Stein-like estimators for causal mediation analysis in randomized trials

Causal mediation analysis aims to estimate natural direct and natural indirect effects under clearly specified assumptions. Traditional mediation analysis based on Ordinary Least Squares assumes an absence of unmeasured causes to the putative mediator and outcome. When these assumptions cannot be justified, instrumental variable estimators can be used in order to produce an asymptotically unbiased estimator of the mediator-outcome link, commonly referred to as a Two-Stage Least Squares estimator. Such bias removal, however, comes at the cost of variance inflation. A Semi-Parametric Stein-Like estimator has been proposed in the literature that strikes a natural trade-off between the unbiasedness of the Two-Stage Least Squares procedure and the relatively small variance of the Ordinary Least Squares estimator. The Semi-Parametric Stein-Like estimator has the advantage of allowing for a direct estimation of its shrinkage parameter. 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.