September 2011

The latest and greatest. I’m still curious how deep into the season we’ll get before the optimal pythagorean exponent will fall below 3. Oakland, for now, is the SRS leader because of an exceptional strength of schedule.

I can now calculate a Homemade Sagarin, but not happy with the results. Perhaps next week.

Median is the median point spread, Pred is the fitted pythagorean expectation, SRS, MOV, and SOS are the simple ranking components.

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.

When I was an undergrad at the University of Guam, all the science majors hung out in the Biology Department office. In part, this was because some of the biologists had licenses to fish and scuba outside the coral reef of Guam, and so you never knew what would be dragged into the building. Another reason was a small but efficient library of science books, one of which was by George Gamow. I wish I recalled the title, as one topic in this book had a powerful influence on me.

It discussed dimensional analysis, and showed an example of using dimensional analysis to derive a formula for some physical process. I’ve long forgotten the analysis and the page, but it left an indelible impression of  the power of accurately accounting for the  physical dimensions of the components of a formula.

On August 15th, Pro Football Focus introduced a new passer rating formula. It is:

Ranking = 4.66667*[ 20*Completions + 20*Drops + Yards in Air +20*Tds – 45*Ints ]/(Attempts – Spikes – Throw Aways)

There are some interesting ideas in this formula, but it seems seriously flawed from my point of view. Complaints in order are:

1. It is double counting yards.

2. It is trying to add two different kinds of yardage metrics in the same formula.

3. It doesn’t seem to understand the origin of the TD and interception terms it actually is using.

4. Items 1 and 3 interact in ways that I suspect the author never intended, yielding a scoring model that seriously undervalues turnovers.

We’ll address each of these issues in turn. As Brian Burke has pointed out and we’ve discussed in more detail here, completions and yardage are related  through the equation yardage = completion*yards per completion. If we note that YPC in the modern NFL is actually 11.4 yards, within a relative error of 9%, the first two terms in the numerator can be rewritten:

20/11.4*[ Yards + Extra Yards] = 20/11.4*Equivalent yards = 1.75*U*Yards

Yards is equal to 11.4*Catches. Extra Yards would be defined as 11.4*Drops, and is equal to the yards a QB would have gotten if  those passes hadn’t been dropped. The sum 11.4*(Catches + Drops) can be defined as Equivalent Yards, the total yards a QB would have gotten without any dropped passes. U, a dimensionless parameter, is Equivalent Yards/Yards. U, pretty much by definition, is greater than or equal to 1.0.

The third term in the numerator, by contrast, is Yards in the Air, the yards a QB is responsible for, or Yards – Yards after the catch. If V is YIA/Yards, then V is a dimensionless positive valued term less than 1. So, not only are there two yardage terms, there are two different kinds of yardage terms. This touches on items 1 and 2. Item 3 will be discussed in a footnote.

To get to item 4, the yardage components in this formula can be combined into a term like this:

20*Completions + 20*Drops + YIA = [1.75*U + V]*Yards

Leading to a numerator like this

4.6667*[ (1.75*U + V)*Yards +20*TDs -45*Ints]

whose functional scoring model becomes this:

(Yards +20/[1.75*U + V]*Tds -45/[1.75*U + V]*Ints)/Equivalent Attempts

I don’t think that was the intended result of the author of this model.

I suspect that U is in the vicinity of 1.1 and V, who knows? Call it 0.5 for the sake of argument.  The term  1.75U + V = 2.425 (which might as well be 2.4) and the core formula then becomes

Yards + 8*Tds – 19*Ints/Equivalent Attempts

So to ask the question that occurs to me, does the author think an interception is only worth about 2 points?


