The competitors are Denver and Seattle, and as stated previously, my model favors Seattle substantially.

Super Bowl
NFC Champion AFC Champion Score Diff Win Prob Est. Point Spread
Seattle Seahawks Denver Broncos 1.041 0.739 7.7

 

Of course by this point my model has been reduced to a single factor, as there is no home field advantage in the Super Bowl and both teams are playoff experienced. Since every season 8 of the 11 games are before the Conference chanpionships and Super Bowl, the model works best for those first eight games. Still, it’s always interesting to see what the model calculates.

At least as interesting is the Peyton Manning factor, a player having the second best season of his career (as measured by adjusted yards per attempt). I thought it would be interesting to try and figure out how much of the value above average of the potent Denver Broncos attack that Peyton Manning was responsible for. We’ll start by looking at the simple ranking of the team, divided into the offensive and defensive components. Simple rankings help adapt for the quality of opposition, which for Denver was below league average.

Denver Broncos Simple Ranking Stats
Margin of Victory Strength of Schedule Simple Ranking Defensive Simple Ranking Offensive Simple Ranking
12.47 -1.12 11.35 -3.31 14.65

 

Narrowed down to the essentials, how much of the 14.65 points of Denver offense (above average) was Peyton Manning’s doing? With some pretty simple stats, we can come up with some decent estimates of the Manning contribution to Denver’s value above average.

We’ll start by calculating Peyton’s adjusted yards per attempt, and do so for the league as a whole. We’ll use the Pro Football Reference formula. Later, we’ll use the known conversion factors for AYA to turn that contribution to points, and the subtract the league average from that contribution.

Passing Stats, 2013
Player(s) Completions Attempts Yards Touchdowns Interceptions AYA
Peyton Manning 450 659 5477 55 10 9.3
All NFL passing 11102 18136 120626 804 502 6.3

 

The difference between Peyton Manning’s AYA and the league average is 3 points. Peyton Manning threw 659 times, averaging about 41.2 passes per game. This compares to the average team passing about 35.4 times a game. To convert an AYA into points per 40 passes, the conversion factor is 3.0. This is math people can do in their head. 3 times 3 equals 9 points. In a game situation, in 2013, where Peyton Manning throws 40 passes, he’ll generate 9 points more offense than the average NFL quarterback. So, of the 14.65 points above average that the Denver Broncos generated, Peyton Manning is at least responsible for 61% of that.

Notes:

There is a 0.5 point difference between the AYA reported by Pro Football Reference and the one I calculated for all NFL teams. I suspect PFR came to theirs by taking an average of the AYA of all 32 teams as opposed to calculating the number for all teams. To be sure, we’ll grind the number out step by step.

The yards term: 120626
The TD term: 20 x 804 = 16080
The Int term: 45 x 502 = 22590

120626 + 16080 – 22590 = 114116

Numerator over denominator is:

114116 / 18136 = 6.29223… to two significant digits is 6.3.

There are two well known adjusted yards per attempt formulas, which easily reduce to simple scoring models. The first is the equation  introduced by Carroll et al. in “The Hidden Game of Football“, which they called the  New Passer Rating.

(1) AYA = (YDs + 10*TDs- 45*INTs)/ ATTEMPTS

And the Pro Football Reference formula currently in use.

(2) AYA  = (YDs +20*TDs – 45*INTs)/ATTEMPTS.

Scoring model corresponding to the THGF  New Passer Rating, with opposition curve also plotted. Difference between curves is the turnover value, 4 points.

Scoring model corresponding to the THGF New Passer Rating, with opposition curve also plotted. Difference between curves is the turnover value, 4 points.

Formula (1) fits well to a scoring model with the following attributes:

  • The value at the 0 yard line is -2 points, corresponding to scoring a safety.
  • The slope of the line is 0.08 points per yard.
  • At 100 yards, the value of the curve is 6 points.
  •  The value of a touchdown in this model is 6.8 points.

The difference, 0.8 points, translated by the slope of the line,  (i.e 0.8/0.08) is equivalent to 10 yards. 4 points, the value of a turnover, is equal to 50 yards. 45 was selected to approximate a 5 yard runback, presumably.

Pro Football Reference AYA formula translated into a scoring model. Difference in team and opposition curves, the turnover value, equals 3.5 points.

