Targeted maximum likelihood estimation for mediation analysis with multiple time-varying mediators

Understanding how an exposure influences an outcome through mediators is essential in medical and epidemiological research, especially when mediators vary over time and influence each other reciprocally. This complex condition, termed causally ordered multiple time-varying mediation, frequently appears in chronic diseases. For instance, in chronic obstructive pulmonary disease, low lung capacity initiates a vicious cycle where dyspnea and physical inactivity reinforce each other, progressively worsening patients' quality of life. However, existing mediation methods often focus on single time-varying mediators or fail to fully decompose the total effect (TE), making them inadequate for capturing such feedback dynamics. To address these limitations, we propose a novel framework that decomposes the TE into path-specific effects (PSEs) for each mediator, offering a precise and clinically relevant analysis. Our approach ensures that the sum of these PSEs equals the TE, resolving interpretative issues in prior methods. For estimation, we derive efficient influence function and employ targeted maximum likelihood estimation, which combines flexibility with strong statistical properties, including multiple robustness, asymptotic normality, and efficiency. Our framework offers powerful solutions for analyzing complex mediation mechanisms in longitudinal data, with substantial applications in clinical research.