@Dis_Yoda is correct. The relationship of the actual race times is not linear. Thus, you can't take mile PR time and multiply by 26.2 to get an estimated marathon finish time.
I think the confusion stems from this comment I made:
So this use of "linear" was not in description of the of the race times multiplied by a linear number, but rather the relationship when looking at the times on a race equivalency look up chart.
This is an example of a race equivalency chart (Daniels VDOT):
View attachment 242264
If the relationship were linear, then your race times would look like this:
View attachment 242263
This person can run a 5:56 mile, a 20:18 5k, and a 3:14:06 marathon. This is not common among recreational runners (I'll get back to this).
More common is a non-linear relationship like this:
View attachment 242262
This person runs a 6:17 mile, but they don't run a 3:24:39 marathon. Instead, they run a 4:34:59 marathon. Thus, the line connecting all of the data points is not linear, but shows a fade as the distance gets longer.
So, if a linear relationship is not common among recreational runners when using a race equivalency calculator, then where did the race equivalency calculator come from?
One of the first running calculators that I am aware of (and most commonly used today) was published in Runner's World in 1977 by Peter Reigel.
Reigel's formula is: t2 = t1 * (d2 / d1)^1.06
t= time
d= distance
So, as the distance increases by double, the pace declines by 6%.
Other formulas:
Reigel #2:
x = (av)^(1/(1-b))
Cameron:
a = 13.49681 - (0.000030363 * old_dist) + (835.7114 / (old_dist^0.7905))
b = 13.49681 - (0.000030363 * new_dist) + (835.7114 / (new_dist^0.7905))
new_time = (old_time / old_dist) * (a / b) * new_dist
Purdy:
P = A(Ts/Tp - B)
where P - is purdy points
Ts - Standard time from tables + time factor
Tp - Performance time to be compared
A, B - the scaling factors.
VO2max:
percent_max = 0.8 + 0.1894393 * e^(-0.012778 * time) + 0.2989558 * e^(-0.1932605 * time)
vo2 = -4.60 + 0.182258 * velocity + 0.000104 * velocity^2
vo2max = vo2 / percent_max
All of these formulas have something in common, they were written some time ago before the latest running boom. Which means much of the data used to generate these formulas was based on well trained athletes at the faster end of the pace spectrum. Sometimes based off world records.
Vickers made an attempt using real-world current data to come up with a better calculator. He takes into account training (using miles per week) as a first attempt at reworking the calculator at the crux of where most calculators fail: the marathon. In most cases, the race equivalency calculator assumes you are under ideal conditions and ideally trained. But for a portion of the running community, they are not well trained for the marathon and thus the calculator will fail in giving them a realistic goal/pacing strategy. Vickers attempted to fix that error in the calculators with his calculation based on several thousand self-reported results. I reviewed his paper back in November last year in my journal (
link).
His forumla is:
Model 1:
View attachment 242286
Model 2:
View attachment 242288
This was my final conclusion on Vickers based on my interpretation of his paper:
There are 310 data points in their model 1 prediction (one other race) and 171 data points in the model 2 prediction. The data is further broken down into percentiles of 5%. So for model 1 that means 15 data points and for model 2 9 data points. Getting a lot smaller, right. So when evaluating the actual data I would conclude that the new model (1 and 2) is better than Riegel for everything in the top 67% of their data set, when evaluating the data as raw data. For model 1 that means everyone faster than a expected marathon of 3:52 should use the new calculator and for model 2 a 3:53. However, if you are slower than a 3:52 or 3:53, then the classic Riegel calculator is still better. If you want to say that avoiding a too fast start is the absolute paramount then the time cutoff is more like 4:11-4:14 (faster use the new calc, and slower use the classic calc). Now remember the NYC and Running in the USA averages? They were roughly 4:11-4:38. So essentially, the average runner should still use the classic calculator because the new calculator isn't as good at predicting average to slower times based on those completed in NYC or Running in the USA. Looks like to me they missed the mark with the original data set, and thus when they created a calculator it badly misjudges the times of those in the bottom 50% of marathon runners (but the classic can do those better, or at least according to the limited data set available in their original values).
But I do urge you to read the full synopsis I did because there was definitely some great things about the paper.
So, what is McMillan (as that was the original questions right? Ugh DopeyBadger and is really long winded answers...
)
To determine, what he uses I did the following. I entered two random marathon times to see what HM output was generated. One generated output could be correct by chance, but having two match means they're very likely the same calculator.
McMillan -
3:00 marathon = 1:25:32 half marathon
5:25:36 marathon = 2:34:43 half marathon
Daniels VDOT -
3:00 marathon = 1:26:20 half marathon
5:25:36 marathon = 2:36:10 half marathon
Hansons -
3:00 marathon = 1:26:20 half marathon
5:25:36 marathon = 2:36:10 half marathon
Reigel -
3:00 marathon = 1:26:20 half marathon
5:25:36 marathon = 2:36:10 half marathon
From this, it shows that he uses a unique formula. This article (
link) from Runner's World in 2014 confirms that it is his own proprietary calculation based on data from real-world samples (not world class).
Hope this helps!