Causality for Complex Continuous-time Functional Longitudinal Studies

The paramount obstacle in longitudinal studies for causal inference is the complex "treatment-confounder feedback." Traditional methodologies for elucidating causal effects in longitudinal analyses are primarily based on the assumption that time moves in specific intervals or that changes in treatment occur discretely. This conventional view confines treatment-confounder feedback to a limited, countable scope. The advent of real-time monitoring in modern medical research introduces functional longitudinal data with dynamically time-varying outcomes, treatments, and confounders, necessitating dealing with a potentially uncountably infinite treatment-confounder feedback. Thus, there is an urgent need for a more elaborate and refined theoretical framework to navigate these intricacies. Recently, Ying (2024) proposed a preliminary framework focusing on end-of-study outcomes and addressing the causality in functional longitudinal data. Our paper expands significantly upon his foundation in fourfold: First, we conduct a comprehensive review of existing literature, which not only fosters a deeper understanding of the underlying concepts but also illuminates the genesis of both Ying (2024)'s and ours. Second, we extend Ying (2024) to fully embrace a functional time-varying outcome process, incorporating right censoring and truncation by death, which are both significant and practical concerns. Third, we formalize previously informal propositions in Ying (2024), demonstrating how this framework broadens the existing frameworks in a nonparametric manner. Lastly, we delve into a detailed discussion on the interpretability and feasibility of our assumptions, and outlining a strategy for future numerical studies.