The effect of an expsore on an outcome of interest is often mediated by intermediate variables. The goal of causal mediation analysis is to evaluate the role of these intermediate variables (mediators) in the causal effect of the exposure on the outcome. In this paper, we consider causal mediation of a baseline exposure on a survival (or time-to-event) outcome, when the mediator is time-dependent. The challenge in this setting lies in that the event process takes places jointly with the mediator process; in particular, the length of the mediator history depends on the survival time. As a result, we argue that the definition of natural effects in this setting should be based on only blocking those paths from treatment to mediators that are not through the survival history. We propose to use a stochastic interventions (SI) perspective, introduced by Didelez, Dawid, and Geneletti (2006), to formulate the causal mediation analysis problem in this setting. Under this formulation, the mediators are regarded as intervention variables, onto which a given counterfactual distribution is enforced. The natural direct and indirect effects can be defined analogously to the ideas in Pearl (2001). In particular, they also allow for a total effect decomposition and an interpretation of the natural direct effect as a weighted average of controlled direct effects. The statistical parameters that should arise are defined nonparametrically; therefore, they have meaningful interpretations, independent of the causal formulations and assumptions. We present a general semiparametric inference framework for these parameters. Using their efficient influence functions, we develop semiparametric efficient and robust targeted substitution-based (TMLE) and estimating-equation-based (A-IPTW) estimators. An IPTW estimator and g-computation estimator will also be presented.
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