Mediation analysis with densities as mediators with an application to iCOMPARE trial

Physical activity has long been shown to be associated with biological and physiological performance and risk of diseases. It is of great interest to assess whether the effect of an exposure or intervention on an outcome is mediated through physical activity measured by modern wearable devices such as actigraphy. However, existing methods for mediation analysis focus almost exclusively on mediation variable that is in the Euclidean space, which cannot be applied directly to the actigraphy data of physical activity. Such data is best summarized in the form of an histogram or density. In this paper, we extend the structural equation models (SEMs) to the settings where a density is treated as the mediator to study the indirect mediation effect of physical activity on an outcome. We provide sufficient conditions for identifying the average causal effects of density mediator and present methods for estimating the direct and mediating effects of density on an outcome. We apply our method to the data set from the iCOMPARE trial that compares flexible duty-hour policies and standard duty-hour policies on interns' sleep related outcomes to explore the mediation effect of physical activity on the causal path between flexible duty-hour policies and sleep related outcomes.