  Cause-effect relationships are typically evaluated by comparing outcome
responses to binary treatment values, representing two arms of a hypothetical
randomized controlled trial. However, in certain applications, treatments of
interest are continuous and multidimensional. For example, understanding the
causal relationship between severity of radiation therapy, summarized by a
multidimensional vector of radiation exposure values and post-treatment side
effects is a problem of clinical interest in radiation oncology. An appropriate
strategy for making interpretable causal conclusions is to reduce the dimension
of treatment. If individual elements of a multidimensional treatment vector
weakly affect the outcome, but the overall relationship between treatment and
outcome is strong, careless approaches to dimension reduction may not preserve
this relationship. Further, methods developed for regression problems do not
directly transfer to causal inference due to confounding complications. In this
paper, we use semiparametric inference theory for structural models to give a
general approach to causal sufficient dimension reduction of a multidimensional
treatment such that the cause-effect relationship between treatment and outcome
is preserved. We illustrate the utility of our proposals through simulations
and a real data application in radiation oncology.
