Robust Learning for Optimal Treatment Decision with NP-Dimensionality
This work addresses the challenge of handling NP-dimensional data in causal inference for treatment regimes, which is incremental by extending existing penalized regression frameworks to ultra-high dimensions.
The paper tackles the problem of identifying important variables for optimal treatment decisions under ultra-high dimensional data with non-polynomial (NP) order dimensionality, proposing a two-step penalized regression method that is robust to model misspecification and demonstrates empirical performance through simulations and a depression dataset application.
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.