Hierarchical Bayesian Joint Modeling of Athlete Performance Trajectories and Latent Peak Ages

Abstract

Determining when athletes reach peak performance is a central question in sports science, with implications for talent identification, training design, and cross-discipline comparisons. Statistical inference is challenging because peak age is not directly observed: competition records are noisy and selective measurements of underlying performance capacity and are often limited to incomplete observation windows. We develop a hierarchical Bayesian joint modeling framework that links a longitudinal model for athlete-specific performance trajectories to a regression submodel for latent peak age. The longitudinal component models observed competition performances as noisy measurements of athlete-specific performance trajectories over age. The peak-age component then treats the maximizer of each trajectory as a latent response in a parametric accelerated failure time (AFT) model, allowing athlete-level covariates to explain earlier or later peak age. The hierarchical structure borrows information across athletes, while joint posterior inference propagates uncertainty from trajectory reconstruction to both individual peak-age estimates and population-level covariate effects. Simulation studies show that, as observation windows become less informative, the joint model preserves substantially smaller regression bias and reflects the information loss through wider uncertainty, whereas pseudo-peak and two-stage alternatives exhibit larger bias. An application to World Athletics ranking data illustrates how the joint model estimates discipline-specific peak-age patterns and covariate effects under incomplete and irregular observation windows. Beyond sports, the proposed framework applies broadly to problems that couple longitudinal trajectories with function-derived latent features.