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

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.Comment: Published at http://dx.doi.org/10.1214/15-AOS1384 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evaluating the Impact of Treating the Optimal Subgroup

Suppose we have a binary treatment used to influence an outcome. Given data from an observational or controlled study, we wish to determine whether or not there exists some subset of observed covariates in which the treatment is more effective than the standard practice of no treatment. Furthermore, we wish to quantify the improvement in population mean outcome that will be seen if this subgroup receives treatment and the rest of the population remains untreated. We show that this problem is surprisingly challenging given how often it is an (at least implicit) study objective. Blindly applying standard techniques fails to yield any apparent asymptotic results, while using existing techniques to confront the non-regularity does not necessarily help at distributions where there is no treatment effect. Here we describe an approach to estimate the impact of treating the subgroup which benefits from treatment that is valid in a nonparametric model and is able to deal with the case where there is no treatment effect. The approach is a slight modification of an approach that recently appeared in the individualized medicine literature

Noise at a Fermi-edge singularity

We present noise measurements of self-assembled InAs quantum dots at high magnetic fields. In comparison to I-V characteristics at zero magnetic field we notice a strong current overshoot which is due to a Fermi-edge singularity. We observe an enhanced suppression in the shot noise power simultaneous to the current overshoot which is attributed to the electron-electron interaction in the Fermi-edge singularity

Tunable graphene system with two decoupled monolayers

The use of two truly two-dimensional gapless semiconductors, monolayer and bilayer graphene, as current-carrying components in field-effect transistors (FET) gives access to new types of nanoelectronic devices. Here, we report on the development of graphene-based FETs containing two decoupled graphene monolayers manufactured from a single one folded during the exfoliation process. The transport characteristics of these newly-developed devices differ markedly from those manufactured from a single-crystal bilayer. By analyzing Shubnikov-de Haas oscillations, we demonstrate the possibility to independently control the carrier densities in both layers using top and bottom gates, despite there being only a nanometer scale separation between them

Super-Learning of an Optimal Dynamic Treatment Rule

We consider the estimation of an optimal dynamic two time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, where the candidate rules are restricted to depend only on a user-supplied subset of the baseline and intermediate covariates. This estimation problem is addressed in a statistical model for the data distribution that is nonparametric, beyond possible knowledge about the treatment and censoring mechanisms. We propose data adaptive estimators of this optimal dynamic regime which are defined by sequential loss-based learning under both the blip function and weighted classification frameworks. Rather than \textit{a priori} selecting an estimation framework and algorithm, we propose combining estimators from both frameworks using a super-learning based cross-validation selector that seeks to minimize an appropriate cross-validated risk. One of the proposed risks directly measures the performance of the mean outcome under the optimal rule. The resulting selector is guaranteed to asymptotically perform as well as the best convex combination of candidate algorithms in terms of loss-based dissimilarity under conditions. We offer simulation results to support our theoretical findings. This work expands upon that of an earlier technical report (van der Laan, 2013) with new results and simulations, and is accompanied by a work which develops inference for the mean outcome under the optimal rule (van der Laan and Luedtke, 2014)