Athletica Vaticana, an interesting initiative

While following results on competitions in Italy on the site atleticalive.it, I ran accross a reference to  "Athletica Vaticana". I was intrigued and I tried to find more about this. 

It turned out that in 2017 the Pontifical Council for Culture gave patronage to Athletica Vaticana. The members of the latter are Vatican citizens or employees. In 2019, Athletica Vaticana received recognition as the first official Vatican sports association.

As Pope Francis himself stated in an interview at the Gazzetta dello Sport, "the athletics team in Vatican is a testimony of the evangelical nature of sport. It is also a way to create a community. One has just to think about the members of the team who come from the various administrations: swiss guards, gardeners, pharmacists, museum employees and also priests. So the Church is reaching out to the stadia".

Athletica Vaticana is already member of the sports community of the Small Countries of Europe (consisting of countries with less than a million citizens). It is affiliated to the Italian Olympic Committee. I don't know if this can lead to the Vatican joining the IOC. I have checked and the Vatican does not figure, yet, among the 206 countries that are currently members of the IOC. But it would be interesting to see the Vatican in the Olympics.

Stair climbing, a very simple cardio test

The ongoing epidemic is a major perturbation to our way of life. Without a possibility to train and compete, for the vast majority of us, living is a permanent frustration. Last year I wrote about a possible way to (try to) keep in shape, linking to an excellent exercise program offered by the National Health Service of UK. In my article on "competition in the times of the epidemic" I broke one of my self-imposed rules and wrote about myself, detailing my personal exasperation due to the lack of training possibilities and competitions for non-olympic sports.

Here I am going, once more, to stray from athletics and propose you something that can be of use, all the more so if the various (inefficient) measures, meant to control the epidemic, are making you missing exercise and be out of shape.

But let us start from the beginning. A team of the University of La Coruña performed a study on a cohort of 165 patients who were known or suspected of a coronary disease. They put them on a treadmill and had them walk or run while they increased gradually the intensity. This test continued until exhaustion of the subjects. The capacity of the participants was measured in metabolic equivalents of task (MET). 

The MET is the measure of the rate at which a person expends energy while performing a given task to the energy expended when sitting quietly. Physiologists have fixed the latter at 3.5 mL of oxygen per kilogram per minute, a quantity roughly equivalent to 1 kilocalorie per hour and per kg of body mass. I have already referred to METs in my article cited above. 

The La Coruña team let their subjects rest for 15-20 min after the first exercise and then asked them to climb 60 stairs (roughly four flights of stairs) at a fast pace. They had to walk (but not run) continuously without stopping. The time to climb the 60 stairs was recorded and the results of the MET measurement were plotted on a graphic (presented below) as a function of the time to climb the stairs. 

(Had I access at the raw data I would have discarded the rightmost point and fitted the remaining ones. This would have changed the value of the parameters of the parabola, but not the conclusions). What is clear from the distribution of the points in the graph is that the subjects who can sustain more than 10 MET of exercise are able to climb the 60 stairs in circa 45 seconds. In fact I believe that the limit in this case is due to the fact that the subjects were not allowed to run. Had running been allowed the ones who could sustain an intensity of exercise of more than 15 MET would have been able to climb much faster. For the group realising 8-10 MET the mean climbing time was of roughly 60 seconds, while those unable to surpass 8 MET were slower, with a mean time exceeding 80 seconds. 

And now comes the interesting part. From previous studies we know that 10 MET during a test are linked with a low mortality rate, less than 10 % in 10 years, whereas the persons who cannot reach 8 MET have a statistical mortality rate of 30 % in 10 years. Measuring the MET is a complicated task requiring the installations of a physiology lab. Having correlated the MET to the climb-time, the La Coruña team is offering everybody a simple tool for the assessment of their cardiovascular condition. Of course, the study was performed on individuals where cardiac problems were suspected. But the research team believe that the correlation between the climb-time and the exercise capacity will be similar in the general population.

So, there you have it. If you wish to test your condition, find a four storey building and climb 60 stairs while measuring your time. If you manage less than 45 seconds you can rest on your laurels you can continue with your current training program. If not, start exercising immediately.   

Theories of scoring: Harder's approach (and a proposed extension)

I have already written a short post on the theories of D. Harder. He is the author of the "Apples to Oranges" monograph where he explains in detail his theory of scoring which allows comparisons between achievements in different sports. The key to this comparison is that you "... compare the number of athletes who reach any given level ..." (proportional to the number of athletes competing in that sport, of course). The quantitative basis of the method is the following. A mark of 100 points is attributed to some performance if a fraction of 0.5 of the population can realise a score equal or better than this. For 200 points, only a fraction of 0.05 of the population can do better than this performance. The next 100 points, i.e. 300, correspond to a performance realised by just 0.005 of the population and so on up to 1000 points where only a fraction 5.10^-10 of the population  (which means less than one person on earth) can realise the corresponding performance. Going in the other direction a performance earning 50 points is one that 84 % of the population should be able to realise, while 0 corresponds to a performance possible for 95 % of the population. 

