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

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.