After finishing my latest series of posts on causal mediation analysis with multiple mediators (you can find them here, here, and here), I had a sinking feeling that I forgot to mention a couple of important points, which means this series is not quite finished yet after all… There are two more issues I want to address: the differences between the post-treatment confounder and sequentially ordered approaches, and a potential Bayesian alternative.

In the second and third posts, I added two figures, both that can be considered Directed Acyclic Graphs (DAGs). Yet, two of those figures were essentially the same, with duty to obey being conditioned on moral alignment:

Does this mean that these two models are essentially the same? The answer is yes and no. Yes, they are the same, in the sense that both of them are accurate representations of the causal model. DAGs are vehicles which can be used to represent causal models (Pearl 2009). In case of two mediators, the post-treatment confounding and sequential ordering cases are causally equivalent. However, this equivalence disappears as soon as more than two variables are present, as the following set of figures demonstrates:

T stands for the treatment, C for a vector of pre-treatment covariates, M is the mediator, L is the post-treatment confounder, and Y is the outcome. The numbering of M indicates the corresponding mediator’s place in the sequential order (1 being the first, 2 the second, etc.). Figure (a) shows the case discussed in the first post when causal independence was assumed. Figure (b) and (c) represent the post-treatment confounder and sequentially ordered cases. Notice, that these DAGs are essentially the same, and they are very similar to the one that was inserted earlier in this post (the only difference is the presence of C in (a) and (b)). By contrast, juxtaposing (d) and (e) presents a different picture. In (d) the post-treatment confounders only predict M, but not each other, whilst in case of (e) M_{1} predicts not only M_{3}, but M_{2} as well.

It is also worth noting that the decomposition for these methods will be quite different. Here is the summary of the esimated direct and indirect effects for each case with two mediators:

- Post-treatment confounding (Imai & Yamamoto 2013; De Stavola et al. 2015): NDE (T -> Y, T -> L -> Y) and NIE (T -> M -> Y, T -> L -> M -> Y)
- Sequential ordering:
- Daniel et al. (2015): NDE (T -> Y), NIE
_{1}(T -> M_{1}-> Y), NIE_{2}(T -> M_{2}-> Y), NIE_{joint}(T -> M_{1}-> M_{2}-> Y) - Steen et al. (2017): NDE (T -> Y), NIE (T -> M
_{1}-> Y, T -> M_{1}-> M_{2}-> Y), PIE (T -> M_{2}-> Y)

- Daniel et al. (2015): NDE (T -> Y), NIE

This also demonstrates that the these various two-way, three-way, and four-way decompositions will provide different effects and the meaning of natural direct (NDE) and indirect effect (NIE) will change accordingly, with the partial indirect effect (PIE) added as well.

The second point I wanted to make is a much briefer one. I have not yet discussed a significant portion of the literature on causal mediation analysis which relies on the Bayesian framework. The excellent article by Daniels et al. (2013) offers a Bayesian alternative for the single mediator case. Even more relevant for the current case is another article by Kim et al. (working paper) where multiple mediators can be estimated. This method permits the estimation of the unique and joint (i.e., M_{1 }+M_{2}) effects of various mediators without any parametric restrictions or constraints on the decomposition. It does not allow the finest decomposition of Daniel et al. (2015), but it still provides a powerful alternative to the ones discussed so far. I will probably come back to this family of methods at some point.

I hope that with these closing thoughts I can let this lingering uneasy feeling go and end the discussion of multiple mediators for some time.

**References**

*Daniel, R. M., B. L. De Stavola, S. N. Cousens, and S. Vansteelandt. 2015. “Causal Mediation Analysis with Multiple Mediators.” Biometrics 71(1):1–14.*

* Daniels, Michael J., Jason A. Roy, Chanmin Kim, Joseph W. Hogan, and Michael G. Perri. 2012. “Bayesian Inference for the Causal Effect of Mediation.” Biometrics 68(4):1028–36.*

* 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.*

* Kim, Chanmin, Michael J. Daniels, and Joseph W. Hogan. workig paper “Bayesian Methods for Multiple Mediators : Relating Principal Stratification and Causal Mediation in the Analysis of Power Plant Emission Controls.” 1–36.*

* Pearl, Judea. 2009. “Causal Inference in Statistics: An Overview.” Statistics Surveys 3(0):96–146.*

* 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.*

* Steen, Johan, Tom Loeys, Beatrijs Moerkerke, and Johan Steen. 2017. “Flexible Mediation Analysis with Multiple Mediators.” American Journal of Epidemiology 186(2):184–93.*