02 July, 2016

A toy model for the dependence of athletic performance on age

As you may have noticed by now, I am always searching the web for sites specialising in athletics. This time I fell upon a dutch site which has several very nice analyses. What attracted my attention was a recent post on the high jump records as a function of age covering a huge span, in the case of men from ages of 2 to 98 and for women from 7 to 95. It was a real treasure trove. I was particularly interested in those data since I had, in 2009, written an article in New Studies in Athletics on "Scoring the athletic performance for age groups" (volume 24:3, pages 63-75) where I was lamenting the dearth of data.

While it is not a matter of repeating the calculations of that publication I was tempted by the appearance of these data into seeing what one can do what I called in my paper the toy model. In this model I assume that an athletic performance is the resultant of several physiological factors, technique as well as psychoemotional factors. The latter lie totally outside the scope of a toy model: they represent the most fickle component since for the same person they may vary within the few hours of a competition. No detailed studies do exist (at least to my knowledge) concerning the evolution of technique with age. Schematically one can distinguish three stages in this evolution. Young athletes spend years acquiring a technique and adapting it to their growing body. When adulthood is reached the technique becomes quasi-optimal for every athlete (but the question of an optimal execution during a given event or attempt is always a crucial one). Finally for aging athletes a decline of technique is to be imputed to the decline of physical qualities. Still, one should keep in mind that an experienced athlete performs with the technique that is best suited to his physical qualities and thus the question of optimality is a delicate one.

In what follows I will neglect the impact of technique to performance (making thus the toy model even more toy-ish) and assume that the performance in a given discipline are maximal aerobic capacity, alactic anaerobic capacity (directly related to speed) and strength. If one considers for instance a 400 m race we may assume that the physiological factors influencing the performance are 40 % aerobic, 40 % anaerobic and 20 % strength (your percentages may vary; mine are given just as an illustration). I represent these qualities by a,l,s and their contributions by α=0.4, β=0.4 and γ=0.2. The total performance q is given by
$$q=\alpha a+\beta l+\gamma s$$
What one needs now is the dependence aerobic and anaerobic capacities as well as strength on age. In fact, since I am going to analyse only the data on high jump for which I do not expect the aerobic capacity to play any role, I will concentrate on just the alactic anaerobic capacity and strength. The two figures below show the  data that will be used in the model. They were taken from the book P. Ceretelli and P.E. di Prampero, Sport, ambiente e limite umano, Ed. Montadori, 1985, p.69,  


as far as the anaerobic capacity is concerned and from the paper of S. Israel, Age-related changes in strength and special groups, in Stength and power in sport, edited by Komi. P.V., Blackwell, 1992,  p.322  
in the case of strength and they give the dependence of said qualities on age. The literature of these subjects is vast but since we are presenting just a toy model the data that can be found in those two monographs do suffice.

The model I will be using is a simple one where the performance is achieved by a simple combination of strength and anaerobic capacity. I did not attempt any sophisticated optimisation but opted for a rough 60 %-40 % combination of these two  factors. The height of the jump is given thus by
$$h=\alpha s+\beta l$$
where α=0.6 and β=0.4. The strength and anaerobic capacities are expressed as a percentage of their maximum and thus the resulting quantity h is a number smaller than 1. Next, h is multiplied by the adequate factor so that at its maximum it coincides with the existing world record. This results to the following diagram of the variation of the best high jump performance with age (thick line) compared to the existing data (points linked by a thin line).



The agreement of the toy-model with the data is quite impressive. The overall shape of the curve is reproduced both in the junior and master's age brackets. One can also use the details of the curve in order to explain on a physiological basis the evolution of the performances. It would appear thus that the observed "plateau" between ages 20 and 30 is due to the fact that a strength increase balances the loss in anaerobic capacity, an equilibrium that cannot be maintained beyond the age of 30, whereupon the decline begins.

However one should not misinterpret the nice agreement as the proof of the validity of the toy model. First, one may remark readily that I have limited my application to the age bracket [10-70]. An application of the model to younger and/or older athletes could (and most certainly would) lead to disagreements. Second, while the model was capable to represent the existing data there is no doubt that alternative models could do equally well. The main virtue of my model is its simplicity in the sense that it captures the basic mechanism underlying the performance in high jump. 

One point that I still find a little bit worrying is that one has the impression that the technique does not play an important role, since it was possible to fit the data with the tacit assumption that the technique was optimal at all ages. I am aware of the fact that it is very difficult to quantify the technique. Also one should make the fine distinction between technique and style. The first is a rather abstract notion: for every athletic gesture there exists an optimal technique for its execution. Style is the application of this technique by each athlete adapted to his physical and physiological capacities. What we see when we watch a performance is style and not technique. Given these difficulties the decision to neglect, as far as the toy-model is concerned, the influence of technique in the performance is just an application of Ockham's razor (or of lex parsimoniae, for those who, like myself, prefer Latin expressions over Saxon ones).

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