Time-dependent mediators in survival analysis: Graphical representation of causal assumptions

We study time-dependent mediators in survival analysis using a treatment separation approach due to Didelez [2019] and based on earlier work by Robins and Richardson [2011]. This approach avoids nested counterfactuals and crossworld assumptions which are otherwise common in mediation analysis. The causal model of treatment, mediators, covariates, confounders and outcome is represented by causal directed acyclic graphs (DAGs). However, the DAGs tend to be very complex when we have measurements at a large number of time points. We therefore suggest using so-called rolled graphs in which a node represents an entire coordinate process instead of a single random variable, leading us to far simpler graphical representations. The rolled graphs are not necessarily acyclic; they can be analyzed by $\delta$-separation which is the appropriate graphical separation criterion in this class of graphs and analogous to $d$-separation. In particular, $\delta$-separation is a graphical tool for evaluating if the conditions of the mediation analysis are met or if unmeasured confounders influence the estimated effects. We also state a mediational g-formula. This is similar to the approach in Vansteelandt et al. [2019] although that paper has a different conceptual basis. Finally, we apply this framework to a statistical model based on a Cox model with an added treatment effect.survival analysis; mediation; causal inference; graphical models; local independence graphs