Wearable devices are often used in clinical and epidemiological studies to monitor physical activity behavior and its influence on health outcomes. These devices are worn over multiple days to record activity patterns, such as step counts recorded at the minute level, resulting in multi-level, longitudinal, high-dimensional, or functional data. When monitoring patterns of step counts over multiple days, devices may record excess zeros during periods of sedentary behavior or non-wear times. Additionally, it has been demonstrated that the accuracy of wearable devices in monitoring true physical activity patterns depends on the intensity of the activities and wear times. While work on adjusting for biases due to measurement errors in functional data is a growing field, relatively less work has been done to study the occurrence of excess zeros along with measurement errors and their combined influence on estimation and inference in multi-level scalar-on-function regression models. We propose semi-continuous modeling approaches to adjust for biases due to zero inflation and measurement errors in scalar-on-function regression models. We provide theoretical justifications for our proposed methods and, through extensive simulations, we demonstrated their finite sample properties. Finally, the developed methods are applied to a school-based intervention study examining the association between school day physical activity with age- and sex-adjusted body mass index among elementary school-aged children.

翻译：可穿戴设备常用于临床与流行病学研究，以监测身体活动行为及其对健康结局的影响。这些设备需连续佩戴多日以记录活动模式（如每分钟步数），从而产生多层级、纵向、高维或函数型数据。在监测多日步数模式时，设备可能在久坐行为或未佩戴时段记录过量零值。此外，研究表明可穿戴设备监测真实身体活动模式的准确性取决于活动强度与佩戴时长。尽管针对函数型数据测量误差偏差校正的研究日益增多，但针对过量零值与测量误差共存现象及其对多层级标量-函数回归模型估计与推断的综合影响研究相对不足。本文提出半连续建模方法，用于校正标量-函数回归模型中零膨胀与测量误差导致的偏差。我们为所提方法提供了理论依据，并通过大量模拟实验验证了其有限样本性质。最终，将所开发方法应用于一项基于学校的干预研究，探究小学适龄儿童在校日身体活动与经年龄性别校正的体重指数之间的关联。