Tanda (2011) - A viewpoint

Giovanni Tanda (2011) looked at the marathon performance of 22 runners who had run a total of 46 marathons (over a 5 year period) at near flat-pace race effort (halfway splits <±4 min) - i.e. near optimal aerobically limited efforts. He looked for correlations between marathon performance time and the following elements of the training diary (warm-ups and recoveries were included):

• number of previous marathons
• number of training days per week
• the mean distance run for each workout
• the longest run each week
• the mean workout distance per week
• the mean training pace

The training data he considered included 11 week, 10 week and 8 week average data leading up to one week before the race. Using the 8 week data he found the best correlations between performance time (marathon finishing time) came from mean training pace and mean distance per week (see Figure 1).

Figure 1. Data taken from Tanda (2011) showing the relationship between performance time for the 46 flat paced marathons and weekly distance in the preceding 8 weeks. 

The distance run per week correlated, albeit with a lot of noise, with a decaying exponential but the majority of the data is clustered around a mean of 60 km per week. He had two performances (possibly just one runner) with times near 2:47 with average distance of close to 110 km per week.
The relationship between training pace and marathon performance was also rather scattered, but can be fitted with a straight line (see Figure 2).

Figure 2. Data take from Tanda (2011) showing the relationship between performance time for the 46 flat paced marathons and average training pace in the preceding 8 weeks.

Although marathon performance time could be predicted from either just the average weekly distance or just the average training pace, the prediction is not terribly good - there is rather a large amount of scatter. However, the two variables combined together produce a better prediction since the faster marathon runners not only tended to run further each week, they also ran faster. In fact it is possible to show how Tanda's dataset is distributed in what one might be refer to as 'training space' (a plot of training pace against distance). I have plotted this in Figure 3.

Figure 3. Data from Tanda (2011) showing the training space that the runners occupied over the 8 week period leading up to the marathon.

This scatter in training space can be used to produce a predicted marathon performance by combining both pace and distance into one equation with the relevant relationships. For his dataset Tanda suggested the following equation (rearranged for race time and average speed):
Race time (min) = 12 + 98.5 * e^{(-km per week/189)}+1390/average speed in km per hour
Figure 4 shows the predicted marathon time plotted against the actual marathon performance.

Figure 4. Data taken from Tanda (2011) showing the predicted marathon time plotted against the actual marathon performance time.

Looking at the equation, one can see that race time is made-up of three components. First, everyone gets 12 mins, this is what one can refer to as an offset. Next there is an exponential function of distance. The 'steepness' of the exponential is set by the number 189 and is what electrophysiologists call the 'length constant' (λ). When km is small (i.e. no training has been done) the term tends to 98.5 - or you get 1 hour 38 mins if you don't do any distance at all. If you run an infinite distance (which is rather hard to actually achieve!) the term tends towards zero. If you run the same distance as λ (i.e. 189 km) you get ~37% of the 1:38 (hh:mm) or about 36 mins added on to your performance pace. The nature of exponential functions is that there are diminishing returns for greater distances. Looking at Tanda's dataset it is hard to have much confidence in the exact values - there are simply no runners with distances between about 85-105 km per week (Figure 3). The final value is for speed and is simply 1390 divided by the average speed in kph.
As I showed in the last post the equation is remarkably well behaved at the extremes. It predicts sensible values for someone who does no running and for an elite runner. But, to demonstrate I have plotted the data for a range of other runners (see next post).
Finally, it is possible to plot the marathon prediction times, as contour lines, on the training parameter space plot (Figure 5).

Figure 5. Tanda's dataset plotted in training parameter space (average 8 week training pace against average daily distance). Each data point has been coloured by marathon performance time (blue for the slowest then 'hotter' colours for the faster performances - red being the fastest). The thick colours lines are the contour lines from the prediction equation. The blue line shows the range of distance and paces that would predict a 3:30 marathon, green are the distance and paces for a 3:15 marathon etc. The thin coloured lines are the age-graded 90% effort lines for the four prediction lines. So, the thin blue lines shows a 90% daily age-graded effort for a 3:30 marathon runner. Note that most 3:00 runners (near the 3:00 thick line) are training at above 90% age-graded effort whereas the 2:45 runner(s) and the 3:30 are training (mostly) below 90% effort. Note that most of the parameter space (to the right of the thick line) contains no runners.

The thick lines are the Tanda prediction contour lines for four different marathon finishing times. The thin lines show 90% age-graded effort for daily training for each of the marathon prediction times. Thus, the 2:45 runners are training below 90% effort - whilst the 3:00 runners are putting in over 90% effort each day. The 3:30 runners are also mostly training below 90% age-graded effort. The Tanda equation can be extrapolated out of the fitted parameter space. That is shown to the right of the thick straight line. No one in Tanda's dataset runs these types of averages. The big question is; "Why?". Is it that the equation fails in this space (i.e. is it impossible to complete a marathon in 2:45 by training at an average of 5 mins per km and 26 km per day for 8 weeks) or is it that no-one does that form of training? This question has some critical consequences. Tanda's dataset shows that marathon runners attempt to get faster by increasing their speed up to a 90% daily age-grade. At this point training is very hard. The 3:00 marathon runners are almost certainly unable to progress further using speed as a training tool.

Next: Filling the training parameter space