An Efficient Two‐Dimensional Functional Mixed‐Effect Model Framework for Repeatedly Measured Functional Data

Advancements in wearable device technology have enabled accelerometers to continuously record minute-by-minute physical activity over consecutive days, yielding curves serially correlated in dense and regular longitudinal design. Motivated by a large-scale cohort of physical activity data throughout a week, the collected repeatedly measured functional data exhibits longitudinal (interday) and functional (intraday) interactions on fine grids. To accommodate this complex data structure and investigate the relationship between health assessment results and weekly physical activity patterns, we propose an innovative and efficient two-dimensional functional mixed-effect model (2dFMM), characterizing the longitudinal and functional cross-variability while incorporating two-dimensional fixed effects and four-dimensional correlation structure in marginal representation. We develop a fast three-stage estimation procedure to provide accurate fixed-effect inference for model interpretability and improve computational efficiency when encountering large datasets. We find strong evidence of intraday and interday varying significant associations between physical activity and mental health assessments among our cohort population, which sheds light on possible intervention strategies targeting daily physical activity patterns to improve school adolescent mental health. Our method is also used in environmental data to illustrate the wide applicability.