Computational solutions for global and closed testing in high-dimensional data

Abstract

This thesis develops computational solutions for multiple testing problems in high-dimensional biomedical data, focusing on global and closed testing procedures to control false discoveries.

Chapter 2 constructs confidence regions by inverting the globaltest, using them to select tuning parameters for ridge regression. This approach selects the least complex model within the confidence region and is computationally efficient, often matching cross-validation in predictive performance.

Chapter 3 proposes a closed testing procedure using the globaltest as a local test for pathway analysis in metabolomics. To overcome the exponential computational burden of closed testing, it introduces novel shortcuts (single-step and iterative branch-and-bound) that enable exact inference while controlling the family-wise error rate, even when feature sets are chosen post hoc.

Chapter 4 presents a method to focus statistical power on prespecified "focus sets" while still providing valid true discovery guarantees for all sets via interpolation. Built from partial procedures and Holm's method, it serves as a conservative shortcut to full closed testing. Chapter 5 improves a permutation-based closed testing shortcut by embedding it in a branch-and-bound algorithm. This iterative approach converges to the exact closed testing result or provides a valid bracket if stopped early, demonstrated in neuroimaging and metabolomics applications.