Cross-direct effects in settings with two mediators

When multiple mediators are present, there are additional effects that may be of interest beyond the well-known natural (NDE) and controlled direct effects (CDE). These effects cross the type of control on the mediators, setting one to a constant level and one to its natural level, which differs across subjects. We introduce five such estimands for the cross-CDE and -NDE when two mediators are measured. We consider both the scenario where one mediator is influenced by the other, referred to as sequential mediators, and the scenario where the mediators do not influence each other. Such estimands may be of interest in immunology, as we discuss in relation to measured immunological responses to SARS-CoV-2 vaccination. We provide identifying expressions for the estimands in observational settings where there is no residual confounding, and where intervention, outcome, and mediators are of arbitrary type. We further provide tight symbolic bounds for the estimands in randomized settings where there may be residual confounding of the outcome and mediator relationship and all measured variables are binary.