Age factors and their little secret

In a previous post of mine I had presented the age factors, which aim at adjusting the performance, correcting for age, and which are tailored to master athletes (although age factors can be, and sometimes are, introduced for young athletes as well). I had given there a simple formula for the age factor F

F=C/(A-a)

where a is the age of the athlete and A,C are two parameters. Typically all three a,A,C are expressed in years and F is a dimensionless number.

Age factors are empirically established by the the various masters sports governing bodies. In order to show how well the simple formula above works I present a fit of the age factor for high jump proposed by the World Masters Athletics association.

The quality of the fit is excellent. So the question that naturally arises is how can such a simple formula be so realistic? The answer is to be found in the way the performances decline with age. Here are the results of the men's masters world high jump records.

We observe that the records follow very closely a straight line, over several decades and performances which vary by a factor of more than two. So if the variation of the performance P can be parametrised as 

P=B(A-a)

the age factor introduced above corrects precisely for the variation with age. 

Once one has the expression of the variation of records with age it is natural to wonder what is the meaning of the quantity A, i.e. the age for which the performance goes to zero. Clearly, this is an indication as to the maximum life span. Assuming that the records are representative of the human race as a whole and that a living person, even at a very advanced age, is capable of locomotion, one can imagine that the total absence of a possibility for an athletic performance occurs only at that person's demise. The records, but even better the age factors, somehow encode this. This is the secret information I refer to in the title. 

I must insist here on the fact that the age factors are better suited for the purpose of the prediction of the maximal possible life span as compared to records. The records are the feats of exceptional individuals, while the age factors are normally established based on solid statistical analyses of performances covering several decades. Still these studies do not usually exceed ages around 70-75. One reason is that the population of athletes beyond those ages is too small to allow for a statistical approach.  Also, as Tanaka and Seals have observed, there is a notable decline in performances around ages 80-85, in particular for endurance performances. This is most closely associated with reductions in exercise training intensity and volume, probably as a consequence of changes in a number of physical and behavioural factors (e.g. increased prevalence of injuries, and reductions in energy, time and motivation to train). Thus neglecting those advanced ages from the statistics on which are based the age factors is the proper way to go, contrary to what one could think. (And the fact that, at least for some events, the dependence of performance on age ceases to be linear is another reason why one should prefer to consider age factors rather than world records).

So what are the predictions for the maximal life span coming form the study of the high jump age factor. From the expression for F we find a value for A equal to 140. In my original post on age factors I had presented a similar analysis for long jump and the value of A obtained was equal to 126. 

Several years ago I wrote a short post on D. Harder's approach to scoring. His system allows, in principle, the comparison of performances not only within a single discipline but between all possible sports. In his monograph "Sports comparisons: you can compare apples to oranges" Harder introduces age factors for various track and field disciplines.  I found, after some "reverse engineering", that the one for high jump corresponds to an an expression with A=134. Similar values are obtained for the other disciplines.

At this point one can wonder whether the interpretation of A based solely on athletics' results is well-founded. After all, one could imagine that track and field events are special and that other disciplines could draw a totally different picture. As it turns out this is not the case. In the graphic below I present the age factor for weightlifting. The value of A obtained from the best fit is 137 in total agreement with the ones obtained from athletics.

It is interesting also to look at the age factors used in swimming. Two different age factors are used (well, probably many more, but those two are the best known ones). One is known as the "finnish formula" and the second one is due to A. Rawson. The finnish formula, referred to as age index is a factor meant for multiplying the swimmer's time. (I really cannot understand why people are working with time when the natural, physical, quantity on which to base all analyses is the velocity). From the expression of the ageing index we find that it goes to zero for an age of 109. The Rawson index is a monstrosity involving a polynomial of 6th degree in the athletes' age. Still it can be fitted by the simple formula given at the beginning of this post. 

The fit results in a value for A of 110. Does the fact that the values for swimming are significantly lower than those of athletics (or weightlifting) mean that A is not universal? I do not think so. I am convinced that the records of swimming have still a substantial margin of progression. In fact pushing the curve of the decline of records upwards, so as to bring the value of A around 125, would correspond to a 100 m world record of 45 s for a 25 year old swimmer, something that is not unrealistic.

So just by looking at the athletic performances we can get an indication of the maximal human life span. The prediction for its value, given the spread of the values, is something in the 120-130 bracket. This means that just by looking at how the human race performs with advancing age we can set a maximum for the possible life duration. 

In his book  "Scale" (which is perfectly accessible to a layman and which I highly recommend) physicist G. West asks the question of the maximal human life span and reaches the conclusion that it should be around 125 years. It is amazing that athletic performances offer an independent estimate for this, in perfect agreement with West's estimate.

PS. While preparing this post I ran across an article on the World Athletics site (it was IAAF at that time) with title "Scoring athletic performances for age groups", which does indeed summarise my work. Unfortunately my name does not appear explicitly in the article, but, still, I was glad that the Athletics' instances did take notice of my work. 

