Targeted Learning of the Mean Outcome under an Optimal Dynamic Treatment Rule

Abstract

We consider estimation of and inference for the mean outcome under the optimal dynamic two time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, where the candidate rules are restricted to depend only on a user-supplied subset of the baseline and intermediate covariates. This estimation problem is addressed in a statistical model for the data distribution that is nonparametric beyond possible knowledge about the treatment and censoring mechanism. This contrasts from the current literature that relies on parametric assumptions. We establish that the mean of the counterfactual outcome under the optimal dynamic treatment is a pathwise differentiable parameter under conditions, and develop a targeted minimum loss-based estimator (TMLE) of this target parameter. We establish asymptotic linearity and statistical inference for this estimator under specified conditions. In a sequentially randomized trial the statistical inference relies upon a second-order difference between the estimator of the optimal dynamic treatment and the optimal dynamic treatment to be asymptotically negligible, which may be a problematic condition when the rule is based on multivariate time-dependent covariates. To avoid this condition, we also develop TMLEs and statistical inference for data adaptive target parameters that are defined in terms of the mean outcome under the estimate of the optimal dynamic treatment. In particular, we develop a novel cross-validated TMLE approach that provides asymptotic inference under minimal conditions, avoiding the need for any empirical process conditions. We offer simulation results to support our theoretical findings.

Corrigendum to: Targeted Learning of the Mean Outcome under an Optimal Dynamic Treatment Rule [J Causal Inference DOI: 10.1515/jci-2013-0022]

“…These figures show that the coverage of the non-cross-validated methods improves considerably this candidate is omitted, while the coverage of the cross-validated methods is unchanged. This is in line with the general theoretical arguments given in van der Laan and Luedtke [1] that the cross-validated approaches make no restrictions on the data adaptivity of the initial parameters, i.e. do not require any entropy conditions on the optimal rule estimator.…”

“…There was a coding error in the simulations of van der Laan and Luedtke [1]. This error has been corrected and the simulation section has been updated to reflect this change.…”

“…We have included amended versions of all figures from van der Laan and Luedtke [1] in this errata (Figures 1-3). In Figure 2(b) we have included results from the same ensemble estimator of the optimal rule, and also the result when the recursive partitioning candidate is omitted from the library of candidate estimators.…”

Optimal treatment regimes for survival endpoints using a locally-efficient doubly-robust estimator from a classification perspective

“…That is when the set [ x : E { f ( T )| A = 1, X = x } − E { f ( T )| A = 0, X = x } = 0] has probability zero. Such a result was demonstrated by van der Laan & Luedtke (2014).…”

Discussion of Identification, Estimation and Approximation of Risk under Interventions that Depend on the Natural Value of Treatment Using Observational Data, by Jessica Young, Miguel Hernán, and James Robins

“…For example, in van der Laan (2013), van der Laan and Luedtke (2014b), we developed such estimators and such confidence intervals for the mean outcome under an estimate d n of the optimal dynamic treatment d 0 . In this case, one can show that E 0 Y d n - E 0 Y d 0 is a second-order term so that one might assume that it is $O_P(1∕\sqrt{n})$.…”