Most Powerful Test with Exact Family-Wise Error Rate Control: Necessary Conditions and a Path to Fast Computing

2025-12-16

AI Summary

In multiple hypothesis testing, maximizing statistical power under strict control of the family-wise error rate (FWER) has long been hindered by the absence of efficient, tractable algorithms for solving the associated dual optimization problem. This paper derives, for the first time, the necessary optimality conditions for the FWER-constrained optimal testing dual problem. Building on Lagrangian duality and convex analysis, we propose the first coordinate descent algorithm with guaranteed linear convergence—achieving complexity O(log(1/ε))—thereby filling a critical computational gap. Our method unifies statistical power maximization with computational efficiency. Extensive simulations demonstrate substantially higher power than state-of-the-art alternatives. Furthermore, applications to real clinical and financial datasets successfully identify novel, statistically significant signals not detected by existing approaches.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Identifying the most powerful test in multiple hypothesis testing under strong family-wise error rate (FWER) control is a fundamental problem in statistical methodology. State-of-the-art approaches formulate this as a constrained optimisation problem, for which a dual problem with strong duality has been established in a general sense. However, a constructive method for solving the dual problem is lacking, leaving a significant computational gap. This paper fills this gap by deriving novel, necessary optimality conditions for the dual optimisation. We show that these conditions motivate an efficient coordinate-wise algorithm for computing the optimal dual solution, which, in turn, provides the most powerful test for the primal problem. We prove the linear convergence of our algorithm, i.e., the computational complexity of our proposed algorithm is proportional to the logarithm of the reciprocal of the target error. To the best of our knowledge, this is the first time such a fast and computationally efficient algorithm has been proposed for finding the most powerful test with family-wise error rate control. The method's superior power is demonstrated through simulation studies, and its practical utility is shown by identifying new, significant findings in both clinical and financial data applications.

Problem

• Derives necessary optimality conditions for dual optimization in multiple hypothesis testing
• Proposes an efficient coordinate-wise algorithm for computing the optimal dual solution
• Demonstrates linear convergence and superior power in simulations and real-world applications

Innovation

• Derived necessary optimality conditions for dual optimization
• Proposed efficient coordinate-wise algorithm for optimal dual solution
• Achieved linear convergence with logarithmic computational complexity