The figure below shows a comparison between the official World Athletics scoring tables and the ones of Harder for men's long jump. The official scoring tables attribute close to 1400 points to the performance corresponding to Harder's 1000 but this is not something crucial. 

It is remarkable that Harder had to use a different slope for his scoring curves in the below 100 points part. This is an attempt in these tables to accommodate a large part of the population. While this goes in the right direction, I believe that it is not quite sufficient. In fact, the part between 0 and 100 points is the weak one in Harder's work. In some events the scoring cannot go all the way down to 0. This is for instance the case for pole vault where there is nothing below 1.20 m corresponding to 80 points, since already in high jump a 1.20 performance obtains 60 points. The treatment of the performances of the 50 % of the population presents definitely a difficulty and would necessitate specific studies. While this is not particularly interesting from a competition oriented point of view, it is a challenge for Harder's approach since the latter is meant to be able to offer comparisons between any two sportsmen (the latter term understood in the broadest possible way).

The reasoning that led me to propose an extension of Harder's approach is that every effort deserves a reward. So a fair scoring should be able to allocate points to any performance: only zero performance should obtain zero points. This led me to the mathematical reformulation of Harder's approach, explained in what follows.

It is a well-established statistical fact that the distribution of human performance in various domains follows a bell-shaped curve (dashed curve in the figure below). The details may vary but the fact remains that most individuals perform close to the median with only a very small percentage registering exceptionally good or bad performances. Given this, the fraction of the population realising a performance better than some threshold x (which is crucial in Harder's approach) should be given by a sigmoidal curve (continuous curve below). 

In my article, The physical basis of scoring the athletic performance published in New Studies in Athletics, 2007, volume 22:3, pages 47-53,  I proposed an analytic expression for this curve:

f(x)=(k+1)/(k+exp(x))

It will be used in what follows in order to introduce a scoring system based on Harder's theory. Going back to the relation of points to performance and interpreting it in the spirit of Harder, i.e. a progress of 100 points means that the fraction of the population realising it is divided by 10, it is natural to introduce a logarithmic relation between points and performance, i.e. p is proportional to log(f), where log is the decimal logarithm. More precisely, I propose the following scoring formula:

p=alog(1+b(exp(cx)-1))

where x is the performance (velocity for track events and length for the field ones). Notice that p is equal to 0 when x is also 0.

How does this expression compare to Harder's curve? By adjusting the parameters a,b,c it is possible to obtain an excellent fit as shown in the figure below. 

So, naturally, the question arises whether the expression I propose could be used in order to approximate the official scoring tables. It turns out that this is indeed the case. By adjusting once more the parameters one obtains the fit shown in the figure below.

The two curves are very close from 100 points upwards but below this mark the expression proposed above curves towards 0. The smallest performance obtaining at least 1 point is 28 cm. 

To put it in a nutshell, I believe that the scoring formula I introduced is a natural extension of Harder's approach. It has the advantage of a closed-form analytical expression and its parameters can be adjusted so as to follow closely the official, World Athletics, scoring curve. Moreover it provides the most fair scoring, since it attributes a non-zero score to any measurable performance. 

On theories of scoring

I have always been a scoring buff. One of my first posts in this blog was on how, when I was just 10 years old, I tried to reconstruct the decathlon scoring tables from the results on the Melbourne, 1956, olympic decathlon. For reasons even myself cannot fathom I am attracted to the concept of scoring and I am regularly revisiting the subject. 

Quantitative measurements allow comparisons of the performance of competitors within the same event but the situation becomes more complicated when different events are involved. Everybody would agree that the performance of a world champion in some event is better than that of an inexperienced competitor in a different event. But we should be able to make this comparison more precise and to distinguish performances on which only specialists could a priori pronounce themselves? And then there is the question of how does one score combined events.

R. Johnson and C.K. Yang 

at the unforgettable 1960 olympic decathlon

Performance scoring is a science and an art. There exist stupid individuals who believe that, since they have mastered the workings of spreadsheets, they can produce scoring tables.  Usually the only thing they succeed in doing is to convince everyone of their incompetence. Developing acceptable scoring tables for athletics took, literally, decades. The theory of scoring and the history of scoring tables is a most interesting topic and I intend to cover it in a series of posts which will be, as always, interspersed with other articles, some of them inspired by current events (crossing my fingers that activity will indeed pick up in the coming months).