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

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