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

A single-index model with a surface-link for optimizing individualized dose rules