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.
摘要：
诸如ActiGraph之类的可穿戴设备现在在研究中通常用于监视或跟踪身体活动。这一趋势与评估身体活动和健康结果之间关系的日益增长的需求相对应。比如肥胖，准确。基于设备的身体活动量度在评估其与标量值结果（例如体重指数）的关联时，最好将其视为函数。在此设置中,函数上标量回归(SoFR)是合适的回归模型。SoFR中的大多数估计方法都假设功能协变量中的测量误差是白噪声。违反这一假设会导致低估模型参数。在这一领域，用于纠正频率方法的测量误差的方法有限，而对于贝叶斯方法则没有。我们提出了一种非参数贝叶斯测量误差校正的SoFR模型，该模型放松了这些模型经常涉及的所有约束假设。我们的估计依赖于允许时变偏置因子的工具变量，与当前的广义矩量法(GMM)方法有很大的不同。我们提出的方法还允许在测量误差校正后对功能协变量进行基于模型的分组。测量误差校正的函数协变量的这种分组允许更容易地解释不同组的差异。我们的方法易于实现，我们在广泛的模拟中证明了它的有限样本属性。最后,我们将我们的方法应用于国家健康和检查调查的数据,以评估美国成年人基于可穿戴设备的体力活动测量值与体重指数之间的关系.
