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.

翻译：纵向研究中因果推断的主要障碍在于复杂的“治疗-混杂反馈”机制。传统纵向分析方法主要基于时间离散变化或治疗措施间断性改变的假设，将治疗-混杂反馈限制在可数范围内。现代医学研究中实时监测技术的应用产生了包含动态时变结局、治疗措施和混杂因素的函数型纵向数据，要求处理可能不可数无穷的治疗-混杂反馈。因此，迫切需要建立更精密完善的理论框架来应对这些复杂性。Ying（2024）近期提出了一个聚焦于研究终点结局的初步框架，初步解决了函数型纵向数据的因果推断问题。本文在其基础上进行四方面拓展：首先，系统综述现有文献，既深化了对基本概念的理解，又阐明了Ying（2024）研究与我们工作的理论渊源；其次，将Ying（2024）框架扩展至完全函数型时变结局过程，纳入具有重要实践意义的右删失和因死亡截断问题；第三，形式化Ying（2024）中先前非正式的命题，论证该框架如何以非参数方式扩展现有理论体系；最后，深入探讨了所提假设的可解释性与可行性，并规划了未来数值研究的实施策略。