Pro Football Reference AYA formula translated into a scoring model. Difference in team and opposition curves, the turnover value, equals 3.5 points.

Formula (2) fits well to a scoring model with the following attributes:

  • The value at the 0 yard line is -2 points, corresponding to scoring a safety.
  • The slope of the line is 0.075 points per yard.
  • At 100 yards, the value of the curve is 5.5 points.
  • The value of a touchdown in this model is 7.0 points.

The difference, 1.5 points, translated by the slope of the line,  (i.e 1.5/0.075) is equivalent to 20 yards. 3.5 points, the value of a turnover, is equal to 46.67 yards. 45 remains in the INT term for reasons of tradition, and the simple fact this kind of interpretation of the formulas wasn’t available when Pro Football Reference introduced their new formula. Otherwise, they might have preferred 40.

Adjusted yards per attempt or adjusted expected points per attempt?

Because these models show a clearly evident relationship between yards and points, you can calculate expected points from these kinds of formulas. The conversion factor is the slope of the line. If, for example, I wanted to find out how many expected point Robert Griffin III would generate in 30 passes, that’s pretty easy, using the Pro Football Reference values of AYA. RG3’s AYA is 8.6, and 0.075 x 30  = 2.25. So, if the Skins can get RG3 to pass 30 times, against a league average defense, he should generate 19.35 points of offense. Matt Ryan, with his 7.7 AYA, would  be expected to generate 17.33 points of offense in 30 passes. Tony Romo? His 7.6 AYA corresponds to  17.1 expected  points per 30 passes.

Peyton  Manning, in his best  year, 2004, with a 10.2 AYA, could have been expected to generate 22.95 points per 30 passes.

This simple relationship is one reason why, even if you’re happy with the correlation between the NFL passer rating and winning  (which is real but isn’t all that great), that  you should sometimes consider thinking in terms of AYA.

A Probabilistic Rule of Thumb.

If you think about these scoring models in a simplified way, where there are only two results, either a TD or a non-scoring result, an interesting rule of thumb emerges. The TD term in equation (1) is equal to 10 yards, or 0.8 points. 0.8/6.8 x 100 = 11.76%, suggesting that the odds of *not* scoring, in formula (1), is about 10%. Likewise, for equation (2) whose TD term is 20, 1.5/7 x 100 = 21.43%, suggesting the odds of *not* scoring, in formula (2), is about 20%.

Ok, this whole article is a kind of speculation on my part. DVOA is generally sold as a kind of generalization of the success rate concept, translated into a percentage above (or below) the norm. Components of DVOA include success rate, turnover adjustments, and scoring adjustments. For now, that’s enough to consider.

Adjusted yards per attempt, as we’ve shown, is derived from scoring models, in particular expected points models, and could be considered to be the linearization of a decidedly nonlinear EP curve. But if I wanted to, I could call AYA style stats the generalization of the yardage concept, one in which scoring and turnovers are all folded into a single number valued in terms of yards per attempt.

So, if I were to take AYA or its fancier cousin ANYA, and replace yards with success rate, and then refactor turnovers and scoring so that turnovers and scoring were scaled appropriately, I would end up with something like the “V” in DVOA. I could then add a SRS style defensive adjustment, and now I have “DV”. If I now calculate an average, and normalize all terms relative to my average, I’d end up with “Homemade DVOA”, wouldn’t I?

The point is, AYA or ANYA formulas are not really yardage stats, they are scoring stats whose units are in yards. So, if really, DVOA is ANYA in sheep’s clothing, where yardage has been replaced by success rate, with some after the fact defense adjustments and normalization from success rate “units”.. well, yes, then DVOA is a scoring stat, a kind of sophisticated and normalized “adjusted net success rate per attempt”.

This is something I’ve wanted to test ever since I got my hands on play-by-play data, and to be entirely  honest, doing this test is the major reason I acquired play-by-play data in  the first place. Linearized scoring models are at the heart of the stats revolution sparked by the book, The Hidden Game of Football, as their scoring model was a linearized model.

