Mediation Formula for a Binary Outcome and a Time-Varying Exposure and Mediator, Accounting for Possible Exposure-Mediator Interaction

Valeri and VanderWeele [1] used the counterfactual framework to extend formulas from Baron and Kenny [2] to account for exposure-mediator interaction in mediation analysis. The results are also extended to the case when the outcome is binary. VanderWeele and Tchetgen Tchetgen [3] identified the randomized interventional analogues of the natural direct and indirect effects in longitudinal settings in presence of time-varying exposure, time-varying mediator, and possibly time-varying confounders where the outcome is continuous. We synthesize the ideas in these two papers to derive expressions for natural direct and indirect effects when the outcome is binary but the exposure, mediator and potential confounders are time-varying. Let Y denote the binary outcome, A(t) denote the exposure measured at time t, M(t) denote the mediator at time t, L(t) denote the potentially time-varying confounders at time t, and V denote the set of time-invariant baseline covariates for t = 1, 2, ..., T . Let Ā(t) = (A(1), ..., A(t)), M̄(t) = (M(1), ...,M(t)), and L̄(t) = (L(1), ..., L(t)) and let Ā ≡ Ā(T ), M̄ ≡ M̄(T ), and L̄ ≡ L̄(T ). The corresponding lower-case letters refer to the realizations of the random variates. With initial baseline covariates V and subsequent temporal order A(t), L(t),M(t), the relationships among A(t), L(t), M(t), Y , and V can be depicted as