  Wearable devices such as the ActiGraph are now commonly used in health
studies to monitor or track physical activity. This trend aligns well with the
growing need to accurately assess the effects of physical activity on health
outcomes such as obesity. When accessing the association between these
device-based physical activity measures with health outcomes such as body mass
index, the device-based data is considered functions, while the outcome is a
scalar-valued. The regression model applied in these settings is the
scalar-on-function regression (SoFR). Most estimation approaches in SoFR assume
that the functional covariates are precisely observed, or the measurement
errors are considered random errors. Violation of this assumption can lead to
both under-estimation of the model parameters and sub-optimal analysis. The
literature on a measurement corrected approach in SoFR is sparse in the
non-Bayesian literature and virtually non-existent in the Bayesian literature.
This paper considers a fully nonparametric Bayesian measurement error corrected
SoFR model that relaxes all the constraining assumptions often made in these
models. Our estimation relies on an instrumental variable (IV) to identify the
measurement error model. Finally, we introduce an IV quality scalar parameter
that is jointly estimated along with all model parameters. Our method is easy
to implement, and we demonstrate its finite sample properties through an
extensive simulation. Finally, the developed methods are applied to the
National Health and Examination Survey to assess the relationship between
wearable-device-based measures of physical activity and body mass index among
adults living in the United States.
