Technical Summary: A Trajectory-Informed Clustering Approach for Information Borrowing in Basket Trials

Problem Statement

Basket trials, which evaluate a single treatment across multiple tumor types or biomarkers simultaneously, face a critical statistical challenge: heterogeneity in treatment efficacy across baskets. While independent analysis of each basket avoids inappropriate information borrowing, it sacrifices statistical efficiency, particularly given the small sample sizes (typically 20–50 patients) common in early-phase oncology. Conversely, complete pooling (assuming all baskets are exchangeable) can inflate false positive rates if baskets differ significantly in response.

Existing solutions, such as hierarchical Bayesian models, allow for partial borrowing but often rely on pre-specified exchangeability structures or covariates. However, in practice, the biomarkers or covariates that define latent subgroups are frequently unknown at trial initiation. Furthermore, previous clustering methods often rely solely on a single summary endpoint, such as the Objective Response Rate (ORR). The authors argue that relying on a single binary proportion can lead to unstable clustering performance, especially in small samples or when response probabilities are near decision thresholds, failing to capture the underlying similarity structure of the baskets.

Methodology

The authors propose a two-stage, data-driven framework that utilizes longitudinal tumor response trajectories to inform information borrowing, rather than relying exclusively on the final ORR.

1. Trajectory Feature Extraction

The method models tumor response as a first-order, time-homogeneous Markov process with states $S = \{CR, PR, SD, PD\}$.

• Transition Probabilities: For each basket $j$, a one-step transition probability matrix $\hat{P}_j$ is estimated from the observed sequences of tumor responses across patients.
• Weighted Final-State Distribution: To account for varying numbers of assessment time points among patients, the authors introduce a weighting scheme based on the empirical distribution of assessment counts. This yields a weighted final-state distribution ($\pi_j$), which estimates the marginal probability of a patient being in a specific response state at their last observed assessment.
• Feature Vector: The clustering feature vector for each basket is constructed as $y_j = (\pi_j^\top, \hat{\theta}_j)^\top$, combining the four components of the weighted final-state distribution with the observed ORR ($\hat{\theta}_j$).

2. Clustering Procedure

• Distance Metric: Dissimilarity between baskets is calculated using the Manhattan distance ($L_1$) between their feature vectors.
• Cluster Selection: The number of clusters is determined using the Silhouette method. The algorithm selects the number of clusters ($u$) that maximizes the average Silhouette index, subject to a minimum cluster size of 2.
• Stability Safeguard: If the maximum average Silhouette index is low ($\bar{s} \leq 0.25$), indicating weak cluster separation, the method defaults to a single-cluster structure (full borrowing) to prevent unstable over-partitioning.

3. Hierarchical Bayesian Modeling

Once clusters are identified, a hierarchical Bayesian model is fitted only to the primary efficacy endpoint (ORR) within each cluster.

• Baskets within the same estimated cluster share a common prior distribution for the log-odds of response, allowing adaptive information borrowing.
• Baskets in different clusters are analyzed independently, preventing inappropriate borrowing across heterogeneous groups.
• The approach is treated as a two-stage algorithm: the clustering step is data-driven, and the subsequent Bayesian inference is conditional on the estimated partition.

Key Contributions

1. Trajectory-Informed Clustering: The paper introduces a novel method for basket trial analysis that leverages longitudinal response trajectories (transition probabilities) to define basket similarity, moving beyond static summary endpoints like ORR.
2. Handling Variable Assessment Schedules: The authors propose a specific weighting mechanism to derive final-state probabilities that accounts for the heterogeneity in the number of assessment time points across patients, a common issue in real-world trial data.
3. Data-Driven Borrowing Groups: The framework identifies latent borrowing groups without requiring pre-specified biomarkers or covariates, addressing a practical limitation in early-phase trials where such information may be unavailable.
4. Separation of Clustering and Borrowing: The method explicitly separates the clustering stage (using full trajectory data) from the borrowing stage (using only the primary endpoint), ensuring that borrowing decisions align with regulatory efficacy definitions while still utilizing trajectory data for grouping.

Simulation Results

The authors evaluated the method through 10,000 simulation replicates across three scenarios: homogeneous baskets (one true cluster), two-cluster heterogeneity, and three-cluster heterogeneity.

• Cluster Recovery: In heterogeneous settings (Scenarios 2 and 3), the proposed trajectory-informed approach recovered the underlying cluster structure more frequently than the "ORR-only" clustering comparator. The ORR-only method struggled to distinguish baskets with similar ORRs but different trajectory patterns.
• Type I Error and Power:
  • In the homogeneous scenario (Scenario 1), the proposed method showed a tendency toward over-partitioning (selecting multiple clusters when only one existed), which led to a slight inflation of the Type I error rate beyond the nominal 5% level.
  • In heterogeneous scenarios, the method maintained Type I error rates generally close to the nominal level while demonstrating improved statistical power compared to independent analyses ("No cluster") and the "One cluster" (full pooling) model.
• Estimation Efficiency: The proposed method yielded narrower 90% credible intervals compared to the non-borrowing approach, indicating improved estimation precision.
• Comparators: The method was compared against independent analysis, full pooling, ORR-only clustering, and an Exchangeable/Non-Exchangeable (EXNEX) model. The EXNEX model in these settings behaved similarly to the independent analysis, showing limited borrowing.

Significance and Limitations

The paper claims that the proposed framework offers a practical solution for information borrowing in basket trials where latent subgroup structures are unknown. By utilizing routinely collected trajectory data, it improves the recovery of true basket groupings and enhances estimation efficiency in heterogeneous settings.

However, the authors modestly acknowledge specific limitations:

• Over-partitioning in Homogeneity: The Silhouette-based implementation tends to over-partition when the true structure is homogeneous, potentially leading to unnecessary complexity and slight Type I error inflation.
• Simplifying Assumptions: The method treats response categories as nominal in the distance calculation and uses a uniform distribution to regularize rows in transition matrices where no outgoing transitions are observed. These choices prioritize numerical stability and simplicity but may not fully reflect the ordinal nature of tumor response or clinically realistic transition patterns.
• Conditional Inference: Because the clustering is estimated before the Bayesian model is fitted, the resulting posterior distributions are conditional on the estimated partition rather than fully marginalizing over cluster uncertainty.

The authors conclude that while the method shows promise for improving borrowing efficiency in heterogeneous settings, future research should explore more robust clustering algorithms to mitigate over-partitioning in homogeneous scenarios.