Last week, Brian Burke provided some excellent data on the value of a first down. I began working on today’s post last offseason, but as you’ll see in a few minutes, I wasn’t quite comfortable with the results. But here’s what I did.

For all teams from 1989 to 2012, I recorded for each team’s average:

• Yards per carry;
• Touchdowns per carry;
• First downs per carry;
• Rush VOA from Football Outsiders (DVOA is FO’s main statistic, but it is adjusted for SOS; VOA is the unadjusted metric).

Then, I ran a series of regressions to help better understand the “proper” weights on the running game. ^[1]Longtime readers may recall that Neil did something similar with passing metrics. First, I used yards per carry and touchdowns per carry as my input, and VOA as my output. The best-fit formula was:

-0.647 + 0.128 * YPC + 4.615 * TD/Carry

Understanding that formula isn’t important. ^[2]Although if you’re curious… Suppose a team averaged 4.50 YPC and rushed for 12 TDs on 450 carries. That team would have a VOA of 5.21%, slightly above average. What we care about is the correlation coefficient (0.65) and the relationship between the YPC variable and the TD/carry variable.

Here we run into our first problem: 4.615 is 36 times as large as 0.128. This would imply that a touchdown is 36 times as valuable as a regular yard (or 35, if you subtract the yard gained on the score). That seems very high, as 20 is the generally accepted standard conversion rate for a touchdown.

What if we introduce first downs per carry into the equation? Then we get this best-fit formula:

-0.761 + 0.081 * YPC + 2.954 * TD/Carry + 1.593 * FD/Carry

Here, the R^2 is 0.72, which is an indication that first downs matter (or, to put it another way, DVOA gives rewards for first downs, as it should). Unfortunately, the TD variable remains very high (it’s now nearly 37 times as large as the YPC variable), but we get the bit of insight I was looking for: a first down is worth 0.54 touchdowns.

The first down variable is about 20 times as large as the YPC variable, but that seems way off to me.  Instead, if we think a TD is really worth 20 yards, this puts the value of a first down at about 10.8 yards. That’s not too far off from Brian’s 8.7 average, which I think makes sense to round to 9.0 once you remember that a few first downs happen on 4th downs, which was ignored in Brian’s analysis.

What helps bridge the gap between the two valuations?  Burke’s method looks at the marginal value of a first down based on an Expected Points model.  What I did with Football Outsiders’ numbers was to try to correlate first downs with rushing value.  But that could lead to overstating the value of a first down, if first downs are correlated with other things (like say, short-yardage success in general).  Teams that are good at rushing for first downs might be better at X, Y, and Z than other teams, and being good at X, Y, and Z could lead to a higher DVOA grade.  As a result, I like Brian’s result better, and I think FO’s numbers serve as a good gut check.  And while the number is a few yards higher, that’s arguably the correct result, and at least the number is higher in the right direction.

As for the idea that touchdowns are worth 36 yards and first downs are worth 19 yards? Well, I’m not so sure about that.  My hunch though, is that Football Outsiders (wisely) cares a lot more about say, success rate, than just generic yards per carry.  So perhaps this represents a devaluation of YPC as much as anything else.

But I’ll open it up to the crowd: what thoughts do you guys have for finding the value of a first down?