  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 (FWER) 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 FWER.
In this work, we investigate the impact of the selection threshold on the
finite sample FWER. We derive power maximizing selection threshold and show
that it is well approximated by an adaptive threshold of Wang et al. (2016). We
study the performance of the proposed procedures in a simulation study, and
apply them to a case-control study examining the effect of fish intake on the
risk of colorectal adenoma.
