Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy

TL;DR

This paper develops statistical methods to estimate and infer the mean outcome under optimal treatment strategies, even when the optimal value is not uniquely defined, with theoretical guarantees and simulations.

Contribution

It establishes a necessary and sufficient condition for the differentiability of the optimal value and proposes methods for inference even when this condition is not met.

Findings

Provided a more general condition for pathwise differentiability.

Developed root-n confidence intervals for the optimal value.

Extended methods to multiple time point treatment problems.

Abstract

We consider challenges that arise in the estimation of the mean outcome under an optimal individualized treatment strategy defined as the treatment rule that maximizes the population mean outcome, where the candidate treatment rules are restricted to depend on baseline covariates. We prove a necessary and sufficient condition for the pathwise differentiability of the optimal value, a key condition needed to develop a regular and asymptotically linear (RAL) estimator of the optimal value. The stated condition is slightly more general than the previous condition implied in the literature. We then describe an approach to obtain root-$n$ rate confidence intervals for the optimal value even when the parameter is not pathwise differentiable. We provide conditions under which our estimator is RAL and asymptotically efficient when the mean outcome is pathwise differentiable. We also outline an extension of our approach to a multiple time point problem. All of our results are supported by simulations.