## Abstract

Suppose that we observe a population of causally connected units. On each unit at each time-point on a grid we observe a set of other units the unit is potentially connected with, and a unit-specific longitudinal data structure consisting of baseline and time-dependent covariates, a time-dependent treatment, and a final outcome of interest. The target quantity of interest is defined as the mean outcome for this group of units if the exposures of the units would be probabilistically assigned according to a known specified mechanism, where the latter is called a stochastic intervention. Causal effects of interest are defined as contrasts of the mean of the unit-specific outcomes under different stochastic interventions one wishes to evaluate. This covers a large range of estimation problems from independent units, independent clusters of units, and a single cluster of units in which each unit has a limited number of connections to other units. The allowed dependence includes treatment allocation in response to data on multiple units and so called causal interference as special cases. We present a few motivating classes of examples, propose a structural causal model, define the desired causal quantities, address the identification of these quantities from the observed data, and define maximum likelihood based estimators based on cross-validation. In particular, we present maximum likelihood based super-learning for this network data. Nonetheless, such smoothed/regularized maximum likelihood estimators are not targeted and will thereby be overly bias w.r.t. the target parameter, and, as a consequence, generally not result in asymptotically normally distributed estimators of the statistical target parameter.

To formally develop estimation theory, we focus on the simpler case in which the longitudinal data structure is a point-treatment data structure. We formulate a novel targeted maximum likelihood estimator of this estimand and show that the double robustness of the efficient influence curve implies that the bias of the targeted minimum loss-based estimation (TMLE) will be a second-order term involving squared differences of two nuisance parameters. In particular, the TMLE will be consistent if either one of these nuisance parameters is consistently estimated. Due to the causal dependencies between units, the data set may correspond with the realization of a single experiment, so that establishing a (e.g. normal) limit distribution for the targeted maximum likelihood estimators, and corresponding statistical inference, is a challenging topic. We prove two formal theorems establishing the asymptotic normality using advances in weak-convergence theory. We conclude with a discussion and refer to an accompanying technical report for extensions to general longitudinal data structures.

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