Physical activity (PA) intervention studies often collect repeated intensity measurements over long observation periods. Quantifying the variation in intervention effects over the study period is critical to evaluating and improving intervention strategies, yet many analyses reduce PA data into scalar summary measures, resulting in limited insights. We propose a functional regression framework, which captures time-varying intervention effects by modeling the entire PA trajectory as a functional observation. From both methodological and practical perspectives, we demonstrate the advantages of function-on-scalar regression (FoSR) over the traditional two-step approach of applying functional principal components analysis (FPCA) followed by regressing scores on covariates. The FoSR is further extended to a function-on-function regression (FoFR) for studying the association of PA across time periods. Methods are applied to daily step counts from the Social incentives to Encourage Physical Activity and Understand Predictors (STEP UP) study, revealing distinct and highly interpretable time-varying effects of three intervention strategies on PA and differences in their sustainability. Our case study highlights the feasibility of functional data analysis techniques for uncovering novel insights in intervention studies with high-dimensional endpoints.

翻译：体力活动（PA）干预研究常需在长期观察期内重复采集强度测量数据。量化干预效果在研究期间的动态变化对评估和优化干预策略至关重要，然而多数分析将PA数据简化为标量汇总测量值，导致洞察力受限。我们提出一种函数回归框架，通过将完整PA轨迹建模为函数观测值，捕捉随时间变化的干预效应。从方法论与实践视角，我们论证了标量响应函数回归（FoSR）相较于传统两步法（先进行函数主成分分析（FPCA），再对协变量回归得分）的优越性。进一步将FoSR扩展为函数对函数回归（FoFR），用于研究不同时间区间PA的关联性。将该方法应用于"社会激励促进体力活动与预测因素研究（STEP UP）"的日步数数据，揭示出三种干预策略对PA具有独特且高度可解释的时变效应及其可持续性差异。本案例研究凸显了函数数据分析技术在探索高维终点干预研究中新洞察的可行性。