  Causal mediation analysis aims to investigate how an intermediary factor,
called a mediator, regulates the causal effect of a treatment on an outcome.
With the increasing availability of measurements on a large number of potential
mediators, methods for selecting important mediators have been proposed.
However, these methods often assume the absence of unmeasured mediator-outcome
confounding. We allow for such confounding in a linear structural equation
model for the outcome and further propose an approach to tackle the mediator
selection issue. To achieve this, we firstly identify causal parameters by
constructing a pseudo proxy variable for unmeasured confounding. Leveraging
this proxy variable, we propose a partially penalized method to identify
mediators affecting the outcome. The resultant estimates are consistent, and
the estimates of nonzero parameters are asymptotically normal. Motivated by
these results, we introduce a two-step procedure to consistently select active
mediation pathways, eliminating the need to test composite null hypotheses for
each mediator that are commonly required by traditional methods. Simulation
studies demonstrate the superior performance of our approach compared to
existing methods. Finally, we apply our approach to genomic data, identifying
gene expressions that potentially mediate the impact of a genetic variant on
mouse obesity.
