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

[Abstract]Wearable devices such as the ActiGraph are now commonly used in research to monitor or track physical activity. This trend corresponds with the growing need to assess the relationships between physical activity and health outcomes, such as obesity, accurately. The device-based physical activity measures are best treated as functions when assessing their associations with scalar-valued outcomes such as body mass index. Scalar-on-function regression (SoFR) is a suitable regression model in this setting. Most estimation approaches in SoFR involve an assumption that the measurement error in functional covariates is white noise. Violating this assumption can lead to under-estimating model parameters. There are limited approaches to correcting measurement error for frequentist methods and none for Bayesian methods in this area. We present a fully non-parametric Bayesian measurement error-corrected SoFR model that relaxes all the constraining assumptions often involved with these models. Our estimation relies on an instrumental variable which is allowed to have a time-varying biasing factor, a significant departure from the current approach. Our method is easy to implement, and we demonstrate its finite sample properties in extensive simulations. Finally, we applied our method to data from the National Health and Examination Survey to assess the relationship between wearable device-based measures of physical activity and body mass index in adults in the United States.