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Defining causal mediation with a longitudinal mediator and a survival outcome

Abstract

In the context of causal mediation analysis, prevailing notions of direct and indirect effects are based on nested counterfactuals. These can be problematic regarding interpretation and identifiability especially when the mediator is a time-dependent process and the outcome is survival or, more generally, a time-to-event outcome. We propose and discuss an alternative definition of mediated effects that does not suffer from these problems, and is more transparent than the current alternatives. Our proposal is based on the extended graphical approach of Robins and Richardson (in: Shrout (ed) Causality and psychopathology: finding the determinants of disorders and their cures, Oxford University Press, Oxford, 2011), where treatment is decomposed into different components, or aspects, along different causal paths corresponding to real world mechanisms. This is an interesting alternative motivation for any causal mediation setting, but especially for survival outcomes. We give assumptions allowing identifiability of such alternative mediated effects leading to the familiar mediation g-formula (Robins in Math Model 7:1393, 1986); this implies that a number of available methods of estimation can be applied.

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Acknowledgements

I would like to thank Odd Aalen, Rhian Daniel, Ilya Sphitser, Mats Stensrud, Susanne Strohmaier, and Stijn Vansteelandt for helpful discussions.

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Appendix

Appendix

The following extension results in a greater range of situations where identifiability can be obtained by including possibly time-varying covariates.

Let the data consist of a time-ordered sequence of measurements A, C(1), M(1), N(1), C(2),  M(2), N(2),  \(\ldots \), \(C(\tau ), M(\tau ), N(\tau )\), where C(t) are sets of covariates. We assume that none of the elements in any C(t) are graphical children of A, or in other words, A is assumed to have no direct effect on C(t). Let further \(C^M(t), C^T(t)\subset C(t)\) such that under \(\text{ do }(A^T, A^M)\)

With analogous reasoning as above, identifiability is then obtained as follows:

$$\begin{aligned}&P(T> t^*; \text{ do }(A^T=a, A^M=a'))\\&\quad = \sum _{{\bar{m}}(t^*) {\bar{c}}(t^*)} \prod _t^{t^*} P(N(t)=0\,|\, N(t^-)=0, {\bar{M}}(t)={\bar{m}}(t),{\bar{C}}^M(t)={\bar{c}}^T(t);\text{ do }( A=a)) \\&\qquad \cdot P(M(t)=m(t)\,|\,N(t^-)=0, {\bar{M}}(t^-)={\bar{m}}(t^-), \bar{C}^T(t^-)={\bar{c}}^T(t^-);\text{ do }( A=a')), \end{aligned}$$
Fig. 7
figure7

Extended system with general time-varying covariates

The assumptions are illustrated in the graph of Fig. 7. Note that, in fact, \(C^T(t)\) could be in time after M(t), if it is possible to have measurements between M(t) and N(t), similarly to C in Fig. 5 (left).

Our assumptions and results are closely related to those of Shpitser (2018). He derives path-specific effects in graphs, first assuming no unobserved variables. This requires the assumption of ‘no recanting witness’ which, in our setting, would be a variable directly affected by both \(A^T\) and \(A^M\)—this would also violate our Assumptions A1.C and A2.C as the conditional independence can then not hold. For graphs with unobserved variables (semi-Markovian graphs), the corresponding criterion is that of ‘no recanting district’, where a district is a sequence of nodes affected by unobserved variables. For instance, in Fig. 5, if C was unobserved, then (N(1), M(2)) would be a district; in fact it would be a ‘recanting district’ because there are directed edges from \(A^T\) as well as from \(A^M\) into nodes of the district. Hence, assumption A1.C and A2.C can be seen as characterising covariates that allow us to avoid recanting districts. In contrast, in Fig. 6, (N(1), N(2)) is a district as frailty is a latent variable; but it is not recanting because there is no edge from \(A^M\) pointing at any nodes in the district. Similar connections to recanting districts are also noted by Vansteelandt et al. (2017). As Shpitser (2018) covers other forms of identifiability, not only based on the g-formula, his results are of greater generality than ours but not immediately applicable to the survival case as they still rely on nested counterfactuals.

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Didelez, V. Defining causal mediation with a longitudinal mediator and a survival outcome. Lifetime Data Anal 25, 593–610 (2019). https://doi.org/10.1007/s10985-018-9449-0

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Keywords

  • Causal inference
  • Mediation analysis
  • Graphical models
  • Causal graphs
  • Path specific effects