  Dimension reduction lies at the heart of many statistical methods. In
regression, dimension reduction has been linked to the notion of sufficiency
whereby the relation of the response to a set of predictors is explained by a
lower dimensional subspace in the predictor space. In this paper, we consider
the notion of a dimension reduction in regression on subspaces that are
sufficient to explain interaction effects between predictors and another
variable of interest. The motivation for this work is from precision medicine
where the performance of an individualized treatment rule, given a set of
pretreatment predictors, is determined by interaction effects.
