Optimal Individualized Treatments in Resource-Limited Settings

Abstract

An individualized treatment rule (ITR) is a treatment rule which assigns treatments to individuals based on (a subset of) their measured covariates. An optimal ITR is the ITR which maximizes the population mean outcome. Previous works in this area have assumed that treatment is an unlimited resource so that the entire population can be treated if this strategy maximizes the population mean outcome. We consider optimal ITRs in settings where the treatment resource is limited so that there is a maximum proportion of the population which can be treated. We give a general closed-form expression for an optimal stochastic ITR in this resource-limited setting, and a closed-form expression for the optimal deterministic ITR under an additional assumption. We also present an estimator of the mean outcome under the optimal stochastic ITR in a large semiparametric model that at most places restrictions on the probability of treatment assignment given covariates. We give conditions under which our estimator is efficient among all regular and asymptotically linear estimators. All of our results are supported by simulations.

Double/debiased machine learning for treatment and structural parameters

“…In subsequent work, van der Laan and Rose () emphasize the use of ML methods to estimate the nuisance parameters for use with the super learner. Much of this work, including recent work such as Luedtke and van der Laan (), Toth and van der Laan () and Zheng et al. (), focuses on formal results under a Donsker condition, though the use of sample splitting to relax these conditions has also been advocated in the targeted maximum likelihood setting, as discussed below.…”

Evaluating the impact of treating the optimal subgroup

“…contains at least 10% of the population. One can show via a change of variables that estimating the mean outcome under such a constrained subgroup is equivalent to estimating the mean outcome under an optimal rule which can treat at most 90% of the population, see [31]. Estimating this alternative constrained parameter is still difficult when the optimal subgroup is nonunique, though there is little risk of degenerate first-order behavior in this case.…”

“…Estimating this alternative constrained parameter is still difficult when the optimal subgroup is nonunique, though there is little risk of degenerate first-order behavior in this case. To construct confidence intervals despite the non-uniqueness of the optimal subgroup, one can combine the results in [31] with the stabilized one-step estimator presented in [17].…”

Interactive Q -Learning for Quantiles

“…A parsimonious, parametric model for c ( H 2 ) trades a restriction on the class of possible regimes for potential gains in interpretability. Note that approaches such as OWL (Zhao et al, 2012) and the methods described in Luedtke and van der Laan (2015) do not require models for m ( H 2 ), while double robust approaches model this term only to increase efficiency and remain consistent for the optimal regime even when the model is misspecified.…”

“…Direct-search and regression-based estimators have been extended to handle survival outcomes (Goldberg and Kosorok, 2012; Huang and Ning, 2012; Huang et al, 2014), high-dimensional data (McKeague and Qian, 2013), missing data (Shortreed et al, 2014), multiple outcomes (Laber et al, 2014b; Linn et al, 2015), and restrictions on treatment resource (Luedtke and van der Laan, 2015). …”