My gut feeling is that this is a formula trying to do too many things. You don’t want to add two different kinds of yardage metrics. So, initially, either dropping the completion + drops terms or getting rid of the YIA terms would yield a formula logically and algebraically sound in its treatment of yardage. A formula like

[11.4*(Completions + Drops) + 20*TDs – 45*Ints]/Equivalent Attempts


[YIA + 20*TDs – 45*Ints]/Equivalent Attempts

or better yet, since Brian Burke’s expected points formulas linearize to a surplus value for TDs of 23.3 yards, and the value of a turnover in yards is about 67 yards, use this:

[YIA + 23.3*TDs – 60*Ints]/Equivalent Attempts [1]

An even better formula, since PFF must have excellent data on how many yards an interception is run back, would be:

(YIA + 23.3*TDs – [ 67 – average net field position relative to original LOS]*Ints)/Equivalent Attempts [2]

So there you have it. With a little work, PFF can have a self consistent formula encompassing many of the new ideas they wish to add to a modern passer rating.

Update 9/27/2011: just noted that average YPC I previously calculated is actually 11.4 ± 0.96, instead of the originally published 14.7. Correcting the math  (which I’ve done) doesn’t affect the argument.


[1] I say this because Chase Stuart’s “derivation” of 20 yards, while it turns out to be a fairly good number, goes through too  many concepts that do not make sense in a world where football is treated as a Markov chain, or alternatively, a finite state machine. Seriously, does anyone believe yardage gained running and yardage gained passing differ? That completely breaks the notion of path independence in a Markov chain. Further, as we explain here and here, the idea that the TD term is “the value of the touchdown” is broken. It’s not something you can measure on the field by calculating, say, the net value of a touchdown relative to the one yard line, as it’s related to total scoring (i.e. TDs plus field goals) of all kinds.

Likewise, the 45 yard term for the interception is based on on the THGF model.  It’s the THGF value of a turnover (4 points or 50 yards) less the net value of field position after the runback (estimated at 5 yards beyond the original LOS).

[2] I’m hesitant to point this out, but yet another variation on these formulas would be to use the dimensionless parameter U or the dimensionless parameter V as a multiplier into the yardage term. Something like

U*YIA or V*11.4*(Catches + Drops)

comes to mind. Just, you’re not really measuring what was actually left on the field, in these instances. You’re measuring what could have been. The use solely of YIA appeals to me,  if the idea is to have a formula that measures the quarterback’s real contribution to scoring.

Update 9/29/2011: U simplifies to (Catches + Drops)/Catches, and as such, U*YIA has a particularly simple, appealing form.

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

First and ten turnover value from Keith Goldner’s Markov chain 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:

Eyeball linearization of BB’s EP plots yield this simplified linear scoring model. The surplus value of a TD = 23.3 yards, and a turnover is valued 66.7 yards.

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.

The earliest possible point you can calculate a simple ranking is in week 2, so I waited for a complete set of games to be played and obtained this result:

2011 Week 2 stats

Note that the New York Jets are the highest ranked squad right now, and since the strength of schedule metric is heavily weighted by a single bad team, both Philadelphia and Pittsburgh are suffering substantially for having played Saint Louis and Seattle, respectively. Median is the median point spread, Pred is the fitted pythagorean expectation, SRS, MOV, and SOS are the simple ranking components.

In the jpeg below, there are some useful 2010 NFL stats.

2010 NFL metrics

Median is the median point spread from 2010. HS is Brian Burke’s Homemade Sagarin metric. I’m not as fond of either of these as I was when I was implementing them. I think that an optimized Pythagorean expectation is a more predictive metric than either of those two. Pythagoreans are in the PRED column, expressed as a winning percentage. Multiply the percentage by 16 to get predicted wins for 2011. SRS, MOV, and SOS are Pro Football Reference’s simple ranking system metrics. SOS is a factor in playoff wins, along with previous playoff experience. Home field advantage is calculated from the Homemade Sagarin metric. Take it for what it’s worth. Other topside metrics are calculated with the Perl CPAN module Sport::Analytics::SimpleRanking, which I authored. The HS was implemented using Maggie Xiong’s PDL::Stats.

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.

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