  In causal mediation analysis, the natural direct and indirect effects
(natural effects) are nonparametrically unidentifiable in the presence of
treatment-induced confounding, which motivated the development of randomized
interventional analogues (RIAs) of the natural effects. The RIAs are easier to
identify and widely used in practice. Applied researchers often interpret RIA
estimates as if they were the natural effects, even though the RIAs could be
poor proxies for the natural effects. This calls for practical and theoretical
guidance on when the RIAs differ from or coincide with the natural effects,
which this paper aims to address. We develop a novel empirical test for the
divergence between the RIAs and the natural effects under the weak assumptions
sufficient for identifying the RIAs and illustrate the test using the Moving to
Opportunity Study. We also provide new theoretical insights on the relationship
between the RIAs and the natural effects from a covariance perspective and a
structural equation perspective. Additionally, we discuss previously
undocumented connections between the natural effects, the RIAs, and estimands
in instrumental variable analysis and Wilcoxon-Mann-Whitney tests.