Impossible Games: the first post-lockdown competition

Garden Clashes are great. But watching athletes run around a 40 m circuit is not the most exciting spectacle, although I confess that I followed it with great interest: that's what withdrawal is doing to you. Having a competition in a real stadium, even one with a stunted program, a scarce participation and a bizarre choice of events feels like a return to normalcy. The Impossible Games in Oslo did just that: they gave us the feeling that Athletics is back on track (sorry for the pun). 

I must say that I was somewhat sceptic when I first read about the Bislett Games metamorphosis into Impossible Games. It turned out that the competition did take place indeed and with quite some success. Of course, you have to make abstraction of the empty stadium and of the fact that only a handful of athletes were participating. On the other hand the organisation, not being a mainstream Diamond League one this time, allowed the organisers to offer events which are only rarely contested (plus one innovation with the team Ingebrigtsen-team Cheruiyot duel). 

The field events were rather poor, with just one jump, a pole vault at a distance duel between Duplantis (who was present in Oslo) and Lavillenie (jumping in his backyard). Seeing Duplantis jump, one had the impression (confirmed by himself) that he is not at the top of his condition. Moreover it is my impression that he is jumping with a stiffer pole and he is not yet fully accustomed to it.

The main event of the competition was the 300 m hurdles solo race of K. Warholm. It was superbly executed and the world record he established with 33.78  was amply merited. For those who wander about the time difference between the 300 and the 400 m hurdles I must point out that the 300 has only one bend compared to two for the 400 m.

Warholm on the way for a world record

I did particularly like the women's 300 m hurdles. S. Petersen prevailed over A. Iuel (an athlete whose progress I am following) 39.42 to 39.44 but it was a close thing. L. Sprunger was third but I did like the idea that she ran with 13 strides between hurdles, while women usually take 15 (in the case of 400 m). Of course her 1.83 m do help. I just hope that she continues with her experimentation with this reduced number of strides. 

Petersen and Iuel at the finish line

Another much expected moment of the competition was the duel at a distance between the norwegian team of the Ingebrigtsen's and the kenyan one led by Cheruiyot and Manangoi. The Oslo race was an occasion for me to see for the first time the Wavelight luminous pacer. It has been touted by World Athletics as an innovation that will enable the athletes to better target a specific pace and also add greater value to the spectator experience. Now the question is, will such a system be allowed in official championships? I guess we'll have to wait for next year. Coming back to the race itself, well, there was practically no race. The three Ingebrigtsen finished well ahead of the first kenyan (Cheruiyot), with Jacob establishing a new European record over 2000 m with 4:50.01. Of course one must take into account that the kenyans were running in Nairobi, a city at 1800 m of altitude and with very bad weather conditions.

Notice the yellow lights on the kerb of the track: they are the Wavelight pacer

The one intriguing thing in this competition was the 10000 m solo race of Th. Johaug. She is last year's norwegian champion over the distance and in Oslo she improved substantially her record with 31:40.69. (The entry standard for the Tokyo Olympics is 31:25.00). Johaug is a world-class cross country skier having won gold in both the World's and the Olympics.  Is she trying to reorient her career? Does this have anything to do with her doping violation for which she had been banned from October 2016 till April 2018? Time will tell. 

The Impossible Games were a most uplifting moment amid the general epidemic-induced moroseness. Now we have to wait for the real start of the season, planned for mid-August.

The Ultimate Garden Clash - Combined Events

On Sunday World Athletics presented us with another edition of the Ultimate Garden Clash. This time it was a competition between three decathletes, World champion and record holder K. Mayer, world champion N. Kaul and world vice-champion M. Uibo. As in the previous Garden Clash events the three athletes were competing in their respective training base. 

I did enjoy the previous Garden Clash events but I found this last one even more interesting. Watching close to a 100 times pole vaults can lack excitement in the long run. The 10+10+5 minutes formula is, and by far, more interesting.

In the first event the athletes had to vault over 4 m. For Mayer and Uibo who have a personal best at 5.45 and 5.40 m respectively that was a pure formality. In fact Mayer was jumping with just a seven stride run-up. Despite having started later by more than 20 seconds Mayer carried the first event with 17 successful jumps (somehow he managed to miss one). Uibo was second with 15 and Kaul (whose personal best is a mere 5.00 m) was third with 14.

Then came the shot put where the athletes had to throw over 12 m. It was a mere formality for everyone since all three have personal bests over 15 m (Mayer's is an impressive 17.08 m). The difficulty here came from the fact that they had to chase the shot and come back to the throwing circle. Mayer again dominated the event with 28 throws while Kaul and Uibo had 22 and 20 respectively. 

