Odds for the 2015 NFL playoff final, presented from the AFC team’s point of view:

SuperBowl Playoff Odds
Prediction Method AFC Team NFC Team Score Diff Win Prob Est. Point Spread
C&F Playoff Model Denver Broncos Carolina Panthers 2.097 0.891 15.5
Pythagorean Expectations Denver Broncos Carolina Panthers -0.173 0.295 -6.4
Simple Ranking Denver Broncos Carolina Panthers -2.3 0.423 -2.3
Median Point Spread Denver Broncos Carolina Panthers -5.0 0.337 -5.0


Last week the system went 1-1, for a total record of 6-4. The system favors Denver more than any other team, and does not like Carolina at all. Understand, when a team makes it to the Super Bowl easily, and a predictive system gave them about a 3% chance to get there in the first place, it’s reasonable to assume that in that instance, the system really isn’t working.

So we’re going to modify our table a little bit and give some other predictions and predictive methods. The first is the good old Pythagorean formula. We best fit the Pythagorean exponent to the data for the year, so there is good reason to believe that it is more accurate than the old 2.37. It favors Carolina by a little more than six points. SRS directly gives point spread, which can be back calculated into a 57.7% chance of Carolina winning. Likewise, using median point spreads to predict the Denver-Carolina game gives Carolina a 66.3% chance of winning.

Note that none of these systems predicted the outcome of the Carolina – Arizona game. Arizona played a tougher schedule and was more of a regular season statistical powerhouse than Carolina. Arizona, however, began to lose poise as it worked its way through the playoffs. And it lost a lot of poise in the NFC championship game.

Odds for the third week of the 2015 playoffs, presented from the home team’s point of view:

Conference Championship Playoff Odds
Home Team Visiting Team Score Diff Win Prob Est. Point Spread
Carolina Panthers Arizona Cardinals -1.40 0.198 -10.4
Denver Broncos New England Patriots 1.972 0.879 14.6


Last week the system went 2-2, for a total record of 5-3. The system favors Arizona markedly, and Denver by an even larger margin. That said, the teams my system does not like have already won one game. There have been years when a team my system didn’t like much won anyway. That was the case in 2009, when my system favored the Colts over the Saints. The system isn’t perfect, and the system is static. It does not take into account critical injuries, morale, better coaching, etc.

Odds for the second week of the 2015 playoffs, presented from the home team’s point of view:

Second Round Playoff Odds
Home Team Visiting Team Score Diff Win Prob Est. Point Spread
Carolina Panthers Seattle Seahawks -1.713 0.153 -12.7
Arizona Cardinals Green Bay Packers -0.001 0.500 0.0
Denver Broncos Pittsburgh Steelers 0.437 0.608 3.2
New England Patriots Kansas City Chiefs -0.563 0.363 -4.2


Last week the system went 3-1 and perhaps would have gone 4-0 if after the Burflict interception, Cincinnati had just killed three plays and kicked a field goal.

The system currently gives Seattle a massive advantage in the playoffs. It says that Green Bay/Arizona is effectively an even match up, and that both the AFC games are pretty close. It favors Denver in their matchup, and the Chiefs in theirs.

One last comment about last week’s games. The Cincinnati-Pitt game was the most depressing playoff game I’ve seen in a long time, both for the dirty play on both sides of the ball, and the end being decided by stupid play on Cincinnati’s part.  It took away from the good parts of the game, the tough defense when people weren’t pushing the edges of the rules, and the gritty play on the part of McCarron and Roethlisberger. There was some heroic play on both their parts, in pouring rain.

But for me, watching Ryan Shazier leading with the crown of his helmet and then listening to officials explain away what is obvious on video more or less took the cake. If in any way shape or form, this kind of hit is legal, then the NFL rules system is busted.

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.


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%.

This is going to be a mixed bag of a post, talking about anything that has caught my eye over the past couple weeks. The first thing I’ll note is that on the recommendation of Tom Gower (you need his Twitter feed), I’ve read Josh Katzowitz’s book: Sid Gillman: Father of the Passing Game.


I didn’t know much about Gillman as a young man, though the 1963 AFL Championship was part of a greatest games collection I read through as a teen. The book isn’t a primer on Gillman’s ideas. Instead, it was more a discussion of his life, the issues he faced growing up (it’s clear Sid felt his Judaism affected his marketability as a coach in the college ranks). Not everyone gets the same chances in life, but Sid was a pretty tough guy, in his own right, and clearly the passion he felt for the sport drove him to a lot of personal success.

Worth the read. Be sure to read Tom Gower’s review as well, which is excellent.

