  Causal mediation analysis is a useful tool for epidemiological research, but
it has been criticized for relying on a "cross-world" independence assumption
that is empirically difficult to verify and problematic to justify based on
background knowledge. In the present article we aim to assist the applied
researcher in understanding this assumption. Synthesizing what is known about
the cross-world independence assumption, we discuss the relationship between
assumptions for causal mediation analyses, causal models, and non-parametric
identification of natural direct and indirect effects. In particular we give a
practical example of an applied setting where the cross-world independence
assumption is violated even without any post-treatment confounding. Further, we
review possible alternatives to the cross-world independence assumption,
including the use of computation of bounds that avoid the assumption
altogether. Finally, we carry out a numerical study in which the cross-world
independence assumption is violated to assess the ensuing bias in estimating
natural direct and indirect effects. We conclude with recommendations for
carrying out causal mediation analyses.
