Application of an Intensive Longitudinal Functional Model with Multiple Time Scales in Objectively Measured Children’s Physical Activity

Abstract

This study proposes an intensive longitudinal functional model with multiple time-varying scales and subject-specific random intercepts through mixed model equivalence that includes multiple functional predictors, one or more scalar covariates, and one or more scalar covariates. An estimation framework is proposed for estimating a time-varying coefficient function that is modeled as a linear combination of time-invariant functions with time-varying coefficients. The model takes advantage of the information structure of the penalty, while the estimation procedure utilizes the equivalence between penalized least squares estimation and linear mixed models. A number of simulations are conducted in order to empirically evaluate the process. In the simulation, it was observed that mean square errors for functional coefficients decreased with increasing sample size and level of association. Additionally, sample size had a greater impact on a smaller level of association, and level of association also had a greater impact on a smaller sample size. These results provide empirical evidence that ILFMM estimates of functional coefficients are close to the true functional estimate (basically unchanged). In addition, the results indicated that the AIC could be used to guide the choice of ridge weights. Moreover, when ridge weight ratios were sufficiently large, there was minimal impact on estimation performance. Studying two time scales is important in a wide range of fields, including physics, chemistry, biology, engineering, economics, and more. It allows researchers to gain a better understanding of complex systems and processes that operate over different time frames. Consequently, studying physical activities with two time scales is critical for advancing our understanding of human performance and health and for developing effective strategies to optimize physical activity and exercise programs. Therefore, the proposed model was applied to analyze the physical activity data from the Active Schools Institute of the University of Northern Colorado to determine what kind of time-structure patterns of activities could adequately describe the relationship between daily total magnitude and kids’ daily and weekly physical activity.