  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.
