Friday, 24 August 2012

Statistical inference and mathematical modelling (Part 3)

First, I want to apologise to any and all of you who were kind enough to show an interest in this blog around the time of my last post over a year ago. As some of you are aware, the last few years have been tough for my family and there have been added distractions involving my career.

But now TFNL is back. And I am continuing my series on modelling sports using numbers - but with a change of sport.

The first two parts of the series - which laid out some of the basic tools of statistical inference - concerned itself with the English Premier League. It was entry level stuff, developing simple but powerful ideas about the relationship between basic numbers like goals scored and conceded and points.

Despite what some would have you believe, however, football statistics like shots and saves, tackles and possession are inadequate in developing a causal understanding of how games are won and lost. I am happy to be proved wrong, but most of the second-level numbers yet produced are highly situational and, as a result, insufficiently predictive. 

Instead, I am turning to the data-rich environment of American sport to lay out some of the powerful examples which will follow. This doesn't mean you have been suddenly marginalised, if NFL, MLB or NBA are not your bag; I will do my best to explain the situations to which I am referring so as any sports fan can follow. It's only that the numbers which pertain to these sports are perfect for my purposes here.

If you are an American sports fan, and specifically an NFL follower or punter, you may be interested to know that your reward for ploughing through my stolid prose may be more than just a better theoretical understanding of the game. Quickly, we will establish statistical truth which will enable you to see the huge difference between a team winning percentage and its underlying efficiency. And I will show how valuable this has been over the past five seasons.

So, with all of that preamble complete, and assuming you are still awake, let's get on with the first example:


3.1 Differentiating between an outcome and an underlying skill

The results of NFL games are strongly correlated to the statistic turnover differential. Turnovers are changes of possession which are not the result of scoring plays. Teams can concede possession either by fumbling the ball and having it be recovered legally by the opposition, or by having a passing play intended for a receiver intercepted by a defensive player. Turnover differential for a team is simply the difference between takeaways (turnovers in which possession is gained) and giveaways (turnovers in which possession is lost).

For instance, a team which makes three interceptions but loses four fumbles is said to have a turnover differential of -1.

Turnovers are highly influential in game outcomes for two reasons:

1) In NFL games, teams have nearly the same number of possessions or drives. And a possession ends with a turnover, limiting the opportunity to score.

2) The change of field position which occurs when turnovers are returned or merely recovered often results in the team inheriting possession in an advanced position towards the opposing goal-line. (For interest: the value of ball possession according to yard-line can be evaluated by the metric Expected Points Scored with the biggest increase-per-yard in the zone after the opponent's 45-yard line.)

3.2 Intercepting the ball: a skill or an outcome?

From a statistical standpoint, a skill can be defined as an ability to control a measurable outcome which persists across time and can survive changing context.

One of the two types of turnovers described above, the interception, is commonly assumed to be a skill. Commentators and analysts refer to defensive players who make multiple interceptions as necessarily being the cause of those interceptions, giving the impression that it is a skill which will result in more interceptions in the future.

This is true to some extent. Some defensive players have better hands, better positional sense and even, I suppose, better eyesight than others. And, as a result, they are indeed more likely to make interceptions than players with lesser capabilities - all other things being equal.

But that last caveat is extremely important. All other things are not ever close to being equal in the different moments in which interceptions are made. And how do I know this without playing the game and experiencing the making of an interception myself? Correlation.

The rate at which any NFL player makes interceptions shows very little correlation from one game to another, one month to another or one season to another. Here, I am referring to correlation in a mathematical sense, measured by the coefficient 'r', as described in Part 2.

Interceptions are better described as an outcome in which skill is only a small factor. And this doesn't just apply to the interceptions made by a player, but by an entire team.

Underneath (apologies it is not HTML) is a list of the total number of interceptions made by every NFL team during the five seasons from 2007-2011 inclusive:


