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.

翻译：功能数据在生物医学研究中日益普遍。尽管功能数据分析方法已发展数十年，针对功能治疗的因果推断研究仍基本处于空白。现有方法通常聚焦于估计因果平均剂量响应函数（ADRF），这需要较强的正性假设且可解释性有限。本研究针对一种新的因果估计量——修正功能治疗策略（MFTP），其核心在于估计当每个个体轻微调整其观测治疗轨迹时的平均潜在结果。该新估计量的主要挑战在于需要对无限维且无密度函数的对象定义平均值。通过提出基于功能主成分分析（FPCA）分解的功能变量总体平均新定义，我们建立了MFTP估计量的因果可识别性。进一步推导了针对MFTP的结果回归、逆概率加权及双重稳健估计量，并在温和正则条件下提供了理论保证。所提估计量通过大量模拟研究得到验证。将MFTP框架应用于美国国家健康与营养调查（NHANES）加速度计数据，我们估计了减少夜间干扰性活动与低活动持续时间对全因死亡率的因果效应。