiency: The fusion approach achieves lower Root Mean Squared Error (RMSE) and posterior variance than RCT-only analysis across all levels of confounding and heterogeneity.
  • Comparison with ML: The method outperforms four flexible machine-learning baselines (Deep Neural Networks, Gradient Boosting, Buckley-James boosting) in both accuracy and interval calibration. While ML methods often produced narrow intervals, they suffered from severe under-coverage (0.00–0.48) or required excessively wide intervals to achieve coverage. The Fusion Forest achieved nominal coverage (0.96) with competitive width.
  • High-Dimensional Covariates: As the number of covariates increased (up to $p=500$), the fusion method maintained lower RMSE and variance compared to the trial-only baseline, demonstrating that the precision gains from borrowing RWD outweigh the cost of searching a larger covariate space.

• Application (HIV Antiretroviral Therapy):

  • The authors fused the ACTG 175 trial (right-censored, ~3 years follow-up) with the Multicenter AIDS Cohort Study (interval-censored, ~25 years follow-up) to estimate the effect of combination therapy vs. ZDV monotherapy.
  • Findings: The fusion estimated an average acceleration factor of 1.65 (95% CI: [1.43, 1.90]), indicating combination therapy extends event-free survival by a factor of 1.65.
  • Individual Estimates: The fusion forest provided significantly sharper individual estimates (average 95% CI width reduced by 62% compared to trial-only). It identified a benefit for nearly every patient (96.4% with >95% posterior probability of benefit), whereas the trial-only analysis was inconclusive for a quarter of patients.
  • Heterogeneity: The treatment benefit was found to vary with CD8 count and race, with higher benefits observed in patients with higher CD8 counts.

Significance and Claims
The paper claims that the Bayesian Fusion Forest is the first method to simultaneously address unmeasured confounding, between-source heterogeneity, and both right- and interval-censoring in the context of heterogeneous treatment effect estimation for survival data.

The authors emphasize that their approach:

1. Relaxes the Unconfoundedness Assumption: By explicitly modeling a confounding function, the method avoids the need to assume RWD is unconfounded, a common and often violated assumption in data fusion.
2. Borrows Strength Appropriately: The hierarchical structure allows for adaptive borrowing of information from RWD, improving precision without compromising causal validity.
3. Provides Calibrated Uncertainty: Unlike many machine-learning predictors that target prediction rather than causal inference, the Bayesian framework provides intrinsic, calibrated uncertainty quantification for individual treatment effects.
4. Delivers Actionable Insights: In the HIV application, the method resolved inconclusive trial results, identifying near-universal benefits of combination therapy that were obscured by the trial's limited follow-up and power.

The authors note that the framework assumes the acceleration factor is time-invariant (dependent on covariates but not time), a standard assumption in AFT models, and suggest that future extensions could incorporate time-dependent effects. The software implementation is made available to facilitate reproducibility.