Team Year Int
Arizona Cardinals 2007 18
Atlanta Falcons 2007 16
Baltimore Ravens 2007 17
Buffalo Bills 2007 18
Carolina Panthers 2007 14
Chicago Bears 2007 16
Cincinnati Bengals 2007 19
Cleveland Browns 2007 17
Dallas Cowboys 2007 19
Denver Broncos 2007 14
Detroit Lions 2007 17
Green Bay Packers 2007 19
Houston Texans 2007 11
Indianapolis Colts 2007 22
Jacksonville Jaguars 2007 20
Kansas City Chiefs 2007 14
Miami Dolphins 2007 14
Minnesota Vikings 2007 15
New England Patriots 2007 19
New Orleans Saints 2007 13
New York Giants 2007 15
New York Jets 2007 15
Oakland Raiders 2007 18
Philadelphia Eagles 2007 11
Pittsburgh Steelers 2007 11
San Diego Chargers 2007 30
San Francisco 49ers 2007 12
Seattle Seahawks 2007 20
St. Louis Rams 2007 18
Tampa Bay Buccs 2007 16
Tennessee Titans 2007 22
Washington Redskins 2007 14
Arizona Cardinals 2008 13
Atlanta Falcons 2008 10
Baltimore Ravens 2008 26
Buffalo Bills 2008 10
Carolina Panthers 2008 12
Chicago Bears 2008 22
Cincinnati Bengals 2008 12
Cleveland Browns 2008 23
Dallas Cowboys 2008 8
Denver Broncos 2008 6
Detroit Lions 2008 4
Green Bay Packers 2008 22
Houston Texans 2008 12
Indianapolis Colts 2008 15
Jacksonville Jaguars 2008 13
Kansas City Chiefs 2008 13
Miami Dolphins 2008 18
Minnesota Vikings 2008 12
New England Patriots 2008 14
New Orleans Saints 2008 15
New York Giants 2008 17
New York Jets 2008 14
Oakland Raiders 2008 16
Philadelphia Eagles 2008 15
Pittsburgh Steelers 2008 20
San Diego Chargers 2008 15
San Francisco 49ers 2008 12
Seattle Seahawks 2008 9
St. Louis Rams 2008 12
Tampa Bay Buccs 2008 22
Tennessee Titans 2008 20
Washington Redskins 2008 13
Arizona Cardinals 2009 21
Atlanta Falcons 2009 15
Baltimore Ravens 2009 22
Buffalo Bills 2009 28
Carolina Panthers 2009 22
Chicago Bears 2009 13
Cincinnati Bengals 2009 19
Cleveland Browns 2009 10
Dallas Cowboys 2009 11
Denver Broncos 2009 17
Detroit Lions 2009 9
Green Bay Packers 2009 30
Houston Texans 2009 14
Indianapolis Colts 2009 16
Jacksonville Jaguars 2009 15
Kansas City Chiefs 2009 15
Miami Dolphins 2009 15
Minnesota Vikings 2009 11
New England Patriots 2009 18
New Orleans Saints 2009 26
New York Giants 2009 13
New York Jets 2009 17
Oakland Raiders 2009 8
Philadelphia Eagles 2009 25
Pittsburgh Steelers 2009 12
San Diego Chargers 2009 14
San Francisco 49ers 2009 18
Seattle Seahawks 2009 13
St. Louis Rams 2009 8
Tampa Bay Buccs 2009 19
Tennessee Titans 2009 20
Washington Redskins 2009 11
Arizona Cardinals 2010 17
Atlanta Falcons 2010 22
Baltimore Ravens 2010 19
Buffalo Bills 2010 11
Carolina Panthers 2010 17
Chicago Bears 2010 21
Cincinnati Bengals 2010 16
Cleveland Browns 2010 19
Dallas Cowboys 2010 20
Denver Broncos 2010 10
Detroit Lions 2010 14
Green Bay Packers 2010 24
Houston Texans 2010 13
Indianapolis Colts 2010 10
Jacksonville Jaguars 2010 13
Kansas City Chiefs 2010 14
Miami Dolphins 2010 11
Minnesota Vikings 2010 15
New England Patriots 2010 25
New Orleans Saints 2010 9
New York Giants 2010 16
New York Jets 2010 12
Oakland Raiders 2010 12
Philadelphia Eagles 2010 23
Pittsburgh Steelers 2010 21
San Diego Chargers 2010 16
San Francisco 49ers 2010 15
Seattle Seahawks 2010 12
St. Louis Rams 2010 14
Tampa Bay Buccs 2010 19
Tennessee Titans 2010 17
Washington Redskins 2010 14
Arizona Cardinals 2011 10
Atlanta Falcons 2011 19
Baltimore Ravens 2011 15
Buffalo Bills 2011 20
Carolina Panthers 2011 14
Chicago Bears 2011 20
Cincinnati Bengals 2011 10
Cleveland Browns 2011 9
Dallas Cowboys 2011 15
Denver Broncos 2011 9
Detroit Lions 2011 21
Green Bay Packers 2011 31
Houston Texans 2011 17
Indianapolis Colts 2011 8
Jacksonville Jaguars 2011 17
Kansas City Chiefs 2011 20
Miami Dolphins 2011 16
Minnesota Vikings 2011 8
New England Patriots 2011 23
New Orleans Saints 2011 9
New York Giants 2011 20
New York Jets 2011 19
Oakland Raiders 2011 18
Philadelphia Eagles 2011 15
Pittsburgh Steelers 2011 11
San Diego Chargers 2011 17
San Francisco 49ers 2011 23
Seattle Seahawks 2011 22
St. Louis Rams 2011 12
Tampa Bay Buccs 2011 14
Tennessee Titans 2011 11
Washington Redskins 2011 13

The correlation coefficient 'r' for the 128 pairs of consecutive seasons for the same team is 0.151. Squaring 'r' results in 0.02 which, as I explained in the earlier part, means that only 2% of the mean variance in interceptions persists from one year to the next. In other words, in the absence of any other information, 98% of a team's total interceptions each year - the outcome - appears to be down to either context, luck or factors which do not persist from one season to the next.

