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 [Lifetime Data Analysis 25, no. 4: 593-610] and based on earlier work by Robins and Richardson [Causality and Psychopathology: Finding the Determinants of Disorders and Their Cures, 103-158. Oxford University Press]. This approach avoids nested counterfactuals and cross-world assumptions which are otherwise common in mediation analysis. The causal model of treatment, mediators, covariates, confounders, and outcome is represented by directed acyclic graphs (DAGs). However, the DAGs tend to be very complex when we have measurements at many 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>δ</mml:mi> <mml:annotation>$\delta$</mml:annotation></mml:semantics> </mml:math> -separation which is the appropriate graphical separation criterion in this class of graphs and analogous to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>d</mml:mi> <mml:annotation>$d$</mml:annotation></mml:semantics> </mml:math> -separation. In particular, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>δ</mml:mi> <mml:annotation>$\delta$</mml:annotation></mml:semantics> </mml:math> -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. [Statistics in Medicine 38, no. 24: 4828-4840], 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.