Evaluating Bias Reduction Methods in Binary Emax Model for Reliable Dose-Response Estimation

Technical Summary: Evaluating Bias Reduction Methods in Binary Emax Model for Reliable Dose-Response Estimation

Problem Statement
The Binary Emax model is a standard tool in Phase II clinical trials for identifying optimal doses based on binary outcomes (e.g., responder vs. non-responder). However, parameter estimation via Maximum Likelihood Estimation (MLE) faces significant challenges in small-to-moderate sample sizes, which are common in early-stage trials. Two primary issues undermine the reliability of standard MLE:

1. Separation: In binary models, complete or quasi-complete separation can occur when covariates perfectly predict outcomes, leading to non-convergence of the estimation algorithm, infinite coefficient estimates, or probabilities of 0 or 1.
2. Non-Monotonicity and Model Violation: While the Emax model assumes a concave, monotonically increasing dose-response relationship, finite sample variability can produce data that appear convex or non-monotonic. This violates model assumptions, causing MLE algorithms to fail or produce unstable estimates with high variance.

Existing software packages (e.g., ClinDR, Dosefinding) utilizing standard MLE do not adequately address separation or these convergence issues. While Firth's bias-reduction method is known to handle separation in Generalized Linear Models (GLMs), its applicability and performance in the specific context of the non-linear Binary Emax model, particularly compared to other penalized approaches, require further investigation.

Methodology
The paper evaluates three bias-reduction strategies applied to the three-parameter logit Emax model:

1. Cox-Snell Bias Correction: A post-hoc analytical correction where the estimated bias term $B(\hat{\theta})$ is subtracted from the MLE. This method relies on the inversion of the Fisher information matrix and higher-order derivatives of the log-likelihood.
2. Firth's Score Modification: A preventive approach that modifies the score function $U(\theta)$ by adding an adjustment term $W_s$ derived from the Jeffreys prior. This aims to produce finite estimates even under separation. However, in non-linear models like Emax, this score modification does not correspond to maximizing a specific penalized likelihood function, limiting the use of certain optimization algorithms (e.g., Newton-Raphson requires a symmetric Hessian).
3. Maximum Penalized Likelihood Estimation (MPLE) with Jeffreys Prior: This method explicitly maximizes a penalized likelihood function $L^*(\theta) = L(\theta)|I(\theta)|^{1/2}$, where $|I(\theta)|^{1/2}$ is the Jeffreys invariant prior. The authors derive the analytical expression for the score function of this penalized likelihood in the context of the logit Emax model. Unlike Firth's score modification in this non-linear setting, MPLE provides an explicit likelihood function, allowing for standard optimization techniques.

The study compares these methods against standard MLE using comprehensive simulation studies across varying sample sizes ($n=50, 100, 150, 200$) and parameter configurations (varying $E_{max}$ and $ED_{50}$). Performance metrics include convergence rates, bias, Mean Squared Error (MSE), variance stability, and confidence interval coverage. A real-world application is demonstrated using data from the TURANDOT Phase II trial.

Key Contributions

• Analytical Derivation: The paper provides the closed-form analytical expressions for the Jeffreys prior penalty and the resulting score function specifically for the binary Emax model with a logistic link, a context where the equivalence between Firth's score modification and Jeffreys penalization (known in canonical GLMs) does not strictly hold.
• Comparative Evaluation: It systematically quantifies the performance of Cox-Snell, Firth's score modification, and MPLE in the specific setting of non-linear binary dose-response modeling, highlighting scenarios where standard methods fail.
• Real-World Validation: The methods are applied to the TURANDOT study, which exhibited a non-monotonic dose-response curve, demonstrating the practical utility of these methods in handling complex, real-world trial data.

Results

• Convergence and Stability: Standard MLE and the Cox-Snell correction frequently failed to converge or produced unstable estimates (e.g., undefined standard errors) in small sample sizes ($n=50$) and when data exhibited separation or non-concave patterns. In contrast, both Firth's method and MPLE consistently produced stable, finite estimates.
• Bias and Variance: Both Firth's and MPLE methods significantly reduced bias compared to MLE. While both were effective, MPLE with Jeffreys prior consistently outperformed Firth's method in terms of stability and lower variance (MSE).
• Robustness to Model Violation: In scenarios where the dose-response relationship deviated from the assumed monotonic concave shape (e.g., convex increasing or non-monotonic data), MPLE demonstrated superior stability. The Cox-Snell method was particularly prone to instability due to its reliance on the inverse of the Fisher information matrix, which becomes ill-conditioned in these scenarios.
• TURANDOT Application: In the analysis of the TURANDOT trial, MLE, Cox-Snell, and Firth's methods yielded unstable standard errors for $\log(ED_{50})$ due to the non-concave data pattern. MPLE provided stable point and variance estimates. Furthermore, MPLE produced probability estimates with significantly smaller variance compared to Firth's method, particularly at lower dose levels.

Significance and Claims
The authors conclude that while Firth's score modification is a robust alternative to MLE for handling separation, MPLE with Jeffreys prior offers a more effective and reliable solution for the binary Emax model. Specifically, MPLE is highlighted as superior for dose-response relationships that deviate from monotonicity, providing stable parameter estimation and lower variance.

The paper modestly notes that while MPLE improves estimation stability, the resulting Wald confidence intervals may be conservative due to the non-quadratic nature of the penalized log-likelihood in small samples. The authors suggest that profile penalized likelihood intervals could offer more precise estimation. Additionally, the paper acknowledges that observed non-monotonicity in real data (like TURANDOT) may stem from non-ignorable missingness rather than just sampling variability, and points to companion work addressing penalized estimation under missing-not-at-random (MNAR) mechanisms.

The work provides a practical, robust alternative for dose-ranging studies where sample sizes are limited and model assumptions may be imperfect, offering an R package (jpemax) to facilitate implementation.