Learning causal effect of physical activity distribution: an application of functional treatment effect estimation with unmeasured confounding

The National Health and Nutrition Examination Survey (NHANES) collects minute-level physical activity data by accelerometers as an important component of the survey to assess the health and nutritional status of adults and children in the US. In this paper, we analyze the NHANES accelerometry data to study the causal effect of physical activity distribution on body fat percentage, where the treatment is a function/distribution. In the presence of unmeasured confounding, we propose to integrate cross-fitting with two methods under the proximal causal inference framework to estimate the functional treatment effect. The two methods are shown practically appealing via both simulation and an NHANES accelerometry data analysis. In the analysis of the NHANES accelerometry data, the two methods also lead to a more intuitive and interpretable causal relationship between physical activity distribution and body fat percentage.