Nonparametric estimation of a covariate-adjusted counterfactual treatment regimen response curve

Flexible estimation of the mean outcome under a treatment regimen (i.e., value function) is the key step toward personalized medicine. We define our target parameter as a conditional value function given a set of baseline covariates which we refer to as a stratum based value function. We focus on semiparametric class of decision rules and propose a sieve based nonparametric covariate adjusted regimen-response curve estimator within that class. Our work contributes in several ways. First, we propose an inverse probability weighted nonparametrically efficient estimator of the smoothed regimen-response curve function. We show that asymptotic linearity is achieved when the nuisance functions are undersmoothed sufficiently. Asymptotic and finite sample criteria for undersmoothing are proposed. Second, using Gaussian process theory, we propose simultaneous confidence intervals for the smoothed regimen-response curve function. Third, we provide consistency and convergence rate for the optimizer of the regimen-response curve estimator; this enables us to estimate an optimal semiparametric rule. The latter is important as the optimizer corresponds with the optimal dynamic treatment regimen. Some finite-sample properties are explored with simulations.