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.

翻译：本文的重点是利用单一指数回归法,模拟和估计共变体和连续治疗变量对结果的相互作用影响。主要动机是在观察研究中估计最佳个别剂量规则。为了模拟病人的共变体和连续治疗变量之间可能的非线性相互作用影响,我们在一个指数处理域使用一个二维的受惩罚的样板回归,该指数被定义为共变体的线性预测。该方法用两种应用和模拟实验来说明。这项工作的独特贡献是专门为互动效应术语定义的共变(单指数)对称化。