ESPN is dealing with the football off season by slowly releasing a list of the “20 Greatest NFL Coaches” (NFL.com does its 100 best players, for much the same reason). I’m pretty sure neither Gillman nor Don Coryell will be on the list. The problem, of course, lies in the difference between the notions of “greatest” and “most influential”. The influence of both these men is undeniable. However, the greatest success for both these coaches has come has part of their respective coaching (and player) trees: Al Davis and Ara Parseghian come to mind when thinking about Gillman, with Don having a direct influence on coaches such as Joe Gibbs, and Ernie Zampese. John Madden was a product of both schools, and folks such as Norv Turner and Mike Martz are clear disciples of the Coryell way of doing things. It’s easy to go on and on here.

What’s harder to see is the separation (or fusion) of Gillman’s and Coryell’s respective coaching trees. Don never coached under or played for Gillman. And when I raised the question on Twitter, Josh Katzowitz responded with these tweets:

Josh Katzowitz : @smartfootball @FoodNSnellville From what I gathered, not much of a connection. Some of Don’s staff used to watch Gillman’s practices, tho.

Josh Katzowitz ‏: @FoodNSnellville @smartfootball Coryell was pretty adament that he didn’t take much from Gillman. Tom Bass, who coached for both, agreed.

Coaching clinics were popular then, and Sid Gillman appeared from Josh’s bio to be a popular clinic speaker. I’m sure these two mixed and heard each other speak. But Coryell had a powerful Southern California connection in Coach John McKay of USC, and I’m not sure how much Coryell and Gillman truly interacted.

Pro Football Weekly is going away, and Mike Tanier has a nice great article discussing the causes of the demise. In the middle of the discussion, a reader who called himself Richie took it upon himself to start trashing “The Hidden Game of Football” (which factors in because Bob Carroll, a coauthor of THGF, was also a contributor to PFW). Richie seems to think, among other things, that everything THGF discussed was “obvious” and that Bill James invented all of football analytics wholesale by inventing baseball analytics. It’s these kinds of assertions I really want to discuss.

I think the issue of baseball analytics encompassing the whole of football analytics can easily be dismissed by pointing out the solitary nature of baseball and its stats, their lack of entanglement issues, and the lack of a notion of field position, in the football sense of the term. Since baseball doesn’t have any such thing, any stat featuring any kind of relationship of field position to anything, or any stat derived from models of relationships of field position to anything, cannot have been created in a baseball world.

Sad to say, that’s almost any football stat of merit.

On the notion of obvious, THGF was the granddaddy of the scoring model for the average fan. I’d suggest that scoring models are certainly not obvious, or else every article I have with that tag would have been written up and dismissed years ago. What is not so obvious is that scoring models have a dual nature, akin to that of quantum mechanical objects, and the kinds of logic one needs to best understand scoring models parallels that of the kinds of things a chemistry major might encounter in his junior year of university, in a physical chemistry class (physicists might run into these issues sooner).

Scoring models have a dual nature. They are both deterministic and statistical/probabilistic at the same time.

They are deterministic in that for a typical down, distance, to go, and with a specific play by play data set, you can calculate the odds of scoring down to a hundredth of a point. They are statistical in that they represent the sum of dozens or hundreds of unique events, all compressed into a single measurement. When divorced from the parent data set, the kinds of logic you must use to analyze the meanings of the models, and formulas derived from those models, must take into account the statistical nature of the model involved.

It’s not easy. Most analysts turns models and formulas into something more concrete than they really are.

And this is just one component of the THGF contribution. I haven’t even mentioned the algebraic breakdown of the NFL passer rating they introduced, which dominates discussion of the rating to this day. It’s so influential that to a first approximation, no one can get past it.

Just tell me: how did you get from the formulas shown here to the THGF formula? And if you didn’t figure it out yourself, then how can you claim it is obvious?

I’ve been looking at this model recently, and thinking.

Backstory references, for those who need them: here and here and here.

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.

If the odds of scoring a touchdown approach 100% as you approach the goal line, then the barrier potential disappears, and the “yards to go” intercept is equal to the value of the touchdown. The values in the PFR model appear to always increase as they approach the goal line. They never go down, the way real values do. Therefore, the model as presented on their pages appears to be a fitted curve, not raw data.

The value they assign the touchdown is 7 points. The EP value of first and goal on the 1 is 6.97 points. 6.97 / 7.00 * 100 = 99.57%. How many of you out there think the chances of scoring a touchdown on the 1 yard line are better than 99%?

More so, the EP value, 1st and goal on the 2 yard line is 6.74. Ok, if the fitting function is linear, or perhaps quadratic, then how do you go 6.74, to 6.97, to 7.00? The difference between 6.74 and 6.97 is 0.23 points. Assuming linearity (not true, as first and 10 points on the other end of the curve typically differ by 0.03 points per yard), you get an extrapolated intercept of 7.20 points.

The PFR model has its issues. The first down intercept seems odd, and it lacks a barrier potential. To what extent this is an artifact of a polynomial (or other curve) fitted to real data remains to be seen.

Update: added a useful Keith Goldner reference, which has a chart giving probabilities of scoring a touchdown.