CAPITAL: Optimal subgroup identification via constrained policy tree search

Abstract

Personalized medicine, a paradigm of medicine tailored to a patient's characteristics, is an increasingly attractive field in health care. An important goal of personalized medicine is to identify a subgroup of patients, based on baseline covariates, that benefits more from the targeted treatment than other comparative treatments. Most of the current subgroup identification methods only focus on obtaining a subgroup with an enhanced treatment effect without paying attention to subgroup size. Yet, a clinically meaningful subgroup learning approach should identify the maximum number of patients who can benefit from the better treatment. In this article, we present an optimal subgroup selection rule (SSR) that maximizes the number of selected patients, and in the meantime, achieves the pre-specified clinically meaningful mean outcome, such as the average treatment effect. We derive two equivalent theoretical forms of the optimal SSR based on the contrast function that describes the treatment-covariates interaction in the outcome. We further propose a constrained policy tree search algorithm (CAPITAL) to find the optimal SSR within the interpretable decision tree class. The proposed method is flexible to handle multiple constraints that penalize the inclusion of patients with negative treatment effects, and to address time to event data using the restricted mean survival time as the clinically interesting mean outcome. Extensive simulations, comparison studies, and real data applications are conducted to demonstrate the validity and utility of our method.

Keywords: constrained policy tree search; optimal subgroup identification; personalized medicine.

FIGURE A1

Illustration of the density function of the contrast function $C(X)$ with a cut point $\eta$ for the prespecified threshold $\delta$

FIGURE A2

Illustration of a simple $L=2$ decision tree with splitting variables $X^{(1)}$ and $X^{(2)}$

FIGURE A3

The estimated optimal subgroup selection tree by CAPITAL under Scenario 2 with $\delta =1.0$ and $n=1000$. Upper left panel: for Replicate No.1. Upper right panel: for Replicate No.2. Lower middle Panel: for Replicate No.3

FIGURE A4

The density function of $C(X)$ within or outside the subgroup under Scenario 2 with $\delta =1.0$ and $n=1000$. Left panel: for Replicate No.1. Middle panel: for Replicate No.2. Right Panel: for Replicate No.3

FIGURE A5

The estimated optimal subgroup selection tree using CAPITAL under the ACTG 175 data. Left panel: for $\delta =0.35$. Right panel: for $\delta =0.40$

FIGURE A6

The estimated optimal subgroup selection tree using CAPITAL under the hematological malignancies data. Left panel: for $\delta =84$. Right panel: for $\delta =108$