  Causal mediation analysis aims to estimate the natural direct and indirect
effects under clearly specified assumptions. Traditional mediation analysis
based on Ordinary Least Squares (OLS) relies on the absence of unmeasured
causes of the putative mediator and outcome. When this assumption cannot be
justified, Instrumental Variables (IV) estimators can be used in order to
produce an asymptotically unbiased estimator of the mediator-outcome link.
However, provided that valid instruments exist, bias removal comes at the cost
of variance inflation for standard IV procedures such as Two-Stage Least
Squares (TSLS). A Semi-Parametric Stein-Like (SPSL) estimator has been proposed
in the literature that strikes a natural trade-off between the unbiasedness of
the TSLS procedure and the relatively small variance of the OLS estimator.
Moreover, the SPSL has the advantage that its shrinkage parameter can be
directly estimated from the data. In this paper, we demonstrate how this
Stein-like estimator can be implemented in the context of the estimation of
natural direct and natural indirect effects of treatments in randomized
controlled trials. The performance of the competing methods is studied in a
simulation study, in which both the strength of hidden confounding and the
strength of the instruments are independently varied. These considerations are
motivated by a trial in mental health evaluating the impact of a primary
care-based intervention to reduce depression in the elderly.
