Large-scale dependent multiple testing via hidden semi-Markov models

Large-scale multiple testing is common in the statistical analysis of high-dimensional data. Conventional multiple testing procedures usually implicitly assumed that the tests are independent. However, this assumption is rarely established in many practical applications, particularly in "high-throughput" data analysis. Incorporating dependence structure information among tests can improve statistical power and interpretability of discoveries. In this paper, we propose a new large-scale dependent multiple testing procedure based on the hidden semi-Markov model (HSMM), which characterizes local correlations among tests using a semi-Markov process instead of a first-order Markov chain. Our novel approach allows for the number of consecutive null hypotheses to follow any reasonable distribution, enabling a more accurate description of complex local correlations. We show that the proposed procedure minimizes the marginal false non-discovery rate (mFNR) at the same marginal false discovery rate (mFDR) level. To reduce the computational complexity of the HSMM, we make use of the hidden Markov model (HMM) with an expanded state space to approximate it. We provide a forward-backward algorithm and an expectation-maximization (EM) algorithm for implementing the proposed procedure. Finally, we demonstrate the superior performance of the SMLIS procedure through extensive simulations and a real data analysis.