A Bayesian semi‐parametric scalar‐on‐function regression 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. 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 assume that the measurement error in functional covariates is white noise. Violating this assumption can lead to underestimating model parameters. There are limited approaches to correcting measurement errors for frequentist methods and none for Bayesian methods in this area. We present a 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 allowing a time‐varying biasing factor, a significant departure from the current generalized method of moment (GMM) approach. Our proposed method also permits model‐based grouping of the functional covariate following measurement error correction. This grouping of the measurement error‐corrected functional covariate allows additional ease of interpretation of how the different groups differ. 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.