Causal mediation analysis with multiple mediators 1. – Causal independence and joint mediation

To kick off the year with an “easy” and “light” topic, I decided to start a series of discussions on causal mediation analysis with multiple mediators. Because the remaining papers in my PhD rely on such techniques, I thought it might make sense to write a brief summary of the different approaches one can take. I am aware that in a rapidly advancing field such as causal inference this post risks becoming obsolete very quickly, at the same time, I hope that this overview can still remain relevant for some upcoming papers (and posts). 

This first post in the series revisits the ideas of causal mediation analysis with a single mediator discussed here. As it has been pointed out there, Structural Equation Modelling has certain limitations that need to be overcome in order to provide reliable causal inference for direct and indirect effects. Causal mediation analysis with a single mediator offers some solutions, provided that certain identifying assumptions are satisfied. From these assumptions the fourth one carries particular relevance here, as it states that there cannot be a post-treatment confounder (L) that could have an effect on the mediator or outcome. In essence, this means that there cannot be another mediator which was affected by the treatment.

To illustrate the problem I will be using an example from the literature of police legitimacy. It is often theorised (Tyler & Jackson 2013; ) that police legitimacy is construed as duty to obey the police (i.e., consent) and moral alignment with the police (i.e., appropriateness). These two aspects of police legitimacy assumed to channel the effects of appropriate police behaviour (i.e., procedural justice and respect for boundaries) to cooperation with the police. There are four ways to handle a similar case where multiple mediators are present.

First, one could assume that duty to obey and moral alignment are causally independent of one another. This case can be depicted by the following figure (which can be considered a Directed Acyclic Graph (DAG)):

In case of causal independence, the mediated pathways can be examined one at a time relying on the usual method. This causal independence is an untestable assumption, though a significant relationship between the two mediators can indicate dependency (Imai & Yamamoto 2013). However, even if there appears to be no association between the mediators, usually further justifications should be given when arguing for causal independence. For instance, variables can be created in a way to assure the orthogonality of mediators (e.g., through exploratory factor analysis with varimax rotation). In other cases, there might be strong theoretical reasons for such independence. Taguri et al. (2018) for example examined the success of two unrelated cavity protection techniques. However, in the social sciences such cases are very rare, usually, the mediators have at least some relationship with each other. Taking the example of this post, it would be very difficult to argue for the independence of duty to obey and moral alignment. Alas, looking at mediators one at a time will also falter provided that there are interactions between the effects of various mediators and the outcome (Lange et al. 2014).

Thus, normally one needs to assume causal dependence between the mediators, which violates the fourth assumption of the sequential ignorability assumption. One way to overcome this issue is to rely on the second approach, and instead of examining the two mediators separately, focus on their joint effect. This proposition can be depicted by the following DAG:

This solution of taking a vector of mediators is robust to unmeasured common causes of mediators, interaction between the treatment and various mediators, interactions between mediators, and can even be used when the mediators assumed to have a causal order (Vanderweele & Vansteelandt 2014; Steen et al. 2017). Reassuringly, the sequential ignorability assumption for a single mediator will still apply, but now for a vector of mediators. However, one of the major limitations of this strategy is that it seriously constrains the purview of the researchers. In the current example, we are interested in the in duty to obey’s and moral alignment’s unique mediating effect on willingness to cooperate with the police. However, estimating such effects means that a new set of identifying assumptions will be needed with new estimation strategies. The upcoming two posts will discuss two such cases: mediation analysis with post-treatment confounders and causally ordered mediators.

References

Huq, A. Z.Aziz H., J. Jackson, and R. J. Trinker. 2017. “Legitimating Practices: Revisiting the Predicates of Police Legitimacy.” British Journal of Criminology (57):1101–22.
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.
Lange, Theis, Mette Rasmussen, and Lau Caspar Thygesen. 2014. “Assessing Natural Direct and Indirect Effects through Multiple Pathways.” American Journal of Epidemiology 179(4):513–18.
Steen, Johan, Tom Loeys, Beatrijs Moerkerke, and Johan Steen. 2017. “Flexible Mediation Analysis with Multiple Mediators.” American Journal of Epidemiology 186(2):184–93.
Taguri, Masataka, John Featherstone, and Jing Cheng. 2018. “Causal Mediation Analysis with Multiple Causally Non-Ordered Mediators.” Statistical Methods in Medical Research 27(1):3–19.
Tyler, Tom R. and Jonathan Jackson. 2013. “Future Challenges in the Study of Legitimacy and Criminal Justice.” Pp. 83–104 in Legitimacy and Criminal Justice – An International Exploration, edited by J. Tankebe and A. Liebling. Wiley.

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