Causality of Functional Longitudinal Data

"Treatment-confounder feedback" is the central complication to resolve in longitudinal studies, to infer causality. The existing frameworks for identifying causal effects for longitudinal studies with discrete repeated measures hinge heavily on assuming that time advances in discrete time steps or treatment changes as a jumping process, rendering the number of "feedbacks" finite. However, medical studies nowadays with real-time monitoring involve functional time-varying outcomes, treatment, and confounders, which leads to an uncountably infinite number of feedbacks between treatment and confounders. Therefore more general and advanced theory is needed. We generalize the definition of causal effects under user-specified stochastic treatment regimes to longitudinal studies with continuous monitoring and develop an identification framework, allowing right censoring and truncation by death. We provide sufficient identification assumptions including a generalized consistency assumption, a sequential randomization assumption, a positivity assumption, and a novel "intervenable" assumption designed for the continuous-time case. Under these assumptions, we propose a g-computation process and an inverse probability weighting process, which suggest a g-computation formula and an inverse probability weighting formula for identification. For practical purposes, we also construct two classes of population estimating equations to identify these two processes, respectively, which further suggest a doubly robust identification formula with extra robustness against process misspecification. We prove that our framework fully generalize the existing frameworks and is nonparametric.