  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
accelerometer-based 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 relationship between PA intensity and BMI obtained from the National
Health and Nutrition Examination Survey data.
