Let the crowd suggest the pace

Meb Keflezighi's win at the Boston Marathon is one of the top art performances I have ever seen. What makes Meb's win so remarkable is that he had no margin of error and was able to perform a perfect act. A major factor for his victory was his choice of running pace, it was right on target from the beginning. It is difficult, even for elite marathoners, to pin down the best race-day pace. On this post I want to present a way to choose the right race-day pace given a (shorter) race effort a few weeks before.

Meb ran the 26.2 miles from Hopkinton to Boston with impeccable technique and a level of masterly rarely seen in the arts, let alone running. It is fair to say that Meb was the underdog in a field that included a handful of sub 2:06:00 marathoners. With a personal record of 2:09:08 (Houston, Jan. 2012), Keflezighi's only chance at the win was to run a personal best and hope for a not so good day for the rest of the field.

The lead pack had covered the first 5 km in 15:09. At this pace the finish time would have been 2:07:51. The second 5 km were ran in 15:20. Had Meb kept running at this pace he would have finished in 2:09:13, right around his best shot for a win. The brilliance of Meb was to keep running at this pace, independently of what others were doing, and hope for the best. Meb followed this plan except for the 20 to 25 km stretch, for which he clocked 14:55 opening a 1 minute gap from the pack. All other splits were ran at paces between 15:10 and 15:18. The chasing pack waited too long to catch up and Meb crossed the finish line first in 2:08:37, his personal best to date, and less than 10 seconds ahead of the runner up.

Talented runners may be able to pick their race-pace by following their gut instinct, but that is not the only option. People training for a marathon usually include a shorter race a few weeks before the big event. Knowing the finishing time at the shorter race, one can predict the right pace for the marathon by leveraging historical data from tens of thousands of runners. The method I propose here is rather simple. Suppose John Doe ran today a half marathon in 1:40:00. This means that John Doe is faster than 75% of runners at the half marathon (see top plot on the figure below).  I propose that John Doe should choose a pace for the marathon such that he is faster than the same 75% of runners. In this case it means that John Doe should aim to finish the marathon in 3:39:14.

The method I am proposing can be stated succinctly: under equivalent race conditions and equivalent training, the finishing times for a runner are such that they divide the distributions of finishing times at the same percentile, independently of the race distance. 

To validate this method, I went ahead and tested how well does it do at predicting the finishing times for actual runners. For the comparison I looked at runners who ran both the 2011 NYC Marathon and the 2011 Grete's Great Gallop half-marathon. I used the half-marathon as the pivot race to predict the finishing times for the marathon.  The following plot shows the results.

These were good races to choose because of the fair and similar weather conditions for both of them, and because the time span between them, 35 days, is long enough for runners to engage the half-marathon at race-effort but not too long for the fitness level of runners to change significantly from the date of the half to the date of the marathon. The field of the Grete's half was large, from the 4974 people who ran it, 2169 also ran the NYC marathon a month after. The predicted times are evenly distributed around the perfect prediction (49.4% of the data lies above the blue line).

The relationship between the marathon and the half marathon times is nearly linear. It is reproduced, rather accurately, by the following formulas:

M=2.50*HM-29.15 (for men)

and

M=2.41*HM-27.02 (for women),

where M stands for the marathon time and HM stands for the half-marathon time, both in minutes. I hope these were convincing arguments for letting the crowd suggest the pace.