[Paper Review] Optimal adaptive two-stage designs for single-arm trial with binary endpoint

Kevin Kunzmann, Meinhard Kieser|arXiv (Cornell University)|May 1, 2016

Statistical Methods in Clinical Trials1 references3 citations

TL;DR

This paper formulates optimal adaptive two-stage designs for single-arm clinical trials with binary endpoints as an integer linear program, enabling exact computation of designs that minimize expected sample size under the null hypothesis without restrictive technical constraints. It resolves pathologies in prior designs—such as non-monotonic sample size functions—while maintaining near-optimal efficiency, using advanced integer programming to ensure interpretability and practical usability.

ABSTRACT

Minimizing the number of patients exposed to potentially harmful drugs in early onco logical trials is a major concern during planning. Adaptive designs account for the inherent uncertainty about the true effect size by determining the final sample size within an ongoing trial after an interim look at the data. We formulate the problem of finding adaptive designs which minimize expected sample size under the null hypothesis for single-arm trials with binary outcome as an integer linear program. This representation can be used to identify optimal adaptive designs which improve previous designs in two ways: Firstly, designs can be found exhibiting lower expected sample size under the null hypothesis than those provided by previous algorithms. Secondly, we explain how integer programming techniques can be exploited to remove pathologies of the optimal and previous solutions arising from the discrete nature of the underlying statistics. The resulting designs are both efficient in terms of expected sample size under the null hypothesis and well interpretable.

Motivation & Objective

• To develop adaptive two-stage designs that minimize expected sample size under the null hypothesis for single-arm trials with binary endpoints.
• To overcome limitations of previous algorithms that required restrictive technical constraints to ensure computational feasibility.
• To address pathologies in sample size functions—such as non-monotonicity—arising from the discrete nature of binomial statistics.
• To improve interpretability and practical adoption of optimal designs by enforcing structural constraints like unimodality and contiguous stopping regions.
• To enable efficient computation using standardized, high-performance integer programming solvers without sacrificing optimality.

Proposed method

• Formulates the design optimization problem as an integer linear program (ILP), representing decision rules and sample size functions as discrete variables.
• Uses the conditional error function principle to maintain type I error control while allowing flexible sample size adaptation after an interim analysis.
• Imposes additional constraints (e.g., unimodal sample size function, contiguous stopping regions) within the ILP framework to eliminate pathological solutions.
• Employs commercial-grade ILP solvers to compute optimal solutions efficiently, avoiding slow branch-and-bound implementations.
• Enables customization of the objective function, such as minimizing expected sample size under a prior distribution or penalizing large maximum sample sizes.
• Validates solutions through comparison with existing methods (e.g., Englert and Kieser, 2013), demonstrating minimal performance loss while improving structural properties.

Experimental results

Research questions

• RQ1 Can optimal adaptive two-stage designs for single-arm binary trials be computed exactly without imposing artificial technical constraints?
• RQ2 How do pathological sample size functions—such as non-monotonicity—arise in optimal designs due to discrete distributions, and can they be eliminated?
• RQ3 To what extent does enforcing structural constraints (e.g., unimodal sample size function) affect the expected sample size under the null hypothesis compared to unconstrained optimal designs?
• RQ4 Can integer programming techniques be used to efficiently compute optimal designs that are both statistically efficient and clinically interpretable?
• RQ5 Is the monotonicity constraint on the conditional error function necessary, or can it be relaxed without significant loss in efficiency?

Key findings

• The proposed ILP formulation enables exact computation of optimal adaptive two-stage designs without any additional technical constraints, achieving true optimality.
• The performance loss in expected sample size under the null hypothesis when enforcing 'niceness' constraints (e.g., unimodal sample size function) is negligible—often less than 1%—demonstrating that the monotonicity constraint in prior work is not unnecessarily restrictive.
• The optimal designs exhibit well-behaved sample size functions, avoiding counterintuitive jumps (e.g., decreasing then increasing), which enhances clinical interpretability and adoption.
• Commercial-grade ILP solvers allow rapid computation of solutions, significantly outperforming naive branch-and-bound approaches in terms of speed and scalability.
• The framework is extensible: additional constraints (e.g., minimum conditional power) or alternative objective functions (e.g., minimizing higher moments of sample size) can be easily incorporated.
• For the parameter setting ρ₀=0.5, ρ₁=0.7, α=0.05, β=0.1, the optimal design requires 47 patients in stage two upon 13 responses, 44 for 14 responses, and 47 for 15 responses—confirming that the non-monotonicity in prior designs is a solvable artifact of discrete optimization.