At this stage of the competition it was clear that Mayer was going to win. The last event was a 5 minutes' shuttle run between two pegs situated at a distance of 20 m. Kaul, being the best of the three over 1500 m, had a definite advantage, and although he did manage to beat Mayer in the end their difference was a meagre 1 lap, 27 to 26. In fact Mayer was keeping up with Kaul up to the very last laps. Uibo had to contend with the third overall place (with 26 laps). Kaul was second, scoring three points more than Uibo, with Mayer dominating the events with 8 points more than the second.

I did like the structure of this competition a lot. It is testing, following the ideas of Gaston Meyer, the overall value of a decathlete (except speed, but I do not see how a speed event can be incorporated to such a repetition-based formula). It is in the spirit of the triathlon events that Mayer would like to see incorporated in major competition circuits, like the Diamond League. While the triathlons Mayer is talking about would be more speed-force oriented ones, a Garden Clash like one is more force-stamina oriented. 

With the adequate changes it could even become a competition event. One can imagine a fixed number of successful attempts, for instance 3, at heights that increase regularly, say by 10 cm every time, the last height determining the points of the athlete. Same for the throwing event, where the athlete will have to succeed repeatedly progressively increasing distances. And I would prefer discus throw which is more difficult technically and will ask the thrower to fetch back the implement from further away. Finally a 1500 m run would complete the event. But of course, as soon as the restrictions are over, we shall forget the Garden Clash events and go back to business as usual. 

I must admit that I am looking forward to the day (sometimes in August if the current planning does not get disrupted) where we'll be able to watch the world elite participate in major competitions. In the meantime I cross my fingers for WA to offer us another Garden Clash combined event, this time with the participation of the best heptathletes. 

Heavy men running

I have already written about Dale Harder and his "Sports comparisons: you can compare apples to oranges" book. It lays the foundation for a universal scoring system, one which would allow comparisons between athletes practicing any sport. Dale is not known only for his "apples to oranges" work. He is also (among others) publishing a newsletter on "Speed and Strength". This year I subscribed to the newsletter and in the January 2020 edition I stumbled upon a very interesting list or records: athletics records by bodyweight. 

The sculptural J. Regis

We all know the 100 m record (and the weight of U. Bolt at the time of the record) and we could easily remember J. Regis and his 98 kg blasting down the track for a 10.15 s 100 m (and a 19.87 s 200 m). But the list goes on and becomes more and more interesting. One finds as best performers over 100 m at 125 and 130 kg none other than the ex-world record holders of discus throw and shot put respectively, Y. Dumchev, with 11.24 s, and U. Beyer, with 11.44 s. The heaviest man with a 12.54 s record is M. Hoebert with 145 kg. And from there things become crazy since several people have established records while carrying another person on their shoulders. We have thus a record of 18.24 s over 100 m with 244 kg and even one at 48.24 s with 357 kg. 

Y. Dumchev (71.86 m WR) confided, years ago, to P.J. Vazel

that he once threw 80 m at practice with a huge wind! 

The longest distance where a record is established by somebody carrying a person on his shoulders is the mile, with 8:30 for 158 kg. The heaviest person to complete a marathon did so in 5 hours and 14:35 while weighing 138 kg. 

D. Harder's list goes on to present also the records for jumps, from lighter to heavier, and also those of throws, from heavier to lighter. I may one day return to analyse those lists but I must say that the use of imperial measures (pounds and feet) is a real torment. I can understand that the US cannot spend the humongous amount of money needed for the conversion. But why on earth people who deal with sport cannot decide to use the metric system since it is the only one used in international competitions? (I remember reading that one reason J. Thomas lost the high jump gold medal in the Rome, 1960, Olympics, was that he did not know what height he was attempting and had each time to ask his coach. I also remember B. Beamon in Mexico realising what he had done only after R. Boston translated his jump into feet). 

I show below a plot of the velocity as a function of the athletes mass. It is clear that over a wide range of masses the dependence is linear.

Similar results are obtained for longer distances up to, and including the marathon. Since the dependence is roughly linear one can perform a straight-line fit and extract the parameters. Once this is done it is easy to answer the question: what would be the maximum mass of an athlete able to complete a certain distance. It is clear that a 150 kg weightlifter can carry a 250 kg weight on his shoulders over a short distance but how about completing a 100 m ? A mile ? A Marathon? Extrapolating the straight lines to zero velocity I found the following:

For short distances, 100 and 200 m, one can complete them even with a great weight. My extrapolations give a limiting weight of over 400 kg for 100 m and around 350 kg for the 200 m. From there the limiting weight diminishes fast and is around 230 kg from 400 m to the mile. For longer distances and up to the Marathon the limiting weight does not exceed 200 kg. This is an amazing result. If the extrapolation is to be taken seriously it would mean that an overweight man, say at around 180 kg could in theory complete a marathon (walking rather than running, but still). This sounds a little bit crazy but on the other hand I would never have believed that a 140 kg man could finish a marathon, until I saw D. Harder's compilation.