  This paper focuses on the problem of modeling and estimating interaction
effects between covariates and a continuous treatment variable on an outcome,
using a single-index regression approach. The primary motivation is to estimate
an optimal individualized dose rule in an observational study. To model
possibly nonlinear interaction effects between patients' covariates and a
continuous treatment variable, we employ a two-dimensional penalized spline
regression on an index-treatment domain, where the index is defined as a linear
projection of the covariates. The method is illustrated using two applications
as well as simulation experiments. A unique contribution of this work is in the
parsimonious (single-index) parametrization specifically defined for the
interaction effect term.
