We’ll start by quoting the Twitter thread between Chris Brown and Trent Dilfer, over the phrase “arm talent” (Some light editing, to improve clarity):

Smart Football ‏@smartfootball 26 Jan — Can we drop the phrase “arm talent”? What happened to “strong arm”?

Trent Dilfer ‏@TDESPN — @smartfootball “Arm talent”, a phrase I started using encompasses the ability to change speeds, trajectory & off balance. Strong=meaningless

Smart Football ‏@smartfootball 26 Jan — @TDESPN Not just you. Just not my favorite phrase; not very descriptive. Understand need for all those things but not sure “arm talent”

Smart Football ‏@smartfootball 26 Jan — @TDESPN Does it. “Talented passer” more descriptive (and grammatical) than “arm talent.” Also to be fair you define the term usually, but

Smart Football ‏@smartfootball 26 Jan — @TDESPN most scout and wannabees throw it around without any backing or understanding. Fine for you to say it’s your term for all of that

Trent Dilfer ‏@TDESPN 26 Jan — @smartfootball understandable, football phrases can be very polarizing if the picture they paint doesn’t make sense. fun conversation

Smart Football ‏@smartfootball 27 Jan — @TDESPN I agree – thanks for engaging. Wasn’t targeted at you and this is partially me as football guy and also as overly pedantic lawyer

Smart Football ‏@smartfootball 27 Jan — @TDESPN as you know, I think you do great work. Look forward to future discussions

I’ll note the phrase “exceptional control” seems to cover what Trent is trying to get at with the phrase “arm talent” as well. And this gets us back to an issue often seen in both coaching and fan circles. Ideas aren’t always born in the minds of the best writers. Some very ordinary folks come up with original, profound, or perhaps just useful concepts and they end up expressing them a little awkwardly. I can’t help but wonder how much more penetration the modern analysis of play by play data would have if we didn’t have to deal with awkward, and sometimes confusing language. If Brian Burke had used the Bill James phrase “Win Shares” instead of Win Probability Added and perhaps “Net Points” instead of Expected Points Added, how much faster would his analysis been assimilated?

In my discussions of expected points curves, I can get gnarled up in the phrase, “the value of a touchdown”, because that has two distinct meanings, depending on your point of view. If you’re thinking about adjusted yards per attempts formulas, that term refers to what I’ve called a “barrier potential” in other contexts. It’s considered the value of the touchdown because of some unfortunate language and usage in The Hidden Game of Football.

The other notion called the “value of a touchdown” is the average score of a touchdown (7 points in general, by logic well discussed here) minus the yardage value of the average kickoff return. For years this was around 6.3 to 6.4 points, because the average return was to about the 27 yard line. This term has to be larger now, with the recent adjustments to the kick off line. This value has meaning in expected points curves, and the Romer/Burke model explicitly uses this notion of the value of a touchdown.

Hopelessly generic terms

The one driving me nuts these days is 5-technique, applied in sloppy fashion to a defensive end of any kind. The term gets used whether or not the defensive end is actually using a 5 technique (on the outside shoulder of the OT), a 4 technique (directly opposing the OT) or a 3 technique (outside shoulder of the OG ). Especially in drafting circles, people start talking about 5 techniques as a draftable position, as opposed to a place you line up and a way you play. More accurate, in drafting circles, would be to talk about ends capable of one or two gap technique, instead of this “5” nonsense.

The “5” nonsense is getting bad enough that confusion is being sold as fact. Despite Jene Bramel’s excellent work on the topic of where defensive linemen line up

Image taken from Jene  Bramel article, Fifth down blog. Standard alignments shown. Note when DL "on" a player, the numbers are even (0,2,4, etc).

Image taken from Jene Bramel article, Fifth down blog. Standard alignments shown. Note when DL “on” a player, the numbers are even (0,2,4,6 etc).

and with the comment:

In a majority of systems, even numbers denote an alignment that is head-up or helmet-to-helmet on an opposing offensive lineman while odd numbers denote an offset alignment, i.e. over the inside or outside shoulder of an opposing lineman.

Pro Football Focus just had to go and mess it up.

Screen capture of the link above. Note the numbering of the "on" positions goes 0,2,5,8. This would  not happen but for the loss of meaning of the phrase "% technique"

Screen capture of the link above. Note the numbering of the “on” positions goes 0,2,5,8. This would not happen but for the loss of meaning of the phrase “5 technique”.

People *pay* to be told these kinds of explanations?

