For that purpose we propose a new online one-step estimator, which is proven to be asymptotically efficient under regularity conditions. This estimator takes as input online estimators of the relevant part of the data generating distribution and the nuisance parameter that are required for efficient estimation of the target parameter. These estimators could be an online stochastic gradient descent estimator based on large parametric models as developed in the current literature, but we also propose other online data adaptive estimators that do not rely on the specification of a particular parametric model.

We also present a targeted version of this online one-step estimator that presumably minimizes the one-step correction and thereby might be more robust in finite samples. These online one-step estimators are not a substitution estimator and might therefore be unstable for finite samples if the target parameter is borderline identifiable.

Therefore we also develop an online targeted minimum loss-based estimator, which updates the initial estimator of the relevant part of the data generating distribution by updating the current initial estimator with the new block of data, and estimates the target parameter with the corresponding plug-in estimator. The online substitution estimator is also proven to be asymptotically efficient under the same regularity conditions required for asymptotic normality of the online one-step estimator.

The online one-step estimator, targeted online one-step estimator, and online TMLE is demonstrated for estimation of a causal effect of a binary treatment on an outcome based on a dynamic data base that gets regularly updated, a common scenario for the analysis of electronic medical record data bases.

Finally, we extend these online estimators to a group sequential adaptive design in which certain components of the data generating experiment are continuously fine-tuned based on past data, and the new data generating distribution is then used to generate the next block of data.

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Subsemble is a flexible ensemble method that partitions a full data set into subsets of observations, fits the same algorithm on each subset, and uses a tailored form of V-fold cross-validation to construct a prediction function that combines the subset-specific fits with a second metalearner algorithm. Previous work studied the performance of Subsemble with subsets created randomly, and showed that these types of Subsembles often result in better prediction performance than the underlying algorithm fit just once on the full dataset. Since the final Subsemble estimator varies depending on the data used to create the subset-specific fits, different strategies for creating the subsets used in Subsemble result in different Subsembles. We propose supervised partitioning of the covariate space to create the subsets used in Subsemble, and using a form of histogram regression as the metalearner used to combine the subset-specific fits. We discuss applications to large-scale data sets, and develop a practical Supervised Subsemble method using regression trees to both create the covariate space partitioning, and select the number of subsets used in Subsemble. Through simulations and real data analysis, we show that this subset creation method can have better prediction performance than the random subset version.]]>

For the sake of presentation, we first consider the case that the treatment/censoring is only assigned at a single time-point, and subsequently, we cover the multiple time-point case. We characterize the optimal dynamic treatment as a statistical target parameter in the nonparametric statistical model, and we propose highly data adaptive estimators of this optimal dynamic regimen, utilizing sequential loss-based super-learning of sequentially defined (so called) blip-functions, based on newly proposed loss-functions. We also propose a cross-validation selector (among candidate estimators of the optimal dynamic regimens) based on a cross-validated targeted minimum loss-based estimator of the mean outcome under the candidate regimen, thereby aiming directly to select the candidate estimator that maximizes the mean outcome. We also establish that the mean of the counterfactual outcome under the optimal dynamic treatment is a pathwise differentiable parameter, and develop a targeted minimum loss-based estimator (TMLE) of this target parameter. We establish asymptotic linearity and statistical inference based on this targeted minimum loss-based estimator under specified conditions. In a sequentially randomized trial the statistical inference essentially only 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 targeted minimum loss based estimators and statistical inference for data adaptive target parameters that are defined in terms of the mean outcome under the {\em 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. For the sake of presentation, in the main part of the article we focus on two-time point interventions, but the results are generalized to general multiple time point interventions in the appendix.

]]>This paper presents a novel targeted maximum likelihood estimator (TMLE) estimator for the parameters of longitudinal static and dynamic marginal structural models.We consider a longitudinal data structure consisting of baseline covariates, time-dependent intervention nodes, intermediate time-dependent covariates, and a possibly time dependent outcome. The intervention nodes at each time point can include a binary treatment as well as a right-censoring indicator. Given a class of dynamic or static interventions, a marginal structural model is used to model the mean of the intervention specific counterfactual outcome as a function of the intervention and time point.Because the true shape of this function is rarely known, the marginal structural model is used as a working model. The causal quantity of interest is defined as the projection of the true function onto this working model. We introduce a new pooled TMLE for the parameters of such marginal structural working models, and compare this estimator to a recently proposed stratified TMLE that is based on estimating the intervention-specific mean separately for each intervention of interest. The performance of the pooled TMLE is compared to the performance of the stratified TMLE and the performance of inverse probability weighted estimators using simulations. Concepts are illustrated using an example in which the aim is to estimate the causal effect of delayed switch following immunological failure of first line antiretroviral therapy among HIV infected patients. Data from the International epidemiological Databases to Evaluate AIDS, Southern Africa are analyzed to investigate this question using both TMLE and IPW estimators. Our results demonstrate practical advantages over an IPW estimator for working marginal structural models for survival, as well as cases in which the pooled TMLE is superior to its stratified counterpart.]]>