Last year, I ran my first marathon ever (at 3:07:52, in the fairly hilly Oslo course), using the FIRST marathon program (which, despite the name, is not necessarily meant for beginners). This year, as the Covid-19 lockdowns started, I decided to go for it again using the same program, but there was one annoyance; I wanted to change target times as it became obvious my initial target got too easy, but there's no way to calculate it electronically.
FIRST comes in the form of a book; you can find an older version of the 10K and marathon programs if you search a bit online, but fundamentally, the way it works is that you declare a 5K personal best (whether true or not), look up a bunch of tempos in a table in the book from that, and then use that to get three runs every week. (You also do cross-training and strength training, or at least that's the idea.) For instance, the book might say that this week's track intervals are 6x 800m, so you go look up your 800m interval times in the table. If you have a 5K PB of 19:30, the book might say that 800m interval times are 2:52 (3:35/km), so off you go running.
The tables are never explained, and they don't match up with the formulas that were published in the earlier versions. There's at least one running calculator that can derive FIRST paces, but it defaults to miles and has a different calculation for marathon pace (which sometimes creates absurd situations like “long tempo” being slower than “marathon pace”), so I really wanted to just get the formulas to input into my own spreadsheets.
Enter regression. I just typed in a bunch of the tables, graphed them, saw that everything was on a dead straight line (R=1.00 for linear regression) and got the constants from there. So without further ado:
If you can run 5K at x seconds per kilometer, the Holy Gospel of FIRST declares that you can run 42.195K at 1.15313x seconds. (I am sure there are more sophisticated models, but perhaps this is good enough?) Incidentally or not, this means an 18:30 5K becomes nearly exactly three hours on a marathon (only two seconds away). (I didn't bother with the 10K and half-marathon estimation paces; there are so many numbers to input).
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That leaves only the track repeats. For this, first take the 5K pace and multiply by 1.00579, leaving what I will call the “reference pace” (RP). I don't know if this constant carries any particular meaning, and obviously, it's nowhere in the book; it's just the slope of the regression. 400m time is 400m at RP, minus 10 seconds. (That is 10 seconds absolute time, not 10 seconds/km. So if you have an 18:30 5K PB, you'll have an 18:36 5K at RP, which is 1:29 400m at RP, which then gives a 1:19 400m.)
Similarly: 600m is -13 seconds, 800m is -16 seconds, 1000m is -18 seconds, 1200m is also -18 seconds, 1600m is -16 seconds, and 2000m (which is specified, but seemingly never used in any of the programs) is -15 seconds. You can see two effects going against each other here; longer intervals mean more seconds to shave off for a given pace, but they also give lower pace, and thus the U-like shape.
And that's all there is to it. Happy running, and may there be a good race close to you!