Eagle Defense

John T Reed has plenty to say about the term Eagle Defense in his Football Dictionary, finally concluding that:

After looking it up in several books, I have a sense that the Eagle defense generally has something to do with shifting the defensive tackle or end outside the weak tackle or tight end and putting a linebacker over or on the weak tackle or tight end. Until the football coaching world gets more precise and consistent, the word “eagle” should be dropped.

It all started when Bart Hubbuch @NYPost_Hubbuch started riffing on Pro Football Focus’s exclusion of Tony Romo in  their top 100 (Top 100 shows are popular now. It’s a way to get some buzz after the draft and before training camp). Somehow this became a back and forth between Hubbuch and Clarence Hill (@clarencehilljr) of the Fort Worth Star Telegram. I piped in, mostly because I don’t care much for the Pro Football Focus passer formula (original discussion here, a cleaner narrative here). Later my clutch of fan contacts made it clear to me that the PFF passer formula ranked Romo #6, so that wasn’t the reason for his exclusion from PFF’s top 100.

But  this somehow morphed into a discussion of the whole ‘loss of faith’ narrative, how during crucial situations, Jason Garrett wasn’t letting Tony Romo ‘do his thing’. The notion is that Jason lost faith in Romo and that two circumstances, in particular, demonstrated  that Jason didn’t trust Tony. The first was a shovel pass during the NE game, and the second was the end of regulation in Arizona.

I’ll note these topics are difficult thing to discuss in the limits of Twitter, and not least in that the games that Bart was arguing about I never saw. I had real life issues on those days, and had no eyeball memory of  either game. I followed some of the New England game on NFL.com towards the end, and never saw any of the Arizona game, but looked a bit at some play by play later.

As a term like ‘loss of faith’ implies some understanding of the mind of Jason Garrett, it can’t be disproven. Likewise, it can’t be demonstrated or proven either. Someone with familiar with Jason, someone more like Clarence Hill Jr, has a better chance to infer states of mind, but proof?

Bart writes articles for the blue collar newspaper in NYC, and pushes an emotional  hook in many of his articles.  As an example, I see words like ‘dirtbag’ used to describe the Falcons offensive line. I’m more an analytics guy,  happier when I can let the numbers tell the story. I don’t need charged rhetoric to make the case for me.

So it was interesting. Bart is missing some points in his determination to ‘demonstrate’ his case. Strategically,  the end of the Arizona game is almost a carbon copy of the end of the 2nd Washington game. Same tactics, same strategy, just badly performed. What Bart sees as a loss of confidence in Romo I see as a poor 2 minute drill with Jason Garrett relying too much on kickers.

The odd call in the New England game.. that wasn’t an end of game call, was it? It was late in the second quarter, if I recall. And to note, Jason Garrett is capable of some really odd calls at crucial times of the game: witness his reverses in red zone situations, where it’s pretty obvious the opposing team will be shooting gaps. And a lot of that can be ascribed to Jason’s newness as a head coach — perhaps.

But, to make the important point, was the game on  the line in the second quarter? The NE-DAL game could have been won or lost on single plays at multiple points in the game. The  more appropriate comparison – Arizona for that matter as well – is to the first New York Giants – Dallas game, which turned on the execution of single plays.

In my mind, all three games are too closely played to lend themselves to a dissection of the Romo/Garrett relationship. You might as well ask what Tom Landry thought of Bob Hayes after  the 1966 championship game, when he was left to block David Robinson.

To ask and answer one last question: if Bart has a more thematic, emotional interest in football, and I have a more analytic interest, why even follow the guy? I’d argue that he’s a worthwhile follow because of his live tweeting of events. His reactions seem genuine, not as emotionally charged as his longer articles or more considered narratives. In moments like those, he’s just a guy, one with .. media credentials, who can get into events I can’t. In those moments, having his feed is like  having another pair of eyes. I’m grateful for the immediacy of his reactions.

Update: late at night on May 5, Bart replied in substance to this post in a series of tweets. We’ll quote the 4 most pertinent of these, reordered so they read like a paragraph:

Saw your blog post. NE play was 5:22 left in 4th — not 2nd. 3rd & 5 from NE 5 in a tie game, JG calls shovel to Choice. Cowboys settle for FG and 16-13 lead, giving Brady more than enough time to produce 20-16 win. Also, “dirtbag” was a quote from Justin Tuck about the Falcons, not my invention. Lastly, I’m from Dallas, used to cover Cowboys for the Dallas Morning News and have known Clarence Hill for 20 years.

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.