This is the second installment in a series of posts on causal mediation analysis with multiple mediators (see the first one here), which discusses how to handle the case of post-treatment confounding.

As earlier, I will carry on using the example of police legitimacy from the previous post. In the procedural justice literature several studies used Structural Equation Modelling (SEM) and defined a correlation between duty to obey and moral alignment, which is usually depicted by a double-headed arrow (e.g., Jackson et al. 2012, Hough et al. 2013). Nevertheless, such specification is meaningless in the causal inference literature where – as implied by their name – Directed Acyclic Graphs rule out the possibility of such double-headed arrows (i.e., they would create a cycle). Therefore instead of a single model, two of them need to be specified, one where moral alignment will be conditional on duty to obey, and a second where duty to obey will be conditional on moral alignment:

As pointed out earlier, the presence of a post-treatment confounder L (i.e., a second mediator) will violate the sequential ignorability assumption of no post-treatment confounders. With L present, no consistent estimates can be computed the usual way for the effect of the treatment (T) on the outcome (Y), because a new T -> L -> Y pathway will emerge. If this so-called backdoor pathway is not controlled for, then it will result in biased estimates. Yet, controlling for L does not address the problem either. If L is controlled for in the model for the mediator (M), this will also block the T -> L -> Y pathway, thus producing biased direct and indirect effects (see: Avin et al. 2005; Daniel et al. 2011).

Surmounting this problem requires *a new set of sequential ignorability assumptions*. Although several of these have been proposed (e.g., Tchetgen Tchetgen & Vanderweele 2014), here the one proposed by Imai and Yamamoto (2013) and slightly modified by De Stavola et al. (2015) will be presented. This modified sequential ignorability assumption posits that controlling for pre-treatment covariates there is:

- No unmeasured confounfounding of the T-Y, T-M, and T-L relationship
- No unmeasured confounding of the M-Y relationship, also controlling for T and L
- No unmeasured confounding of the L-Y relationship, also controlling for T
- No unmeasured confounder Z that was affected by the treatment

From this new set of identifying assumptions, the first one is automatically satisfied in case of a randomised treatment. The second assumption makes the mediator conditional on the post-treatment confounder. The third assumption is akin to the one for the single mediator case, but here for the post-treatment confounder. Finally, the first assumption is also a reiteration of the original assumption and restates that there is no unmeasured post-treatment confounder (i.e., another mediator) that has not been considered. Notably, L does not need to be a single mediator, it can be considered a vector of mediators, which allows the inclusion of several mediators.

However, simply rephrasing the original sequential ignorability assumption can be hardly sufficient to make the natural direct and indirect effects estimable. In addition, parametric restrictions are also necessary, which will be very similar to the ones that are required for SEMs. Thus, to assure the additivity of the effects, the *linearity assumption* needs to apply. In addition, *a loosened version of effect homogeneity needs to be satisfied* (i.e., on average, not for each unit). As shown by De Stavola et al. (2015) given these parametric restrictions, assumption 3 will be automatically satisfied.

Using this method, the natural direct effects will encompass L’s effect on Y that does not go through M (T -> L -> Y), whilst L’s mediated effect through M will be added to the natural indirect effect (T -> L -> M -> Y). I will return to the details of the necessary identification test and estimation in a later, separate post. In the next, final post in this series, I will discuss the case of causal mediation analysis with sequentially ordered mediators.

**References**

*Avin, Chen, Ilya Shpitser, and Judea Pearl. 2005. “Identifiability of Path-Specific Effects.” Proceedings of the International Joint Conferences on Artifical Intelligence 34:163–92.
Daniel, Rhian M., Bianca L. De Stavola, and Simon N. Cousens. 2011. “Gformula – Estimating Causal Effects in the Presence of Time-Varying Confounding or Mediation Using the G-Computation Formula.” The Stata Journal 11(4):479–517.
Hough, Mike, Jonathan Jackson, and Ben Bradford. 2013. “Legitimacy, Trust and Compliance: An Empirical Test of Procedural Justice Theory Using the European Social Survey.” Pp. 326–53 in Legitimacy and Criminal Justice – An International Exploration, edited by J. Tankebe and A. Liebling. Oxford University Press.
Imai, Kosuke and Teppei Yamamoto. 2013. “Identification and Sensitivity Analysis for Multiple Causal Mechanisms: Revisiting Evidence from Framing Experiments.” Political Analysis 21(2):141–71.
Jackson, Jonathan et al. 2012. “Why Do People Comply with the Law?” British Journal of Criminology 52(6):1051–71.
Stavola, Bianca L. De, Rhian M. Daniel, George B. Ploubidis, and Nadia Micali. 2015. “Practice of Epidemiology Mediation Analysis With Intermediate Confounding : Structural Equation Modeling Viewed Through the Causal Inference Lens.” 181(1):64–80.
Tchetgen Tchetgen, Eric J. and Tyler J. VanderWeele. 2014. “Identification of Natural Direct Effects When a Confounder of the Mediator Is Directly Affected by Exposure.” Epidemiology. 25(2):282–91.
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