  Mediation analysis has become an important tool in the behavioral sciences
for investigating the role of intermediate variables that lie in the path
between a randomized treatment and an outcome variable. The influence of the
intermediate variable on the outcome is often explored using structural
equation models (SEMs), with model coefficients interpreted as possible
effects. While there has been significant research on the topic in recent
years, little work has been done on mediation analysis when the intermediate
variable (mediator) is a high-dimensional vector. In this work we present a new
method for exploratory mediation analysis in this setting called the directions
of mediation (DMs). The first DM is defined as the linear combination of the
elements of a high-dimensional vector of potential mediators that maximizes the
likelihood of the SEM. The subsequent DMs are defined as linear combinations of
the elements of the high-dimensional vector that are orthonormal to the
previous DMs and maximize the likelihood of the SEM. We provide an estimation
algorithm and establish the asymptotic properties of the obtained estimators.
This method is well suited for cases when many potential mediators are
measured. Examples of high-dimensional potential mediators are brain images
composed of hundreds of thousands of voxels, genetic variation measured at
millions of SNPs, or vectors of thousands of variables in large-scale
epidemiological studies. We demonstrate the method using a functional magnetic
resonance imaging (fMRI) study of thermal pain where we are interested in
determining which brain locations mediate the relationship between the
application of a thermal stimulus and self-reported pain.
