Continuing with the example of police legitimacy, there are some who have argued that the two aspects of police legitimacy follow a specific order, and that duty to obey is actually an outcome of moral alignment with the police (e.g., Huq et al 2017). In this case, the effect of a procedural justice/legality treatment (T) will be mediated by moral alignment (M1) and duty to obey (M2) in sequence, towards the outcome of willingness to cooperate with the police (Y). In graphical terms this would look like this:
However, such a causal chain of mediators also violates the sequential ignorability assumption for a single mediator, which means that a new set of assumptions needs to be devised. Daniel et al. (2015) and Steen et al. (2017) both proposed a very similar set of assumptions, where controlling for pre-treatment covariates there are:
- No unmeasured confounding of T-Y, T-M1, and T-M2
- No unmeasured confounding of the M1-Y relationship, also controlling for T
- No unmeasured confounding of the M2-Y relationship, also controlling for T and M1
- No unmeasured M1-Y, M1-M2, or M2-Y confounder L1 or L2 that was affected by T
From this updated sequential ignorability assumption, the first is automatically satisfied in case of randomised treatment. The second assumption is very similar to the one made for a single mediator, while the third includes controlling for the first mediator. Finally, the fourth mediator reiterates that there cannot be post-treatment confounders that have not been considered. In other words, all potential mediators need to be accounted for in the model.
Yet again, simply stating these assumptions will not allow estimating the natural direct and indirect effects, other parametric assumptions are also necessary. Here, however, Daniel et al. (2015) and Steen et al. (2017) will differ from one another. The method offered by Daniel et al. (2015) requires the linearity assumption (similar to Structural Equation Models), which permits the additivity of effects. In addition, a sensitivity parameter κ2 (kappa squared) also needs to be estimated, which stands for the correlation of certain potential outcomes, and can have a marked influence on the results (I will return to this matter in a future post). This technique will permit the finest possible decomposition, with estimates for the natural direct effect (T -> Y), the natural indirect effect for mediator 1 (T -> M1 -> Y), the natural indirect effect for mediator 2 (T->M2->Y), and finally, the two mediators jointly mediated effect (T -> M1 -> M2 -> Y).
The method suggested by Steen at al. (2017) offers a slightly different decomposition: here the natural direct effect remains the same (T -> Y), whilst the natural indirect effect for the first mediator will incorporate both the joint effect and its own effect (T -> M1 ->Y and T -> M1 -> M2 -> Y). For the second mediator, only a partial indirect effect will be estimated (T -> M2 -> Y). “Front-loading” the joint effect of the first mediator means that the finest decomposition of the earlier model will not be possible anymore. Sacrificing the partitioning of the effects comes with some gains on the modelling side: the linearity assumption does not need to apply, which makes the model more flexible. Also, there is no need for the sensitivity parameter κ2 which makes the interpretation of the results more straightforward.
Overall, your data and research question should determine which method you rely upon. The first one is more akin to Structural Equation Modelling and provides the finest possible decomposition. The second one is cruder, as it does not allow the separate estimation of the joint effects, but it provides much more flexibility with the modelling. As a last cautionary note, it must be emphasised that neither of these methods can assess whether the chosen causal order is the correct one. The veracity of the model always needs to be theory- (e.g., what does the existing evidence suggest?) and/or data-driven (e.g., is there clear temporal order between the mediators?). As with the earlier post, I will return to the nitty-gritty of the estimation in a later post.
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.
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.
Steen, Johan, Tom Loeys, Beatrijs Moerkerke, and Johan Steen. 2017. “Flexible Mediation Analysis with Multiple Mediators.” American Journal of Epidemiology 186(2):184–93.