We consider challenges that arise in the estimation of the value of 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 asymptotically linear (RAL) estimator of this parameter. 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. In particular, we develop an estimator that, when properly standardized, converges to a normal limiting distribution. We provide conditions under which our estimator is RAL and asymptotically efficient when the mean outcome is pathwise differentiable. We outline an extension of our approach to a multiple time point problem in the appendix. All of our results are supported by simulations.
Biostatistics | Statistical Methodology | Statistical Theory
Luedtke, Alexander R. and van der Laan, Mark J., "Statistical Inference for the Mean Outcome Under a Possibly Non-Unique Optimal Treatment Strategy" (December 2014). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 332.