Controlling the False Discovery Rate in Complex Multi-Way Classified Hypotheses

In this article, we propose a generalized weighted version of the well-known Benjamini-Hochberg (BH) procedure. The rigorous weighting scheme used by our method enables it to encode structural information from simultaneous multi-way classification as well as hierarchical partitioning of hypotheses into groups, with provisions to accommodate overlapping groups. The method is proven to control the False Discovery Rate (FDR) when the p-values involved are Positively Regression Dependent on the Subset (PRDS) of null p-values. A data-adaptive version of the method is proposed. Simulations show that our proposed methods control FDR at desired level and are more powerful than existing comparable multiple testing procedures, when the p-values are independent or satisfy certain dependence conditions. We apply this data-adaptive method to analyze a neuro-imaging dataset and understand the impact of alcoholism on human brain. Neuro-imaging data typically have complex classification structure, which have not been fully utilized in subsequent inference by previously proposed multiple testing procedures. With a flexible weighting scheme, our method is poised to extract more information from the data and use it to perform a more informed and efficient test of the hypotheses.