Robust learning for optimal treatment decision with NP-dimensionality

Abstract

In order to identify important variables that are involved in making optimal treatment decision, Lu, Zhang and Zeng (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 robust procedure for estimating 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.

High-dimensional inference for personalized treatment decision

“…The proposed method can be extended to observation studies with high-dimensional covariates, where the propensity score model needs to be correctly specified from data. Related work on propensity score estimation can be found in [16]. However, the derivation of the limiting distribution of our desparsified estimator for the treatment-covariates interaction coefficients would be much more involved and requires further investigation.…”

“…We adopt the robust regression approach of [16] and transform the interaction from $A_i\left({\beta }_0^T{\tilde{X}}_i\right)$ to $\left(A_i-\pi \left(X_i\right)\right)\left({\beta }_0^T{\tilde{X}}_i\right)$, where π ( X i ) = E ( A i | X i ) is the propensity score. Because E ( A i − π | X i ) = 0, the transformed interaction is orthogonal to the baseline function μ ( X i ) given X i .…”

“…Because E ( A i − π | X i ) = 0, the transformed interaction is orthogonal to the baseline function μ ( X i ) given X i . By doing so, we can protect the estimation of β 0 from the effect of baseline function misspecification [8, 16]. For simplicity of presentation, we consider a completely randomized study with prespecified propensity score, i.e.…”

“…Existing methods can be broadly partitioned into regression-based or classification-based approaches. Popular regression-based approaches include Q-learning [21, 11, 3, 6, 7, 17] and A-learning [13, 10, 8, 16, 15]. Q-learning models the conditional mean of the outcome given covariates and treatment while A-learning directly models the interaction between treatment and covariates that is sufficient for treatment decisions.…”

Deep advantage learning for optimal dynamic treatment regime

“…Lu et al [2013] considered model selection for estimating optimal treatment regime via penalized least square. Shi et al [2016] extended Lu’s method to cases where the propensity score is unknown. They studied the theoretical properties of the proposed estimator given the number of covariates is of the non-polynomial (NP) order of the sample size.…”