  In situations with non-manipulable exposures, interventions can be targeted
to shift the distribution of intermediate variables between exposure groups to
define interventional disparity indirect effects. In this work, we present a
theoretical study of identification and nonparametric estimation of the
interventional disparity indirect effect among the exposed. The targeted
estimand is intended for applications examining the outcome risk among an
exposed population for which the risk is expected to be reduced if the
distribution of a mediating variable was changed by a (hypothetical) policy or
health intervention that targets the exposed population specifically. We derive
the nonparametric efficient influence function, study its double robustness
properties and present a targeted minimum loss-based estimation (TMLE)
procedure. All theoretical results and algorithms are provided for both
uncensored and right-censored survival outcomes. With offset in the ongoing
discussion of the interpretation of non-manipulable exposures, we discuss
relevant interpretations of the estimand under different sets of assumptions of
no unmeasured confounding and provide a comparison of our estimand to other
related estimands within the framework of interventional (disparity) effects.
Small-sample performance and double robustness properties of our estimation
procedure are investigated and illustrated in a simulation study.
