  Functional data is a powerful tool for capturing and analyzing complex
patterns and relationships in a variety of fields, allowing for more precise
modeling, visualization, and decision-making. For example, in healthcare,
functional data such as medical images can help doctors make more accurate
diagnoses and develop more effective treatment plans. However, understanding
the causal relationships between functional predictors and time-to-event
outcomes remains a challenge. To address this, we propose a functional causal
framework including a functional accelerated failure time (FAFT) model and
three causal approaches. The regression adjustment approach is based on
conditional FAFT with subsequent confounding marginalization, while the
functional-inverse-probability-weighting approach is based on marginal FAFT
with well-defined functional propensity scores. The double robust approach
combines the strengths of both methods and achieves a balance condition through
the weighted residuals between imputed observations and regression adjustment
outcomes. Our approach can accurately estimate causality, predict outcomes, and
is robust to different censoring rates. We demonstrate the power of our
framework with simulations and real-world data from the Alzheimer's Disease
Neuroimaging Initiative (ADNI) study. Our findings provide more precise
subregions of the hippocampus that align with medical research, highlighting
the power of this work for improving healthcare outcomes.
