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Unlike the commonly used parametric regression models such as mixed models,that can easily violate the required statistical assumptions and result ininvalid statistical inference, target maximum likelihood estimation allows morerealistic data-generative models and provides double-robust, semi-parametricand efficient estimators. Target maximum likelihood estimators (TMLEs) for thecausal effect of a community-level static exposure were previously proposed byBalzer et al. In this manuscript, we build on this work and presentidentifiability results and develop two semi-parametric efficient TMLEs for theestimation of the causal effect of the single time-point community-levelstochastic intervention whose assignment mechanism can depend on measured andunmeasured environmental factors and its individual-level covariates. The firstcommunity-level TMLE is developed under a general hierarchical non-parametricstructural equation model, which can incorporate pooled individual-levelregressions for estimating the outcome mechanism. The second individual-levelTMLE is developed under a restricted hierarchical model in which the additionalassumption of no covariate interference within communities holds. The proposedTMLEs have several crucial advantages. First, both TMLEs can make use ofindividual level data in the hierarchical setting, and potentially reducefinite sample bias and improve estimator efficiency. Second, the stochasticintervention framework provides a natural way for defining and estimatingcasual effects where the exposure variables are continuous or discrete withmultiple levels, or even cannot be directly intervened on. Also, the positivityassumption needed for our proposed causal parameters can be weaker than theversion of positivity required for other casual parameters.

Targeted Maximum Likelihood Estimation of Community-based Causal Effect of Community-Level Stochastic Interventions