Causal Inference for Complex Continuous-Time Longitudinal Studies

The existing causal inference frameworks for identifying causal effects for longitudinal studies typically assume that time advances in discrete time steps. However, medical studies nowadays with either irregular visit times or real-time monitoring have posed threats to the existing frameworks, rendering them invalid or to the very least, inefficient usage of the data. Therefore more general and advanced theory around causal inference for longitudinal data when confounders and treatments are measured continuously across time is needed. We develop a framework to identify causal effects under a user-specified treatment regime for continuous-time longitudinal studies. We provide sufficient identification assumptions including generalized consistency assumption, sequential randomization assumption, positivity assumption, and a novel ``achievable'' assumption designed for continuous time. Under these assumptions, we propose a g-computation process and an inverse probability weighting process, which suggest a g-computation formula and an inverse probability weighting formula for identification. For practical purposes, we also construct two classes of population estimating equations to identify these two processes, respectively, which further suggest a doubly robust formula that identifies causal effects under the user-specified treatment regime with extra robustness against process misspecification.