A simple and powerful method for large-scale composite null hypothesis testing with applications in mediation analysis

Large-scale mediation analysis has received increasing interest in recent years, especially in genome-wide epigenetic studies. The statistical problem in large-scale mediation analysis concerns testing composite null hypotheses in the context of large-scale multiple testing. The classical Sobel's and joint significance tests are overly conservative and therefore are underpowered in practice. In this work, we propose a testing method for large-scale composite null hypothesis testing to properly control the type I error and hence improve the testing power. Our method is simple and essentially only requires counting the number of observed test statistics in a certain region. Non-asymptotic theories are established under weak assumptions and indicate that the proposed method controls the type I error well and is powerful. Extensive simulation studies confirm our non-asymptotic theories and show that the proposed method controls the type I error in all settings and has strong power. A data analysis on DNA methylation is also presented to illustrate our method.