On optimal two-stage testing of multiple mediators

Mediation analysis in high-dimensional settings often involves identifying potential mediators among a large number of measured variables. For this purpose, a two-step familywise error rate procedure called ScreenMin has been recently proposed (Djordjilovi\'c et al. 2019). In ScreenMin, variables are first screened and only those that pass the screening are tested. The proposed threshold for selection has been shown to guarantee asymptotic familywise error rate. In this work, we investigate the impact of the selection threshold on the finite sample familywise error rate. We derive a power maximizing selection threshold and show that it is well approximated by an adaptive threshold of Wang et al. (2016). We illustrate the investigated procedures on a case-control study examining the effect of fish intake on the risk of colorectal adenoma.