The simplicity of the model they presented, the ability to derive it from pure reason (as opposed to hard core number crunching) makes me want to name it in some way that denotes the fact: perhaps Standard model or Common model, or Logical model. Yes, scoring the ‘0’ yard line as -2 points and  the 100 as 6, and everything in between as a linearly proportional relationship between those two has to be regarded as a starting point for all sane expected points analysis. Further, because it can be derived logically, it can be used at levels of play that don’t have 1 million fans analyzing everything: high school play, or even JV football.

From the scoring models people have come up with, we get a series of formulas that are called adjusted yards per attempt formulas. They have various specific forms, but most operate on an assumption that yards can be converted to a potential to score. Gaining yards, and plenty of them, increases scoring potential, and as Brian Burke has pointed out, AYA style stats are directly correlated with winning.

With play-by-play data, converted to expected points models, some questions can now be asked:

1. Over what ranges are expected points curves linear?

2. What assumptions are required to yield linearized curves?

3. Are they linear over the whole range of data, or over just portions of the data?

4. Under what circumstances does the linear assumption break down?

We’ll reintroduce data we described briefly before, but this time we’ll fit the data to curves.

Linear fit is to formula Scoring Potential = -1.79 + 0.0653*yards. Quadratic fit is to formula Scoring Potential = 0.499 + 0.0132*yards + 0.000350*yards^2. These data are "all downs, all distance" data. The only important variable in this context is yard line, because this is the kind of working assumption a linearized model makes.

Fits to curves above. Code used was Maggie Xiong's PDL::Stats.

One simple question that can change the shape of an expected points curve is this:

How do you score a play using play-by-play data?

I’m not attempting, at this point, to come up with “one true answer” to this question, I’ll just note that the different answers to this question yield different shaped curves.

If the scoring of a play is associated only with the drive on which the play was made, then you yield curves like the purple one above. That would mean punting has no negative consequences for the scoring of a play. Curves like this I’ve been calling “raw” formulas, “raw” models. Examples of these kinds of models are Kieth Goldner’s Markov Chain model, and Bill Connelly’s equivalent points models.

If a punt can yield negative consequences for the scoring of a play, then you get into a class of models I call “response” models, because the whole of the curve of a response model can be thought of as

response = raw(yards) – fraction*raw(100 – yards)

The fraction would be a sum of things like fractional odds of punting, fractional odds of a turnover, fractional odds of a loss on 4th down, etc. And of course in a real model, the single fractional term above is a sum of terms, some of which might not be related to 100 – yards, because that’s not where the ball would end up  – a punt fraction term would be more like fraction(punt)*raw(60 – yards).

Raw models tend to be quadratic in character.  I say this because Keith Goldner fitted first and 10 data to a quadratic here. Bill Connelly’s data appear quadratic to the eye. And the raw data set above fits mostly nicely to a quadratic throughout most of the range.

And I say mostly because the data above appear sharper than quadratic close to the goal line, as if there is “more than quadratic” curvature less than 10 yards to go. And at the risk of fitting to randomness, I think another justifiable question to look at is how scoring changes the closer to the goal line a team gets.

That sharp upward kink plays into  how the shape of response models behaves. We’ll refactor the equation above to get at, qualitatively, what I’m talking about. We’re going to add a constant term to the last term in the response equation because people will calculate the response differently

response = raw(yards) – fraction*constant*raw(100 – yards)

Now, in this form, we can talk about the shape of curves as a function of the magnitude of “constant”. As constant grows larger,  the more the back end of the curve takes on the character of the last 10 yards. A small constant and you yield a less than quadratic and more than linear curve. A mid sized constant yields a linearized curve. A potent response function yields curves more like  those of David Romer or Brian Burke, with more than linear components within 10 yards on both ends of the field. Understand, this is a qualitative description. I have no clues as to the specifics of how they actually did their calculations.

I conclude though, that linearized models are specific to response function depictions of equivalent point curves, because you can’t get a linearized model any other way.

So what is our best guess at the “most accurate” adjusted yards per attempt formula?

