A fully Bayesian semi-parametric scalar-on-function regression (SoFR) with measurement error using instrumental variables

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.