Unlike the commonly used parametric regression models such as mixed models, that can easily violate the required statistical assumptions and result in invalid statistical inference, target maximum likelihood estimation allows more realistic data-generative models and provides double-robust, semi-parametric and efficient estimators. Target maximum likelihood estimators (TMLEs) for the causal effect of a community-level static exposure were previously proposed by Balzer et al. In this manuscript, we build on this work and present identifiability results and develop two semi-parametric efficient TMLEs for the estimation of the causal effect of the single time-point community-level stochastic intervention whose assignment mechanism can depend on measured and unmeasured environmental factors and its individual-level covariates. The first community-level TMLE is developed under a general hierarchical non-parametric structural equation model, which can incorporate pooled individual-level regressions for estimating the outcome mechanism. The second individual-level TMLE is developed under a restricted hierarchical model in which the additional assumption of no covariate interference within communities holds. The proposed TMLEs have several crucial advantages. First, both TMLEs can make use of individual level data in the hierarchical setting, and potentially reduce finite sample bias and improve estimator efficiency. Second, the stochastic intervention framework provides a natural way for defining and estimating casual effects where the exposure variables are continuous or discrete with multiple levels, or even cannot be directly intervened on. Also, the positivity assumption needed for our proposed causal parameters can be weaker than the version of positivity required for other casual parameters.
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