In my data above, fitting a response model to a line yields an equation. Turning the values of that fit into an equation of the form:

AYA = (yards + α*TDs – β*Ints)/Attempts

Takes a little algebra. To begin, you have to make a decision on  how valuable your touchdown  is going to be. Some people use 7.0 points, others use 6.4 or 6.3 points. If TD = 6.4 points, then

delta points = 6.4 + 1.79 – 6.53 = 1.79 + 0.07 = 1.86 points

α = 1.86 points/ 0.0653 = 28.5 yards

turnover value = (6.53 – 1.79) + (-1.79) = 6.53 – 2*1.79 = 2.95 points

β = 2.95 / 0.0653 = 45.2 yards

If TDs = 7.0 points, you end up with α = 37.7 yards instead.

It’s interesting that this fit yields a value of an interception (in yards) almost identical to the original THGF formula. Touchdowns are more close in value to the NFL passer rating than THGF’s new passer rating. And although I’m critical of Chase Stuart’s derivation of the value of 20 for  PFR’s AYA formula, the adjustment they made does seem to be in the right direction.

So where does the model break down?

Inside the 10 yard line. It doesn’t accurately depict  the game as it gets close to the goal line.  It’s also not down and distance specific in the way a more sophisticated equivalent points model can be. A stat like expected points added gets much closer to the value of an individual play than does a AYA style stat. In terms of a play’s effect on winning, then you need win stats, such as Brian’s WPA or ESPNs QBR to break things down (though I haven’t seen ESPN give us the QBR of a play just yet, which WPA can do).

Update: corrected turnover value.

Update 9/24/11: In the comments to this link, Brian Burke describes how he and David Romer score plays (states).

Summary: The NFL passer rating can be considered to be the sum of two adjusted yards per attempt formulas, one cast in units of yards and the other using catches as a measure of yards. We show, in this article, how to build such a model by construction.

My previous article has led to some very nice emails back and forth with the Pro Football Focus folks. In thinking about ways to explain the complexities of the original NFL formula,  it occurred to me that there are two yardage terms because the NFL passer rating can be regarded as the sum of two adjusted yards per attempt formulas. Once you begin thinking in those terms, it’s not all that hard to derive an NFL style formula.

Our basic formula will be

<1> AYA = (yards + α*TDs – β*Ints)/Attempts

The Hidden Game of Football’s new passer rating is a formula of this kind, with α = 10 and β = 45. Pro Football Reference’s AY/A has an α value of 20 and a β value of 45. On this blog, we’ve shown that these formulas are tightly associated with scoring models.

Using the relationship Yards = YPC*Catches, we then get

<2> AYA = (YPC*Catches + α*TDs – β*Ints)/Attempts

Since the point of the exercise is to end up with an NFL-esque formula, we’ll multiply both sides of equation <2> with 20/YPC.

<3> 20*AYA/YPC = (20*Catches + 20*α*TDs/YPC – 20*β*Ints/YPC)/Attempts

Now, adding equations <1> and <3>, we now  have

<4> (20/YPC + 1)*AYA = (20*Catches + Yards + [20/YPC + 1]*α*TDs – [20/YPC + 1]*β*Ints)/Attempts

and if we now define RANKING as the left hand side of equation <4>, A as [20/YPC + 1]*α and B as [20/YPC + 1]*β, formula <4> becomes

RANKING = (20*Catches + Yards + A*TDs – B*Ints)/Attempts

Look familiar? This is the same form as the NFL passer  rating, when stripped of its multiplier and the additive coefficient. To complete the derivation, multiply both sides of the equation by 100/24 and then add 50/24 to both sides. You end up with

RANKING = 100/24*[(20*Catches + Yards + A*TDs - B*Ints)/Attempts] + 50/24

which is the THGF form of the NFL passer rating, when A = 80 and B = 100.

If YPC equals 11.4, then the conversion coefficient (20/YPC + 1) becomes 2.75. The relationship between the scoring model coefficients α and β and the NFL style passer model coefficients A and B become

A = 2.75*α
B = 2.75*β

Just for the sake of argument, we’re going to set alpha to 25, pretty close to  the 23.3 that we get from a linearized Brian Burke model, and beta we’ll set to 60, 6.7 yards less than  the 66.7 yards we calculated from the linearized Brian Burke scoring model. using those values, we get 68.75 for A and 165 for B. Rounding the first value to the nearest 10 and rounding B down a little, our putative NFL style model becomes:

RANKING = (20*Catches + Yards + 70*TDs – 160*Ints)/Attempts

Note that formulas <1> and <2> do not contribute equally to the final sum. Equation <2> is weighted by the factor (20/YPC)/(20/YPC + 1) and equation <1> is weighted by the factor 1/(20/YPC + 1). When YPC is about 11.4 yards, then the contribution of equation <2> to the total is about 63.6% and equation <1> adds about 35.4% to the total. Complaints that the NFL formula is heavily driven by completion percentage are correct.

