Derivative curve estimation in longitudinal studies using P-splines

The estimation of curve derivatives is of interest in many disciplines. It allows the extraction of important characteristics to gain insight about the underlying process. In the context of longitudinal data, the derivative allows the description of biological features of the individuals or finding change regions of interest. Although there are several approaches to estimate subject-specific curves and their derivatives, there are still open problems due to the complicated nature of these time course processes. In this article, we illustrate the use of P-spline models to estimate derivatives in the context of longitudinal data. We also propose a new penalty acting at the population and the subject-specific levels to address under-smoothing and boundary problems in derivative estimation. The practical performance of the proposal is evaluated through simulations, and comparisons with an alternative method are reported. Finally, an application to longitudinal height measurements of 125 football players in a youth professional academy is presented, where the goal is to analyse their growth and maturity patterns over time.