Robust Optimal Design of Two-Armed Trials with Side Information

Significant evidence has become available that emphasizes the importance of personalization in medicine. In fact, it has become a common belief that personalized medicine is the future of medicine. The core of personalized medicine is the ability to design clinical trials that investigate the role of patient covariates on treatment effects. In this work, we study the optimal design of two-armed clinical trials to maximize accuracy of statistical models where the interaction between patient covariates and treatment effect are incorporated to enable precision medication. Such a modeling extension leads to significant complexities for the produced optimization problems because they include optimization over design and covariates concurrently. We take a robust optimization approach and minimize (over design) the maximum (over population) variance of interaction effect between treatment and patient covariates. This results in a min-max bi-level mixed integer nonlinear programming problem, which is notably challenging to solve. To address this challenge, we introduce a surrogate model by approximating the objective function for which we propose two solution approaches. The first approach provides an exact solution based on reformulation and decomposition techniques. In the second approach, we provide a lower bound for the inner optimization problem and solve the outer optimization problem over the lower bound. We test our proposed algorithms with synthetic and real-world data sets and compare it with standard (re-)randomization methods. Our numerical analysis suggests that the lower bounding approach provides high-quality solutions across a variety of settings.