Partial Ordering Bayesian Logistic Regression Model for Phase I Combination Trials and Computationally Efficient Approach to Operational Prior Specification

Technical Summary: Partial Ordering Bayesian Logistic Regression Model for Phase I Combination Trials

Problem Statement
Phase I clinical trials aim to identify the Maximum Tolerated Dose (MTD) or Maximum Tolerated Combination (MTC) within a target toxicity level (TTL). While the Continual Reassessment Method (CRM) and its extension, the Partial Ordering CRM (POCRM), are widely used for single-agent and combination trials respectively, they rely on one-parameter models. The POCRM, in particular, assumes a specific ordering of dose combinations and uses a one-parameter power model. This structure is criticized for being overly restrictive in complex settings, such as:

1. Control Group Integration: In trials randomizing patients between experimental dose combinations and a Standard of Care (SoC) control, accurate toxicity estimation is required at multiple points on the toxicity surface (specifically at the SoC and the MTC). One-parameter models often fail to fit these distinct points simultaneously.
2. Complexity of Interaction: In multi-drug or dose-schedule trials, understanding the shape of the combination-toxicity surface is often necessary beyond simply locating the MTC.
3. Computational Burden of Calibration: Designing these trials requires specifying hyperparameters (e.g., toxicity skeletons, prior distributions). The conventional "grid search" calibration method, which evaluates all parameter combinations across multiple scenarios, suffers from multiplicative computational costs that grow exponentially with the number of parameters and scenarios.

Methodology
The paper proposes two primary methodological advancements: the Partial Ordering Bayesian Logistic Regression Model (POBLRM) and a Cyclic Calibration Approach.

1. The POBLRM Model
The POBLRM extends the POCRM by replacing the one-parameter power model with a two-parameter logistic regression model while retaining the partial ordering framework.

• Model Structure: Under a specific ordering $s$, the toxicity probability $\psi_s(d_i, \theta)$ is modeled as:
  $$\text{logit}(\psi_s(d_i, \theta)) = \theta_1 + \theta_2 \tilde{d}_i$$
  where $\theta = (\theta_1, \theta_2)^T$ are model parameters and $\tilde{d}_i$ are standardized dose levels derived from prior toxicity estimates (skeleton).
• Randomized Setting Adaptation: In randomized trials (experimental arm vs. SoC), the model imposes a constraint $d_0 = 0$ for the control arm. This ensures that updates to the model based on the experimental arm do not alter the toxicity estimate at the SoC, allowing the model to fit both the control and the MTC accurately.
• Prior Distributions and Efficiency: The authors address the computational cost of estimating two-parameter models under multiple orderings.
  • They introduce a pseudo prior (based on Beta distributions derived from pseudo-cohorts) which allows for closed-form likelihood calculations and avoids numerical integration or MCMC.
  • They establish a link between the conventional Normal prior and the pseudo prior by minimizing the Kullback-Leibler (KL) divergence between the resulting posterior toxicity curves. This allows researchers to specify intuitive Normal priors while utilizing the computationally efficient pseudo prior for simulations and ordering selection (via AIC).

2. Cyclic Calibration Approach
To address the computational inefficiency of calibrating design parameters (skeleton spacing, prior means/variances), the paper proposes a "cyclic calibration" method.

• Mechanism: Instead of evaluating the full grid of parameter combinations (multiplicative complexity), the algorithm updates parameters one-at-a-time in a cycle. It searches for the optimal value of one parameter while holding others fixed, then moves to the next.
• Scenario Selection: To further reduce costs, the method suggests calibrating against a subset of scenarios rather than the full set. The subset is chosen based on scenario complexity, measured by the PO-benchmark (an upper bound on the probability of correct selection). Specifically, the authors propose using only the scenarios with the lowest and highest benchmark performance (e.g., "Ben2").
• Convergence: The algorithm stops when a full cycle yields no improvement in the geometric mean Probability of Correct Selection (PCS), guaranteed to converge to a local maximum.

Key Results
Numerical evaluations were conducted on a $3 \times 3$ dual-agent combination trial setting ($N=45$ patients) and a randomized setting.

• Performance vs. POCRM:
  • In non-randomized settings, POBLRM performs similarly to POCRM in terms of PCS (Mean PCS $\approx$ 44%).
  • In randomized settings (with a control arm), POBLRM significantly outperforms POCRM. The mean PCS increased from 38% (POCRM) to 45% (POBLRM). The POBLRM successfully identified the SoC when all combinations were overly toxic (78% selection rate vs. 45% for POCRM) and selected higher doses when all were safe, whereas the one-parameter model struggled to fit both extremes.
• Performance vs. 2BLRM:
  • Compared to the two-dimensional Bayesian Logistic Regression Model (2BLRM), POBLRM showed similar average PCS but offered more balanced performance across scenarios. The 2BLRM exhibited high variance, performing well in easy scenarios but poorly in "antagonistic" scenarios where drug interactions reduce toxicity.
  • Computational Efficiency: POBLRM simulation times were roughly double that of POCRM (1.36s vs 0.6s) but dramatically faster than 2BLRM (95.42s).
• Calibration Efficiency:
  • The Cyclic2 approach (cyclic calibration using only 2 scenarios) achieved a mean PCS within 0.5% of the Grid20 approach (full grid search over 20 scenarios).
  • This resulted in a computational reduction of over 2,500-fold (and up to 5,000-fold in other investigations) compared to full grid search, while maintaining operational characteristics.

Significance and Claims
The paper claims that the POBLRM provides a necessary evolution for Phase I combination trials, particularly those involving control groups or complex dose-schedule interactions where one-parameter models are insufficient. By combining the flexibility of a two-parameter logistic model with the partial ordering framework, it offers accurate toxicity estimation at multiple points on the surface without the extreme computational burden of fully flexible multi-dimensional models like 2BLRM.

Furthermore, the proposed cyclic calibration and scenario reduction strategies offer a practical solution to the "computationally expensive" nature of modern adaptive trial design. The authors assert that these methods allow for the specification of operational priors that are robust across a wide range of scenarios without requiring prohibitive computational resources, making sophisticated Bayesian designs more accessible for real-world clinical trials. The work is motivated by and applied to specific oncology and COVID-19 trials (e.g., AGILE platform), demonstrating its utility in real-world statistical collaboration.