Estimating causal effects of functional treatments with modified functional treatment policies

Functional data are increasingly prevalent in biomedical research. While functional data analysis has been established for decades, causal inference with functional treatments remains largely unexplored. Existing methods typically focus on estimating the causal average dose response functional (ADRF), which requires strong positivity assumptions and offers limited interpretability. In this work, we target a new causal estimand, the modified functional treatment policy (MFTP), which focuses on estimating the average potential outcome when each individual slightly modifies their treatment trajectory from the observed one. A major challenge for this new estimand is the need to define an average over an infinite-dimensional object with no density. By proposing a novel definition of the population average over a functional variable using a functional principal component analysis (FPCA) decomposition, we establish the causal identifiability of the MFTP estimand. We further derive outcome regression, inverse probability weighting, and doubly robust estimators for the MFTP, and provide theoretical guarantees under mild regularity conditions. The proposed estimators are validated through extensive simulation studies. Applying our MFTP framework to the National Health and Nutrition Examination Survey (NHANES) accelerometer data, we estimate the causal effects of reducing disruptive nighttime activity and low-activity duration on all-cause mortality.

💡 Research Summary

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This paper addresses the largely unexplored problem of causal inference when the treatment itself is a functional object, such as a high‑frequency activity curve recorded over a day. Existing work in this area has focused on estimating the average dose‑response functional (ADRF), defined as the expected potential outcome when every individual in the population receives the same treatment trajectory a(·). The ADRF approach suffers from three major drawbacks: (1) it requires a very strong positivity assumption—every possible trajectory must be feasible for every subject—which is unrealistic for infinite‑dimensional treatments; (2) the estimand is itself a function, making interpretation, visualization, and communication to clinicians difficult; and (3) many trajectories of interest are simply not observed in the data, rendering the ADRF scientifically irrelevant.

To overcome these limitations, the authors introduce a new causal estimand called the Modified Functional Treatment Policy (MFTP). Instead of forcing a common treatment on all units, the MFTP defines a deterministic modification rule q(X, A(·)) that maps each subject’s observed treatment curve A(·) and covariates X to a slightly altered curve. For example, a policy might halve nighttime activity for every individual, or increase overall activity by 10 % for those with low baseline levels. The causal target is then μ_q = E