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Inference for Individual Mediation Effects and Interventional Effects in Sparse High-Dimensional Causal Graphical Models
We consider the problem of identifying intermediate variables (or mediators)
that regulate the effect of a treatment on a response variable. While there has
been significant research on this topic, little work has been done when the set
of potential mediators is high-dimensional and when they are interrelated. In
particular, we assume that the causal structure of the treatment, the potential
mediators and the response is a directed acyclic graph (DAG). High-dimensional
DAG models have previously been used for the estimation of causal effects from
observational data and methods called IDA and joint-IDA have been developed for
estimating the effects of single interventions and multiple simultaneous
interventions respectively. In this paper, we propose an IDA-type method called
MIDA for estimating mediation effects from high-dimensional observational data.
Although IDA and joint-IDA estimators have been shown to be consistent in
certain sparse high-dimensional settings, their asymptotic properties such as
convergence in distribution and inferential tools in such settings remained
unknown. We prove high-dimensional consistency of MIDA for linear structural
equation models with sub-Gaussian errors. More importantly, we derive
distributional convergence results for MIDA in similar high-dimensional
settings, which are applicable to IDA and joint-IDA estimators as well. To the
best of our knowledge, these are the first distributional convergence results
facilitating inference for IDA-type estimators. These results have been built
on our novel theoretical results regarding uniform bounds for linear regression
estimators over varying subsets of high-dimensional covariates, which may be of
independent interest. Finally, we empirically validate our asymptotic theory
and demonstrate the usefulness of MIDA in identifying large mediation effects
via simulations and application to real data in genomics.Comment: 35 pages (supplementary material - 26 pages), 5 tables, 4 figure