Using the values α = 20 and β = 45, which are values found in Pro Football Reference’s version of adjusted yards per attempt, we then get values of A and B that are 55 and 123.75 respectively. Rounding down to the nearest 10, and plugging these values into the NFL style formula yields

RANKING = (20*Catches + Yards + 50*TDs – 120*Ints)/Attempts

Note that the two models in question have smaller A values than the core of the traditional NFL model (80) and larger B values than the traditional NFL model (100). This probably reflects the times. The 1970s were a defensive era. It was harder to score then. As it becomes harder to score, the magnitude of the TD term should increase. TD/Interception ratios were smaller in the 1950s, 1960s, and 1970s. As interceptions were more a part of the job, perhaps their effect wasn’t as valued when the original NFL formula was constructed.

Afterward: in many respects, this article is just the reverse of the arguments here. However, the proof by construction yields some useful formulas, and in my opinion, is easier to explain.

Update: more exhaustive derivation of the NFL passer rating.

The value of a touchdown is a phrase used in formulas like this one

PASSER RANKING = (yards + 10*TDs – 45*Ints)/attempts

where the first thing that comes to mind is that the TD is worth 10 yards and the interception is worth 45 yards. But is it? A TD after all, is worth about 7 points, and in The Hidden Game of Football formulation, a turnover is worth 4 points. Therefore, a TD is worth considerably more than a turnover, but the formula values the TD less. How is that?

Well, let me reassure you that in the new passer rating of the Hidden Game of Football, the value of a touchdown is a constant, equal to 6.8 points or 85 yards. The interception of 4 points is usually valued at 45 yards instead of 50, because most interceptions don’t make it back to the line of scrimmage.

The field itself is zero valued at the 25 yard line. That means once you get to the one yard line, you have one yard to go of field and the TD is worth an additional 10 yards of value. That’s where the 10 comes from. It’s not the value of the touchdown, but the additional value of the touchdown not measured on the field itself.

But what does this additional term actually mean?

Figure 1. The basic linear scoring model of THGF. TD = 6, linear slope = 0.08 points/yard. The probability of a score goes to 1.0 as the goal line is approached.

Figure 2. The model of THGF's new passer rating. The difference between y value at 100 yards and TD equals 0.8 points or 10 yards. Maximum probability of a score approaches 75/85.

If you check out the figures above, Figure 1 is introduced in The Hidden Game  of Football on page 102, and features in just about all the descriptions of worth up until page 186, where we run into this text. The authors appear to be carving out a new formula from the refactored NFL formula they introduce in their book.

Awarding a 80 yard bonus for a touchdown pass makes no sense either. It’s like treating every TD pass as though it were a 80-yard bomb. Yet, the majority of touchdown passes are from inside the 25 yard line.

It’s not the bonus we’re objecting to-after all, the whole point of throwing a pass is to get the ball into the end zone-but the size of the bonus is way out of kilter. We advocate a 10 yard bonus for each touchdown pass. It’s still higher than the yardage on a lot of TD passes, but it allows for the fact that yardage is a lot harder to get once a team gets inside the opponent’s 25.

and without quite saying so, the authors introduce the model in Figure 2. To note, the value of the touchdown and the yardage value merge in Figure 1, but remain apart in Figure 2. This value, which I’ve called a barrier potential previously, is the product of a chance to score that’s less than a 1.0 probability as you reach the goal line.  If your chances maximize at merely 80%, you’ll end up with a model with a barrier potential.

If I have an objection to the quoted argument, it’s that it encourages the whole notion of double counting the touchdown “yardage”. The appropriate way to figure out the slope of any linear scoring model is by counting all scoring at a particular yard line, or within a particular part of the field (red zone scoring, for example, which could  be normalized to the 10 yard line). These are scoring models, after all, not touchdown models.

Where did 6.8 come from, instead of 7?

