eAppendix for : ” SAS macro for causal mediation analysis with survival data ”

We let Ta and Ma denote respectively the values of the time-to-event outcome and mediator that would have been observed had the exposure A been set to level a. We let Tam denote the value of the time-to-event outcome that would have been observed had the exposure, A, and mediator, M , been set to levels a and m, respectively. The average controlled direct effect comparing exposure level a to a∗ and fixing the mediator to level m on the mean survival time ratio scale is defined by CDEa,a∗(m) = E[Tam]/E[Ta∗m]. The average natural direct effect is then defined by NDEa,a∗(a ∗) = E[TaMa∗ ]/E[Ta∗Ma∗ ]. The average natural indirect effect can be defined as NIEa,a∗(a) = E[TaMa ]/E[TaMa∗ ], which compares the effect of the mediator at levels Ma and Ma∗ on the survival outcome when exposure A is set to a. Controlled direct effects and natural direct and indirect effects within strata of C = c are then defined by: CDEa,a∗|c(m) = E[Tam|c]/E[Ta∗m|c], NDEa,a∗|c(a ∗) = E[TaMa∗ |c]/E[Ta∗Ma∗ |c] and NIEa,a∗|c(a) = E[TaMa |c]/E[TaMa∗ |c] respectively. For an arbitrary time-to-event variable V , let λV (t) and λV (t|c) denote the hazard or hazard conditional on covariates c at time t, that is the instantaneous rate of the event conditional on V ≥ t. The causal effects can also be defined on the hazard ratio scale, replacing E[·] with λ(·).