Two-sample Testing with Block-wise Missingness in Multi-source Data

Multi-source and multi-modal datasets are increasingly common in scientific research, yet they often exhibit block-wise missingness, where entire modalities are systematically absent in some sources or no single source contains all modalities. This structured missingness poses major challenges for two-sample hypothesis testing. Standard approaches, such as imputation or complete-case analysis, may introduce bias or suffer efficiency loss, especially under missingness-not-at-random mechanisms. To address this challenge, we propose the Block-Pattern Enhanced Test, a general framework for constructing two-sample testing statistics that explicitly accounts for block-wise missingness. We show that the framework yields valid tests under a new condition allowing for missing-not-at-random mechanism. Building on this general framework, we further propose the Block-wise Rank In Similarity graph Edge-count (BRISE) test, which accommodate heterogeneous modalities using rank-based similarity graphs. Theoretically, we establish that the null distribution of BRISE converges to a $χ^2$ distribution, and that the test is consistent both in the standard asymptotic regime and in the high-dimensional low-sample-size setting under mild conditions. Simulation studies demonstrate that BRISE controls the type-I error rate and achieves strong power across a wide range of alternatives. Applications to two real-world datasets with block-wise missingness further illustrate the practical utility of the proposed method.