Functional Causal Mediation Analysis with Zero-inflated Count Data

Abstract

Causal mediation analysis decomposes the effect of an exposure on an outcome into a component operating directly and a component operating through an intermediate variable. In many biomedical and behavioural studies, the mediator evolves over time and is naturally represented as a function, while the outcome is a count with excess zeros. These features create methodological challenges for identification, estimation, and robustness assessment, particularly when mediator processes are sparsely and irregularly observed, and the outcome model is nonlinear due to zero inflation. This thesis develops a functional causal mediation framework for zero-inflated count outcomes, together with estimation procedures and robustness tools tailored to the functional setting.

Chapter 2 lays the conceptual and methodological foundations for functional mediation with zero-inflated counts. Within the potential outcomes framework, it defines the total effect, natural direct and indirect effects when the mediator is a time-varying process observed sparsely and irregularly, and clarifies the identifying assumptions required for natural-effect decompositions. The chapter develops a simulation-based implementation of the mediation formula under function-on-scalar regression models for the mediator and functional zero-inflated count models for the outcome. The proposed framework is illustrated in a MIMIC-IV application among patients undergoing coronary artery bypass grafting, where the use of sex as a non-manipulable treatment is discussed, and a principal stratification framework is adopted to handle post-treatment selection into the surgical cohort. Postoperative processes of physiological measurements are studied as candidate mechanisms relating sex to subsequent rehospitalisation outcomes within the always-CABG principal stratum.

Chapter 3 addresses computational and inferential limitations of simulation-based mediation implementations by developing a marginal functional mediation approach built on the functional marginalised zero-inflated Poisson model. By modelling the marginal mean of the zero-inflated count outcome directly, the resulting framework yields closed-form expressions for total, natural direct, and natural indirect effects on both incidence-rate-ratio and risk-difference scales. This analytic structure supports fast computation, transparent decomposition of time-local contributions through inner-product representations, and delta-method standard error estimation. Extensive simulation studies across multiple sample sizes and mediator complexities demonstrate that the proposed estimators exhibit good performance in moderate to large samples. The methodology is applied to the Wisconsin Smokers' Health Study 2, evaluating whether time-varying craving during the first two weeks after cessation mediates the effect of varenicline versus nicotine patch on subsequent cigarette use, which is highly concentrated at zero.

Chapter 4 revisits causal interpretation and robustness when natural-effect identification assumptions are vulnerable in realistic observational settings. In particular, it addresses concerns about unmeasured baseline mediator--outcome confounding, and exposure-induced mediator--outcome confounding. To assess the robustness of natural effect estimates under potential unmeasured baseline mediator--outcome confounding, we explore both a simulation-based Gaussian copula approach and a closed-form characterisation approach to sensitivity analysis. As natural effects are generally unidentifiable in the presence of post-treatment confounding, the chapter further develops interventional estimands that avoid cross-world potential outcomes and are more directly connected to policy-relevant mediator-distribution interventions. These ideas are illustrated through the MIMIC-IV CABG analysis, providing a robustness-oriented perspective on conclusions drawn from functional mediators and zero-inflated rehospitalisation outcomes.

Collectively, the thesis contributes a set of causal estimands, modelling tools, and inferential procedures for mediation problems in which mediators evolve over time and outcomes display zero inflation, and it demonstrates their practical value through simulation evidence and applications to large-scale electronic health records and intensive longitudinal trial data.