Targeted learning ensembles for optimal individualized treatment rules with time-to-event outcomes

Abstract

We consider estimation of an optimal individualized treatment rule from observational and randomized studies when a high-dimensional vector of baseline variables is available. Our optimality criterion is with respect to delaying expected time to occurrence of an event of interest (e.g., death or relapse of cancer). We leverage semiparametric efficiency theory to construct estimators with desirable properties such as double robustness. We propose two estimators of the optimal rule, which arise from considering two loss functions aimed at (i) directly estimating the conditional treatment effect (also know as the blip function), and (ii) recasting the problem as a weighted classification problem that uses the 0-1 loss function. Our estimated rules are super learning ensembles that minimize the cross-validated risk of a linear combination in a user-supplied library of candidate estimators. We prove oracle inequalities bounding the finite sample excess risk of the estimator. The bounds depend on the excess risk of the oracle selector and a doubly robust term related to estimation of the nuisance parameters. We discuss some important implications of these oracle inequalities such as the convergence rates of the value of our estimator to that of the oracle selector. We illustrate our methods in the analysis of a phase III randomized study testing the efficacy of a new therapy for the treatment of breast cancer.

SurvITE: Learning Heterogeneous Treatment Effects from Time-to-Event Data

“…[58] performed a simulation study using different survival prediction models as base-learners for a twomodel approach to estimating the difference in median survival time. Based on ideas from the semi-parametric efficiency literature, [26] and [59] propose estimators that target the (restricted) mean survival time directly and consequently do not output estimates of the treatment-specific hazard or survival functions. We consider the ability to output treatment-specific predictions an important feature of a model if the goal is to use model output to give decision support, given that it allows the decision-maker to trade-off relative improvement with the baseline risk of a patient.…”

“…As in e.g. [59], we let T a denote the potential event time that would have been observed had treatment a been assigned, and C = t max been externally set. Then, the following assumptions are implied by the DAG: Then, we can write…”

“…Assumptions 1.a, 1.c and 2.a are standard within the treatment effect estimation literature [7,9]; assumptions 1.b and 2.b are standard within the literature with survival outcomes [59,26]. Assumption 2.c is needed only if we aim to estimate λ a (t|x) for all t, otherwise it would suffice to follow a convention such as setting λ a (t|x) = 1 whenever P(N T (τ −1) = 0|A = a, X = x) = 0.…”

Optimizing Precision and Power by Machine Learning in Randomized Trials, with an Application to COVID-19

“…Correction for this first order bias will allow us to establish normality of the estimators. Specifically, for any estimate η we have the following first order expansion around the true parameter value θ(η), proved in Lemma 1 in the Supplementary materials of Díaz et al (2018):…”

Resampling‐based confidence intervals for model‐free robust inference on optimal treatment regimes

“…Applications in medicine, public policy, internet marketing, and other scientific areas often require estimating an individualized treatment rule (or regime, policy) to maximize the potential benefit. Several successful methods have been developed for estimating an optimal treatment regime, including Q‐learning (Watkins and Dayan, 1992; Murphy, 2005b; Chakraborty et al ., 2010; Qian and Murphy, 2011; Song et al ., 2015), A‐learning (Robins et al ., 2000; Murphy, 2003, 2005a; Moodie and Richardson, 2010; Shi et al ., 2018), model‐free methods (Robins et al ., 2008; Orellana et al ., 2010; Zhang et al ., 2012; Zhao et al ., 2012, 2015; Athey and Wager, 2017; Linn et al ., 2017; Zhou et al ., 2017; Zhu et al ., 2017; Lou et al ., 2018; Qi et al ., 2018; Wang et al ., 2018), tree or list‐based methods (Laber and Zhao, 2015; Cui et al ., 2017; Zhu et al ., 2017; Zhang et al ., 2018), targeted learning ensembles approach (Díaz et al ., 2018), among others.…”