Robust Learning for Optimal Treatment Decision with NP-Dimensionality

TL;DR

This paper introduces a robust two-step penalized regression method for optimal treatment decision-making in ultra-high dimensional data, addressing model misspecification and providing theoretical guarantees and empirical validation.

Contribution

It develops a novel two-step estimation procedure using non-concave penalized regressions for NP-dimensional data, with proven asymptotic properties and practical application.

Findings

Method achieves selection consistency and oracle properties.

Simulation results demonstrate superior performance.

Application to STAR*D data validates effectiveness.

Abstract

In order to identify important variables that are involved in making optimal treatment decision, Lu et al. (2013) proposed a penalized least squared regression framework for a fixed number of predictors, which is robust against the misspecification of the conditional mean model. Two problems arise: (i) in a world of explosively big data, effective methods are needed to handle ultra-high dimensional data set, for example, with the dimension of predictors is of the non-polynomial (NP) order of the sample size; (ii) both the propensity score and conditional mean models need to be estimated from data under NP dimensionality. In this paper, we propose a two-step estimation procedure for deriving the optimal treatment regime under NP dimensionality. In both steps, penalized regressions are employed with the non-concave penalty function, where the conditional mean model of the response given predictors may be misspecified. The asymptotic properties, such as weak oracle properties, selection consistency and oracle distributions, of the proposed estimators are investigated. In addition, we study the limiting distribution of the estimated value function for the obtained optimal treatment regime. The empirical performance of the proposed estimation method is evaluated by simulations and an application to a depression dataset from the STAR*D study.