  Social and behavioral scientists are increasingly employing technologies such
as fMRI, smartphones, and gene sequencing, which yield 'high-dimensional'
datasets with more columns than rows. There is increasing interest, but little
substantive theory, in the role the variables in these data play in known
processes. This necessitates exploratory mediation analysis, for which
structural equation modeling is the benchmark method. However, this method
cannot perform mediation analysis with more variables than observations. One
option is to run a series of univariate mediation models, which incorrectly
assumes independence of the mediators. Another option is regularization, but
the available implementations may lead to high false positive rates. In this
paper, we develop a hybrid approach which uses components of both filter and
regularization: the 'Coordinate-wise Mediation Filter'. It performs filtering
conditional on the other selected mediators. We show through simulation that it
improves performance over existing methods. Finally, we provide an empirical
example, showing how our method may be used for epigenetic research.
