Four-Way Decomposition

The four are: the controlled direct effect (neither mediation nor interaction — the effect with the mediator fixed at a reference), the reference interaction (interaction but no mediation — the joint effect that operates even without the exposure changing the mediator), the mediated interaction (both mediation and interaction — requires the exposure to change the mediator and the two to interact), and the pure indirect effect (mediation but no interaction — the effect transmitted through the mediator absent any interaction). They are mutually exclusive and sum to the total effect, so together they account for every part of the exposure's impact through these two lenses.

It refines them. The familiar two-way split into natural direct and natural indirect effects can be recovered by grouping the four components: the natural direct effect equals the controlled direct effect plus the reference interaction, and the natural indirect effect equals the pure indirect effect plus the mediated interaction. The four-way version goes further by revealing how much of each natural effect is due to interaction versus 'pure' mechanism. This is why VanderWeele describes it as a unification — the standard mediation quantities are sums of its building blocks.

Beyond consistency and positivity, the decomposition requires four no-unmeasured-confounding conditions: no unmeasured confounding of the exposure-outcome relationship, of the mediator-outcome relationship, of the exposure-mediator relationship, and — critically — no mediator-outcome confounder that is itself affected by the exposure. The last condition is the one most often violated in observational social-epidemiologic data; when it fails, the natural-effect components are not identified and interventional or path-specific decompositions are needed instead. Sensitivity analysis, such as E-values for the mediation assumptions, is strongly recommended.