Dynamic Treatment Effects under Functional Longitudinal Studies

Abstract: Establishing causality is a fundamental goal in fields like medicine and social sciences. While randomized controlled trials are the gold standard for causal inference, they are not always feasible or ethical. Observational studies can serve as alternatives but introduce confounding biases, particularly in complex longitudinal data, where treatment-confounder feedback complicates analysis. The challenge increases with Dynamic Treatment Regimes (DTRs), where treatment allocation depends on rich historical patient data. The advent of real-time healthcare monitoring technologies, such as MIMIC-IV and Continuous Glucose Monitoring (CGM), has popularized Functional Longitudinal Data (FLD). However, there is yet no investigate of causal inference for FLD with DTRs. In this paper, we address it by developing a population-level framework for functional longitudinal data, accommodating DTRs. To that end, we define the potential outcomes and causal effects of interest. We then develop identification assumptions, and derive g-computation, inverse probability weighting, and doubly robust formulas through novel applications of stochastic process and measure theory. We further show that our framework is nonparametric and compute the efficient influence curve using semiparametric theory. Last, we illustrate our framework's potential through Monte Carlo simulations.