Background: Obesity results from a chronic imbalance between energy intake and energy expenditure. Total energy expenditure for all physiological functions combined can be measured approximately by calorimeters. These devices assess energy expenditure frequently (e.g., in 60-second epochs), resulting in massive complex data that are nonlinear functions of time. To reduce the prevalence of obesity, researchers often design targeted therapeutic interventions to increase daily energy expenditure. Methods: We analyzed previously collected data on the effects of oral interferon tau supplementation on energy expenditure, as assessed with indirect calorimeters, in an animal model for obesity and type 2 diabetes (Zucker diabetic fatty rats). In our statistical analyses, we compared parametric polynomial mixed effects models and more flexible semiparametric models involving spline regression. Results: We found no effect of interferon tau dose (0 vs. 4 µg/kg body weight/day) on energy expenditure. The B-spline semiparametric model of untransformed energy expenditure with a quadratic term for time performed best in terms of the Akaike information criterion value. Conclusions: To analyze the effects of interventions on energy expenditure assessed with devices that collect data at frequent intervals, we recommend first summarizing the high dimensional data into epochs of 30 to 60 minutes to reduce noise. We also recommend flexible modeling approaches to account for the nonlinear patterns in such high dimensional functional data. We provide freely available R codes in GitHub.

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 1

Physical activity (PA) is an important risk factor for many health outcomes.Wearable-devices such as accelerometers are increasingly used in biomedical studies to understand the associations between PA and health outcomes. Statistical analyses involving accelerometer data are challenging due to the following three characteristics: (i) high-dimensionality, (ii) temporal dependence, and (iii) measurement error. To address these challenges we treat accelerometerbased measures of physical activity as a single function-valued covariate prone to measurement error. Specifically, in order to determine the relationship between PA and a health outcome of interest, we propose a regression model with a functional covariate that accounts for measurement error. Using regression calibration, we develop a two-step estimation method for the model parameters and establish their consistency. A test is also proposed to test the significance of the estimated model parameters. Simulation studies are conducted to compare the proposed methods with existing alternative approaches under varying scenarios. Finally, the developed methods are used to assess the 1