Modelling Athletic Ageing Relative to an Estimated Performance Envelope

2026-08-06

AI Summary

This study addresses the challenge of modeling sparse and irregular longitudinal performance data in athletes, where existing approaches often focus on population-average aging trends while neglecting individual deviations from their latent peak performance—termed the performance envelope. The authors propose RACE, a two-stage framework that first estimates an age-conditioned high-performance envelope across the population and then employs STAR (Shape Translation And Rotation), a nonlinear mixed-effects model, to represent individual trajectories as low-dimensional geometric transformations of this envelope—capturing shifts in level, timing, and pace. A key innovation lies in jointly modeling the performance envelope and individual aging parameters, embedding the correlation between level and pace within the random-effects covariance structure, thereby revealing how the envelope’s geometry governs parameter identifiability. Applied to MLB Statcast data, the method successfully disentangles ability level from aging pace, explaining why burst initiation rate identifies timing shifts whereas sprint speed does not.

Abstract

Athletic careers yield sparse, irregular longitudinal series: few seasons per athlete, incomplete paths, and selection into continued play. Scientific interest often centres on proximity to peak attainable performance-a ceiling-rather than on the mean trajectory, and on how that proximity co-varies with the rate of decline. Standard tools address only pieces of this problem. Linear mixed models describe average ageing; functional principal components describe dominant modes of variation; shape-invariant models register curves about a mean template; growth charts estimate population centiles but stop short of individual latent trajectories. We develop RACE (Relative Aging Curves via Envelopes), a two-stage framework that estimates a population performance envelope as an age-conditional high centile and then models each athlete as a low-dimensional geometric transformation of that envelope. STAR (Shape Translation And Rotation) is the Stage 2 mixed model, with parameters for level, timing, and tempo. Envelope geometry determines which of these parameters are identifiable when careers are short: near-linear envelopes identify level and tempo only, whereas curved envelopes identify all three. Embedding STAR in a nonlinear mixed-effects hierarchy makes the level-tempo association a parameter of the random-effect covariance rather than a post-hoc correlation of separate fits. Simulations ask whether that association is recoverable under sparsity, how geometry governs identifiability, and how sensitive results are to envelope misspecification. Applied to Major League Baseball Statcast sprint speed and bolt rate, the analysis demonstrates how the proposed embedding separates athletic level and ageing tempo in Functional Ageing Space, and shows why timing is identifiable for bolt rate but not for sprint speed.

Problem

• athletic ageing
• performance envelope
• longitudinal data
• individual trajectories
• peak performance

Innovation

• performance envelope
• nonlinear mixed-effects model
• functional ageing
• geometric transformation
• athletic trajectory