Sciolism Rocks

Michael Phelps may now be the all-time record holder for gold medals, but I wish I were “The World’s Greatest Athlete”. That’s the title given to the winner of the olympic decathlon, first bestowed upon Jim Thorpe by King Gustav I after the 1912 olympics.

It’s not difficult to see why the title is apt. The decathlon is a series of ten events spread over two days:

At the end of the decathlon, there are ten scores, one for each event. Beware, though, of the illusion of scoring. Finding ten ways to measure something is not the same as having ten things to measure.

People in educational measurement and psychometrics have been faced with this dilemma for a hundred years or so. In 1901, Karl Pearson was among the scientists that felt that there was a single type of “intelligence”. He developed the method of Principal Components Analysis (PCA) as a way of taking several test scores and condensing them down to a single score (reducing dimensionality if you want to sound splashy). Pearson called this component g and let it represent the theoretical idea of a single measure of intelligence.

The idea of a single intelligence quotient is now old-fashioned and generally not adhered to by professional educators. But the mathematical technique is still useful. The decathlon is a good example : despite having ten measurements on each athlete, are we really measuring ten different things? Most people can see that we aren’t. Here’s what I mean.

I’ll use JMP to examine the decathlon scores from the last three Olympics:1996 in Atlanta, 2000 in Sydney, and 2004 in Athens. If I look at a plot of 100m times vs. 400m times, you get a noticeable correspondence:

It shouldn’t be surprising that there is a high correlation (r = 0.611, p < 0.001) between these two scores. People that are good at the 100m dash are also good at the 400m dash. Thus, these two scores are, in a sense, measuring the same thing. And they’re not the only two that have a strong correlation. Dig the Discus Throw and the Shot Put (r = 0.6572, p < 0.001):

(The literary part of me wants to emphasize that “putting” is a real verb that implies “throwing in a pushing motion”, as opposed to what baseball pitchers do. The thing that is put is called the “shot”.)

Now that I’m all excited by a couple of bivariate graphs, I’m going whole-hog and looking at all possible two-by-two event comparisons. JMP’s Scatterplot Matrix lets me look at all these bivariate graphs at once.

You can see the strong positive correlations that we’ve already discussed, but there are some strong negative ones as well. Up at the top left corner, you can see a fairly strong negative correlation between the Long Jump and the 100m. Why ? Because the goals are different for the two events: you want a small score (time) on the 100m but a big score (distance) on the Long Jump.The negative correlations means that people who are good at one are also good at the other.

Since it’s a bit difficult to pick out the strong and weak correlations with the scatterplot matrix, JMP has a color map that shows the same information. Bright red indicates a strong correlation, bright blue indicates a strong negative correlation, and gray shows no relationship.

The bright red diagonal isn’t a surprise (every variable is perfectly correlated with itself). Looking at the slightly paler red, we see that the 100m, 400m, and 100m hurdles are all correlated. If I cluster similar correlations together, you can see that there are some clumps.

This is where the real PCA starts. We’ve got lots of correlations, and we want to see if some are measuring the same thing. I’ve got some suspicions about the data: that there may be two components (Track, and Field) or that there may be three components (running, throwing, jumping), or possible more. Look at these color maps and see how many clusters you see.

It’s fairly easy to see how two clumps form. There is a cluster of Track events in the upper left. The rest (sort of) cluster into Field events. The 1500m doesn’t seem to follow the rules, but I’ll have more about that in a bit.

It’s also possible to see three clusters in this map that correspond to running, jumping, and throwing. Again, the 1500m seems to break the pattern.

There’s some difficult mathematics involved in actually computing these clusters mathematically (”I can use eigenvalues and singular value decompositions.” Makes me sound exciting, right?) but as you’ve just seen, the spirit of the analysis is just looking at clumps of correlations, seeing if several scores actually measure a single thing.

When JMP does PCA, it produces a loadings plot. This lets you see which events tend to group into each cluster. Take, for example, the loadings plot for our two-clump example.

This plot echos our clustered color map. 400m, 100m, and 110m hurdles all point along the positive x-axis. Discus and Shot put point along the y.

Note that the 1500m seems to be off on its own. Throughout our groupings, we’ve noticed that this event doesn’t behave like the others. My theory is that

• It is a long run — three times as long as any of the others.
• It is the last event on the second day, thus illustrates the state of athletes after they are tired. Contrast that to the 100m and 400m, which are the first events on their respective days.
• It is placed on a day where skills matter much more than endurance, yet it is an endurance event.

And the ordering of these events must be preserved in order for world records and comparisons to stand.

---

The 2008 Olympic decathlon takes place August 20 and 21 in National Stadium.