Optimal Control of Directional False Discovery Rates in Large‐Scale Testing

ABSTRACT

The high‐throughput biomedical technology enables measurement of thousands of gene expression levels contemporaneously. A major task in analyzing these gene expression data is to identify both over‐expressed and under‐expressed genes. The popular two‐group models select the non‐null genes without further classifying them as overexpression or underexpression. Consequently, two‐group decision rules are unable to constrain the numbers of falsely discovered over‐expressed or under‐expressed genes respectively. We propose a general three‐group model that allows dependence between the test statistics and develop a decision rule that separately controls the two types of false discoveries. We show that the optimal decision rule in our three‐group model has a special monotonic structure. By making use of this monotonic structure, we can linearize the two‐directional false discovery rate constraints. We prove that our decision rule optimizes the expected number of true discoveries while controlling the proportions of falsely discovered over‐expressed and under‐expressed genes at desired levels simultaneously. The data‐driven versions of the proposed procedures are suggested, and their consistency is established. Comparisons with state‐of‐the‐art approaches and applications to genomic studies show that our procedures work well.