Estimating Optimal Treatment Rules with an Instrumental Variable: A Semi-Supervised Learning Approach

Individualized treatment rules (ITRs) are regarded as a promising recipe to deliver better policy interventions. One key ingredient in optimal ITR estimation problems is to estimate average treatment effect conditional on a subject's covariate information, which is often challenging in observational studies due to the universal concern of unmeasured confounding. Instrumental variables (IVs) are widely-used tools to infer treatment effect when there is unmeasured confounding between the treatment and outcome. In this work, we propose a general framework to approach the ITR estimation problem with a valid IV. Just as Zhang et al. (2012) and Zhao et al. (2012) cast the ITR estimation problem in non-IV settings into a supervised classification problem, we recast the ITR estimation problem with a valid IV into a semi-supervised classification problem consisting of a supervised and an unsupervised part. The unsupervised part stems naturally from the partial identification nature of an IV in terms of identifying the treatment effect. We define a new notion of optimality called "IV-optimality". A treatment rule is said to be IV-optimal if it minimizes the maximum risk with respect to the putative IV and the set of IV identification assumptions. We propose a statistical learning method that estimates such an IV-optimal rule, design computationally-efficient algorithms, and prove theoretical guarantees. We apply our method to studying which moms would better deliver their premature babies at hospitals with high-level neonatal intensive care units (NICUs).