The value of a turnover is a topic addressed in The Hidden Game of Football, noting that the turnover value consists of the loss of value by the team that lost the ball and the gain of value by the team that recovered the ball. To think in these terms, a scoring model is necessary, one that gives a value to field position. With such a model then, the value is
Turnover = Value gained by team with the ball + Value lost by team without the ball
In the case of the classic models of THGF, that value is 4 points, and it is 4 points no matter what part of the field the ball is recovered.
That invariance is a product of the invariant slope of the scoring model. The model in THGF is linear, the derivative of a line is a constant, and the slopes, because this model doesn’t take into account any differences between teams, cancel. That’s not true in models such as the Markov chain model of Keith Goldner, the cubic fit to a “nearly linear” model of Aaron Schatz in 2003, and the college expected points model (he calls his model equivalent points, but it’s clearly the same thing as an expected points model) of Bill Connelly on the site Football Study Hall. Interestingly, Bill’s model and Keith’s model have a quadratic appearance, which guarantees better than constant slope throughout their curves. Aaron’s cubic fit has a clear “better than constant” slope beyond the 50 yard line or so.
Formula with slopes exceeding a constant result in turnover values that maximize at the end zones and minimize in the middle of the field, giving plots that Aaron calls the “Happy Turnover Smile Time Hour”. As an example, this is the value of a turnover on first and ten (ball lost at the LOS) for Keith Goldner’s model
And this is the piece of code you can use to calculate this curve yourself.
Note also, the models of Bill Connelly and Keith have no negative expected points values. This is unlike the David Romer model and also unlike Brian Burke’s expected points model. I suspect this is a consequence of how drives are scored. Keith is pretty explicit about his extinction “events” for drives in his model, none of which inherit any subsequent scoring by the opposition. In contrast, Brian suggests that a drive for a team that stalls inherits some “responsibility” for points subsequently scored.
A 1st down on an opponent’s 20 is worth 3.7 EP. But a 1st down on an offense’s own 5 yd line (95 yards to the end zone) is worth -0.5 EP. The team on defense is actually more likely to eventually score next.
This is interesting because this “inherited responsibility” tends to linearize the data set except inside the 10 yard line on either end. A pretty good approximation to the first and ten data of the Brian Burke link above can be had with a line that is valued 5 points at one end, -1 points at the other. The value of the slope becomes 0.06 points, and the value of the turnover becomes 4 points in this linearization of the Advanced Football Stats model. The value of the touchdown is 7.0 points minus subsequent field position, which is often assumed to be 27 yards. That yields
27*0.06 – 1.0 = 1.62 – 1.0 = 0.62 points, or approximately 6.4 points for a TD.
This would yield, for a “Brianized” new passer rating formula, a surplus yardage value for the touchdown of 1.4 points / 0.06 = 23.3 yards.
The plot is below:
Update 9/29/2011: No matter how much I want to turn the turnover equation into a difference, it’s better represented as a sum. You add the value lost to the value gained.