Whereas before I was thinking  it was 6 points for the TD and 0.8 points for the extra point, I’m now thinking it came from the same notions that drove the score value of 6.4 for Romer and 6.3 for Burke. It’s 7 points less the value of the runback. I’ve used 6.4 points to derive scoring models for PFR’s aya and the NFL passer rating, but on retrospect, those aren’t appropriate uses. These models tend to zero in value around 25 yards, whereas the Romer model has much higher initial slopes and reaches positive values faster than these linear models.

This value can be calculated, but the formula that results can’t be calculated directly. It can be solved iteratively, though, with a pretty short piece of code

Figure 3. Perl code to solve for slope, effective TD value and y value at 100 yards in linear scoring models.

Figure 4. Solving for barriers of 10 and 20 yards.

And the solution is close enough to 6.8 that it’s easy enough to ignore the difference. Plugging 7 points for the touchdown, 20 and 29.1 yards respectively for the barrier potential yields almost no changes in the touchdown value for  the PFR aya model and the NFL passer rating formula, and we end up with these scoring model plots.

Figure 5. PFR aya amended model. TD = 7 points, slope = 0.075 points/yard, y at 100 = 5.5 points.

Figure 6. Amended NFL prf scoring model. TD = 7.05 points, slope = 0.07 points/yard, y at 100 = 5.0 points.

After the previous post in this series, I realized there is a scoring model buried within the NFL passer rating formula. Pretty much any equation of the form

RATE = (yards + a*TDs – b*(INTS + FUMBLES) – sacks)/plays

implies the existence of one of these models. Note that this form suggests a single barrier potential for touchdowns, while there equally well could be one for the 0 yardage side (“the sack side”) of the equation. To plot the one suggested by Pro Football Reference adjusted yards per attempt formula,

RATE = (yards + 20*TDs – 45*Ints)/attempts

we see this

Pro Football Reference's AYA statistic as a scoring potential model. The barrier potential represents the idea that scoring chances do not become 100% as the opponents goal line is neared.

The refactored NFL passer rating has the form

RATE = 100/24*2.75[( yards + 29.1*TDs - 36.4*Ints)/attempts]  + 50/24

when the completion and yards terms are combined using yards per completion as a constant. The term in brackets is a scoring model. To figure out the model, some algebra is needed to determine the value of the line at 100 yards.

0.291(x + 2 ) + (x + 2) = 6.4 + 2 = 8.4

1.291 x + 2.582 = 8.4

1.291x = 5.818

x ≈ 4.5

This yields a slope of 0.065, a barrier potential of 1.9 points or so, and a value for a turnover of 2.5 points. Plotted, it looks like this

NFL passer rating interpreted in terms of an internal scoring potential model.

and is not all that much different from the implied model in the PFR aya formula.

To get to the idea that the barrier potential represents a difference between a model that allows a 100% chance to score, and a model that has an imperfect chance of scoring, we’re going to build a scoring potential model from just a single data point. Understand, as a line has two points, and -2 at 0  yards is generally assumed, the slope of the line can be determined by solving for the expected points at a single yard line.

If on first down at the 1 yard line, you have an 80% change of scoring a touchdown and a 15% chance of scoring a field goal, and a 5% chance of just losing possession, then solving for the expected points on first and one,  you get

expected points = 0.8*6.4 + 0.15*3 = 5.57 points

value of yards at 100 = 5.57*100/99 ≈ 5.63 points

barrier potential = 6.4 – 5.63 = 0.77 points =  10.1 yards

turnover value = 5.63 – 2 = 3.63 points ≈ 47.6 yards

and expressed as a passer ranking formula, you might get something like

RATE = (yards + 10.1*TDs – 48*Int)/attempts

and plotted, look something like this:

Scoring potential model derived from assuming 80% chance of TD and 15% of FG on first and one.

The synthetic first and one data above differ little from the real first and one data given here, but PFR’s adjusted yards per attempt is a formula that averages data over all downs, as opposed to being the data for a single down.

Conclusions

The size of the barrier potential is a measure of how hard it is to score. The smaller the barrier potential, the easier it is to score. When the barrier potential is zero, scoring approaches 100% as the team approaches the goal line. Therefore, in more realistic scoring models, barrier potentials tend to appear.

