Testing composite null hypotheses with high-dimensional dependent data: a computationally scalable FDR-controlling procedure

Abstract: Testing composite null hypotheses arises in various applications, such as mediation and replicability analyses. The problem becomes more challenging in high-throughput experiments where tens of thousands of features are examined simultaneously. Existing large-scale inference methods for composite null hypothesis testing often fail to explicitly incorporate the dependence structure, producing overly conservative or overly liberal results. In this work, we first develop a four-state hidden Markov model (HMM) to model a bivariate $p$-value sequence from replicability analysis with two studies, accounting for local feature dependence and study heterogeneity. Building on the HMM, we propose a multiple testing procedure that controls the false discovery rate (FDR). Extending the HMM to model the $p$-values from $n$ studies requires a computational cost of exponential order of $n$. To address this challenge, we introduce a novel e-value framework that reduces the computational cost to quadratic growth in the number of studies while maintaining FDR control. We show that the proposed method asymptotically controls the FDR and exhibits higher power numerically than competing methods at the same FDR level. In a real data application to genome-wide association studies (GWAS), our method reveals new biological insights that are overlooked by existing methods.