Modelling Athletic Ageing Relative to an Estimated Performance Envelope

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.