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Testing composite null hypotheses arises in various applications, such asmediation and replicability analyses. The problem becomes more challenging inhigh-throughput experiments where tens of thousands of features are examinedsimultaneously. Existing large-scale inference methods for composite nullhypothesis testing often fail to explicitly incorporate the dependencestructure, producing overly conservative or overly liberal results. In thiswork, we first develop a four-state hidden Markov model (HMM) to model abivariate $p$-value sequence from replicability analysis with two studies,accounting for local feature dependence and study heterogeneity. Building onthe HMM, we propose a multiple testing procedure that controls the falsediscovery rate (FDR). Extending the HMM to model the $p$-values from $n$studies requires a computational cost of exponential order of $n$. To addressthis challenge, we introduce a novel e-value framework that reduces thecomputational cost to quadratic growth in the number of studies whilemaintaining FDR control. We show that the proposed method asymptoticallycontrols the FDR and exhibits higher power numerically than competing methodsat the same FDR level. In a real data application to genome-wide associationstudies (GWAS), our method reveals new biological insights that are overlookedby existing methods.

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