Minimum free energy randomized design to improve covariate balance

2026-06-09

AI Summary

This study addresses the long-standing lack of a quantifiable trade-off criterion between covariate balance and randomization robustness in randomized controlled trials. Inspired by the second law of thermodynamics, it introduces the concept of free energy into experimental design and proposes a randomization strategy that minimizes free energy—simultaneously minimizing covariate imbalance and maximizing allocation entropy—to formally characterize the optimal compromise between these competing objectives. Building on this framework, the authors develop a theoretically guaranteed dynamic allocation algorithm integrated with finite-sample variance decomposition techniques. Numerical experiments demonstrate that the proposed method outperforms existing strategies in both statistical efficiency and robustness, effectively controlling covariate imbalance, mitigating the impact of unobserved heterogeneity on mean squared error, and preserving minimax efficiency.

Abstract

``Block what you can and randomize what you cannot'' is the core principle for treatment effect estimation in randomized controlled trials. Although a wealth of allocation strategies has been developed, an explicit trade-off between the covariate balance achieved by blocking and the robustness guaranteed by randomization is seldom quantified. Motivated by the second law of thermodynamics, this work posits a new criterion that lowers the covariate imbalance while maximizing the entropy that quantifies contrast and allocation diversity. The resulting optimal strategy, termed the minimum free energy randomized design, is then derived, thereby formally achieving such a trade-off. To facilitate practical implementation, we further develop a computationally efficient dynamic allocation algorithm with theoretical guarantees. Using a finite-sample variance decomposition, the proposed randomization strategy is shown to control covariate imbalance while preventing unobserved heterogeneity from dominating the mean squared error, thus retaining minimax efficiency under the prescribed design constraints. Extensive numerical simulations demonstrate that our method achieves superior statistical efficiency and greater robustness than existing approaches.

Problem

• covariate balance
• randomization
• blocking
• treatment effect estimation
• minimax efficiency

Innovation

• minimum free energy
• randomized design
• covariate balance
• entropy maximization
• dynamic allocation