Summary: Replacing Wentz with Foles removes about 6.5 points of offense from the Philadelphia Eagles, turning a high flying offense into something very average.

Last night the Atlanta Falcons defeated the LA Rams. Now we’re faced with the prospect of the Falcons playing the Eagles. I have an idiosyncratic playoff model, one I treat as a hobby. It is based on static factors, the three being home field advantage, strength of schedule, and previous playoff experience. And since it values the Eagles as 0.444 and the Falcons as 1.322, the difference is -0.878 (win probability in logits). The inverse logit of -0.878 is 0.294, which is the probability of the Eagles winning, and an estimated point spread would be a 6.5 point advantage for the Falcons.

Another question that a Falcons or Eagles fan might have is how much is Carson Wentz worth as a QB, in points scored? We can use the adjusted yards per attempt stat of Pro Football Reference to estimate this, and also to estimate how much Carson Wentz is better than Foles. We have made these kinds of analyses before for Matt Ryan and Peyton Manning.

Pro Football Reference says that Carson Wentz has a AYA of 8.3 yards per attempt. Nick Foles has a AYA of 5.4. Now lets calculate the overall AYA for every pass thrown in the NFL. Stats are from Pro Football Reference.

(114870 yards + 20*741 TDs – 45*430 Ints) / 17488 Attempts
(114870 yards + 14820 TD “yards” – 19350 Int “yards”) / 117488 Attempts
110340 net yards / 17488 yards
6.31 yards per attempt to three significant digits

So about 6.3 yards per attempt. Carson Wentz is 2 yards per attempt better than the average. Nick Foles is 0.9 yards less than the average. The magic number is 2.25 which converts yards per attempt to points scored per thirty passes. So Carson, compared to Foles, is worth 2.9 * 2.25 = 6.5 points per game more than Foles, and 4.5 points more than the average NFL quarterback.

This doesn’t completely encompass Carson Wentz’s value, as according to ESPN
‘s QBR stat
, he account for 10 expected points on the ground in 13 games, so he nets about 0.8 points a game on the ground as well.

Now, back to some traditional stats. The offensive SRS assigned to Philadelphia by PFR is 7.0 with a defensive SRS of 2.5. If Carson Wentz is worth between 6.5 and 7.3 points per game, then it reduces Philadelphia’s offense to something very average, about 0.5 to -0.3. That high flying offense is almost completely transformed by the loss of their quarterback into an average offense.

Note: logits are to probabilities as logarithms are to multiplication. Rather than multiplying probabilities and using transitive rules, you just add the logits and convert back. Logarithms allow you to add logarithms of numbers rather than multiplying them.

Summary: with some calculations based on adjusted yards per attempt, Matt Ryan’s value as a passer in the 2016 season can be shown to be almost 9 points a game more than the average QB.

Mark Zinno is a host on a sports talk show, 92.9 the Game, in the 7pm ET time slot. Often booted out of the slot by Atlanta Hawks games, he nonetheless has been a dogged supporter of Matt Ryan. This isn’t new, btw. Even in years where Matt Ryan wasn’t at his best, he would doggedly argue that Matt Ryan was an elite quarterback, and said repeatedly that compared to an average NFL team, that Atlanta was blessed.

So, we’re dedicating this blog post to Mark Zinno.

It’s hard to understand the scope of what Matt Ryan has done until you look at his adjusted yards per attempt in 2016. Pro Football Reference lists it as 10.1, which is one of the highest I’ve seen, and comparable to Peyton Manning’s 2004 season, where PM’s AYA was 10.2. Looking a little further, you can see that PFR ranks this the 4th best performance in history. Aaron Rogers is in the top 4, and for some reason, so is Nick Foles.

The value in using AYA is that you can build an expected points curve that satisfies all the requirements of the AYA function, and then use the slope of that curve to relate yards to points. Don’t worry, I did that long ago, and the result is documented here. The simple take home is the magic conversion 2.25, which converts AYA from yards to “expected points generated per 30 passes”.

Then, using the 2016 annual data from Pro Football Reference, you can calculate  what the average QB did, by calculating an AYA using the overall season’s statistics.  So the formula is:

(123639 yards + 20*786 TD – 45*415 Ints)/  18295 attempts

(123639 yards + 15720 “TD” yards – 18675 “Int” yards) / 18295 attempts

120684 yards / 18295 attempts

6.60 AYA to 3 significant digits.

Now things become simpler. Matt Ryan generated 10.1*2.25 = 22.7 points per 30 attempts, while Joe QB generated 14.8 points per 30 attempts. The difference, rounded to a whole number, suggests that Matt Ryan was worth about 8 more points in 30 attempts than the average NFL QB this season.

That doesn’t entirely encompass his per game value. Matt threw 534 attempts  this season for an average of 33.4 passes per game. So his per game value, to the nearest tenth of a point, was more like 8.8 points a game more than the average quarterback.

But if the numbers baffle you, then the simple take home is that Matt’s statistical efficiency in 2016 is comparable to the best single season Peyton Manning ever had.

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.

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.

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.

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. 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.