A Globally Calibrated Bayesian Optimal Phase II Design for Adaptive Enrichment Trials

Technical Summary: A Globally Calibrated Bayesian Optimal Phase II Design for Adaptive Enrichment Trials

Problem Statement
In phase II oncology trials, adaptive enrichment strategies allow a study to continue in a prespecified biomarker-positive subgroup if the experimental treatment shows insufficient activity in the general "all-comer" population. A critical statistical challenge arises when applying standard Bayesian Optimal Phase II (BOP2) designs to such branching procedures. If the all-comer and biomarker-positive components are calibrated separately, the probability of making at least one false-positive efficacy conclusion (global type I error) for the entire adaptive procedure is inflated. Existing adaptive enrichment methods have primarily focused on randomized confirmatory trials or seamless phase II/III designs, often differing from the single-arm, all-comer-futility-gated setting required here.

Methodology
The authors propose a pathwise, globally calibrated BOP2 framework for single-arm adaptive enrichment trials with binary (and extendable to complex categorical) endpoints. The design operates as follows:

1. Trial Structure: The trial begins by enrolling all-comers. At prespecified interim analyses, the trial monitors for futility in the all-comer population. If the all-comer futility boundary is crossed, enrollment in the general population stops, and the trial transitions to an enrichment path, enrolling only biomarker-positive patients.
2. Decision Rules: The design utilizes BOP2-type power-family thresholds for futility monitoring. For a population $h$ (all-comer $A$ or biomarker-positive $+$) at sample size $s$, the futility boundary $b_h(s)$ is defined such that the trial stops if the posterior probability $P(\theta_h \le p_0 | \text{data})$ exceeds a threshold $C_h(s) = 1 - \lambda_h (s/N_h)^{\gamma_h}$.
3. Global Calibration: Unlike separate calibration, the tuning parameters $(\lambda_A, \gamma_A, \lambda_+, \gamma_+)$ are selected jointly. The objective is to control the global type I error rate, defined as the probability of claiming efficacy in either the all-comer population ($R_A$) or the biomarker-positive subgroup ($R_+$), under a point global null hypothesis ($H_{0,global}: \theta_+ = \theta_- = p_0$).
4. Handling Randomness: The calibration explicitly accounts for the random number of biomarker-positive patients available at the moment the trial transitions to the enrichment path. To ensure prospective implementation, decision rules are prespecified for every attainable first-assessment sample size and all subsequent interim looks.
5. Exact Evaluation: For binary endpoints, the authors derive an exact finite-state recursive enumeration of the complete adaptive procedure. This method calculates the probabilities of all possible trajectories (all-comer path vs. enrichment path) without relying on Monte Carlo simulation, enabling precise calibration and operating characteristic evaluation.

Key Contributions

• Unified Framework: The paper extends the BOP2 framework from population-specific monitoring to a globally calibrated, branching adaptive enrichment procedure. It treats the all-comer and biomarker-positive efficacy decisions as components of a single global rejection event.
• Exact Enumeration: The derivation of an exact recursive enumeration allows for the precise calculation of global type I error and operating characteristics, eliminating Monte Carlo error in the evaluation of the binary endpoint design.
• Prospective Implementation: The design retains the simplicity of prospectively tabulated decision boundaries. No recalibration is required during the trial based on observed biomarker prevalence or interim outcomes.
• Extension to Categorical Endpoints: The framework is extended to complex categorical endpoints using a Dirichlet–multinomial formulation, with calibration evaluated via simulation (detailed in the Supplementary Material).

Results
The authors evaluated the design through exact enumeration under a scenario with a null response probability $p_0 = 0.20$, target $p_1 = 0.40$, and nominal type I error $\alpha = 0.10$.

• Type I Error Control: Across a range of biomarker-positive prevalences ($\pi \in \{0.4, \dots, 0.8\}$), the proposed globally calibrated design maintained global type I error rates between 8.5% and 9.0%, staying below the 10% threshold. In contrast, a comparator design using separately calibrated BOP2 components exhibited inflated global type I error rates ranging from 12.5% to 14.5%.
• Power and Path Dynamics: Under alternative scenarios, the probability of claiming efficacy generally increased with the biomarker-positive response probability. The biomarker-positive prevalence influenced the route of the efficacy claim: higher prevalence increased the likelihood of an all-comer claim (by raising the marginal response rate) and decreased the reliance on the enrichment path.
• Comparison: While the componentwise comparator occasionally showed higher apparent power, this was attributed to its failure to control the global type I error rate, rendering the comparison invalid under comparable error constraints.

Significance
The paper claims that the proposed design offers a principled alternative to combining separately calibrated phase II designs after an all-comer futility decision. By jointly calibrating the thresholds, the design preserves the prospective simplicity and tabulated nature of BOP2 while extending rigorous type I error control to the complete branching decision procedure. This allows the target population to be adapted based on emerging evidence of subgroup-specific activity without sacrificing statistical validity. The authors emphasize that the framework enables development decisions to be supported by accumulating evidence while maintaining strict control over the probability of false-positive conclusions across the entire adaptive strategy.