It is entirely possible that the larger barrier potentials of the NFL passer formula merely reflect the times in which the model was created. The 1970s was an era dominated by defense and a running game. It was harder to score then. It would be interesting to calculate scoring rates for first and one situations from, say, 1965 to 1971, when the NFL passer formula was created, and see if the implied formula actually matches the data of the times.

Other issues these models suggest: since they are easy to construct with very modest data sets, they can be individualized for college and high school conferences, leagues, and even teams. They suggest trends that can be useful for analyzing particular times and ages. Note that as scoring gets harder and barrier potentials grow larger, the value of  the turnover grows less. It’s not that hard also, to set up an equation representing a high scoring team with one that doesn’t score much at all. Since the slope  of the line of the low scoring team is less than that of the high scoring team, turnover value becomes dependent on field position, as the slopes don’t cancel. The turnover becomes more valuable towards the goal line of the low scoring team.

It’s a simple function of algebra, that two variables, related by a constant, are really only one independent parameter. Mixing the two variables in the formula really means only one is actually important, and if you add this kind of misbuilt formula into a nonlinear least squares curve fitter, usually the covariance between these terms will calculate out to a value of 1. As Brian Burke has pointed out here, there is a relationship between yardage and completions in the NFL.

yardage  = completions x yards per completion

This is used as a fundamental part of the argument against  the NFL passer rating, usually stated in the form “completions are counted twice“. But is that  true? The more compelling notion to me is that if yards per completion is a de facto constant, there really is only one independent variable here, not two. And if so, no one should care which one of the two is actually used.

One of the nice thing about Sports Reference sites are their consistent use of tables that allow users to sort data along a column of interest. So if we go to the Pro Football Reference 2010 passer stats, and sort the Y/C column, we get this result:

Neat, huh? The highest value of Y/C is about 13.2, the smallest about 9.9 and the median has to be about 11.8 or so. Interesting how much of the data set is encompassed by the value 11.5 ± 1.5. Just playing with these numbers by eye, we end up with a chart of maxima, minima, and median values over the last 4 years of:

Year Maximum Minimum Median
2010 13.3 9.9 11.8
2009 13.4 9.8 11.4
2008 13.4 8.6 11.4
2007 12.7 9.7 11.3

If you then take every NFL quarterback who had 100 or more completions from 2007 to 2010 and calculate the average YPC and the standard deviation of that value, you get 11.41 YPC ± 0.92. A physicist might not see that as a constant, but in the biological sciences, a relative error of 8% is a pretty tightly determined value. And if we repeat the calculation from 2001 to 2010,  then we get 11.40 YPC ± 0.96.

In the modern context,  you just about could rewrite the NFL passer formula to be

RATE = 100/24 * [ (Completions * 31.4 + Tds * 80 - ints * 100)/attempts] + 50/24

or

RATE = 100/24 * [ (2.75*yards + Tds * 80 - ints * 100)/attempts] + 50/24

That wasn’t true back in 1971, when the passer formula was invented. The spread of values in YPC was considerably wider.

The formula hadn’t quite degenerated yet. There could be passers who threw for lots of completions or passers who threw really long passes. The evolution of the pass rush and pass rushers hadn’t placed such an emphasis on shorter drops and quicker patterns in that day and age.

More mathematical transformations.

Let’s take the second form of the NFL formula above, throw away that useless constant and useless first multiplier and divide the remaining core by 2.75, to scale everything to  units of yards. Please remember that in THGF, a yard has a linear value with regard to expected points, and 1 yard = 0.08 points. Interceptions were deemed to be worth 4 points. Anyway, the formula becomes:

CORE RATE = (yards + 29.1*TD – 36.4*Int)/attempts

The value 36.4 yards comes out to 2.9 points, via the THGF scale, and a touchdown valued at 29.1 yards is just about 2.3 points of value. The NFL passer formula, transformed in this way, is not all that far removed from Pro Football  Reference’s adjusted yards per attempt (see also here). I hope this kind of explanation might help people understand why  the old dog of a formula retains a useful core that actually tracks wins fairly well.

Aside: please note that more sophisticated treatments of data show a nonlinear relationship between net expected points and yards to go, and on those terms, the value of an interception becomes dependent on field position.

Update: link and grammar fixes.

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