But it's massively important to recognise this does not mean every defense is created equal. Far from it.

Instead, the number of interceptions made by a team each year is a poor guide to the future mainly because the sample-size is far too small for the underlying skills (defensive positioning, hand-eye coordination and whatever else) to be expressed  in the difference between the largest and smallest result.

It is clear we need another measure of these same skills which does a better job in this regard, an event which is related to the making of an interception but which happens more frequently for it to be more descriptive about the aspects of a team which do persist over time.

This is one of the most common and powerful devices of modelling sports: we model the probability of an important but infrequent event with a more frequently occurring variable to which it is correlated.

I can't stress the importance of this enough. In mathematical modelling of football, for instance, the ambition would be to model the probability of goals (infrequent) by variables such as shots, tackles, field position (more frequent) so that the underlying skills of goal-scoring are separated from the outcome in order to predict the likelihood of goals (and, vitally, goal difference in a game) better than is possible from past goals alone. So far, however, I am not aware that anyone has got there.

And here is where American sport is so powerful. An interception - which is an outcome with a similar frequency in the NFL to a goal in football - can indeed be modelled better with correlated variables than merely by past interceptions (or interceptions as a percentage of passes, for the advanced among you).

I am going to use just one of those variables - passes defended - to show the power of this technique. The official gamebook of the NFL (via NFL.com) awards a pass defended to a player (and hence a team) when he deflects the path of the ball resulting in an incomplete pass or an interception made by another player. That is, at least one defensive player touches the ball in flight.

While the best (or perhaps 'luckiest') teams intercept as many as 30-odd passes in a 16-game season, they may defend as many as 129 passes, like the 2011 Green Bay Packers. It doesn't seem a huge difference in quantity, but it matters.

3.3 Thinking in terms of expectation

For every NFL team in the period 2007-2011 (out data-set for all examples), the correlation between passes defended and interceptions = 0.685.

That is, there is a reasonably strong relationship between the number of times a defensive teams gets their hands on the ball in flight and the number of interceptions it makes. But, remember, the number of interceptions made varies tremendously from season to season.

We can use the relationship between passes defended and interceptions to create a metric Expected Interceptions (xINT) which is simply the number of interceptions a team would make if it turned its own number of passes defended into interceptions at a league-average rate.

Expected variables (often denoted by the addition of a small 'x') are tremendously useful in understanding the role of randomness and context in outcomes which are not wholly dependent on skill.

Let's now use:

1) the totals of passes defended for each of the teams in our data-set and
2) the knowledge that the NFL average rate at which they become interceptions is 17.7%

to add to the earlier list a column xINT for the interceptions each team would have made had they turned passes defended into interceptions (the table is sorted by xINT):

Team Year Pdef Int % xINT
Green Bay Packers 2011 129 31 24.0 22.9
Green Bay Packers 2009 126 30 23.8 22.3
Baltimore Ravens 2008 125 26 20.8 22.2
San Francisco 49ers 2011 124 23 18.5 22.0
Baltimore Ravens 2011 122 15 12.3 21.6
Chicago Bears 2008 121 22 18.2 21.5
San Diego Chargers 2007 119 30 25.2 21.1
Philadelphia Eagles 2009 117 25 21.4 20.7
Houston Texans 2011 114 17 14.9 20.2
Philadelphia Eagles 2010 113 23 20.4 20.0
Cincinnati Bengals 2009 112 19 17.0 19.9
Green Bay Packers 2008 110 22 20.0 19.5
Green Bay Packers 2010 110 24 21.8 19.5
Kansas City Chiefs 2010 110 14 12.7 19.5
Buffalo Bills 2009 109 28 25.7 19.3
New Orleans Saints 2009 109 26 23.9 19.3
Pittsburgh Steelers 2010 109 21 19.3 19.3
Tennessee Titans 2007 108 22 20.4 19.1
Arizona Cardinals 2009 108 21 19.4 19.1
Philadelphia Eagles 2008 107 15 14.0 19.0
Pittsburgh Steelers 2008 107 20 18.7 19.0
Baltimore Ravens 2007 106 17 16.0 18.8
New Orleans Saints 2008 106 15 14.2 18.8
Tennessee Titans 2008 106 20 18.9 18.8
Oakland Raiders 2011 106 18 17.0 18.8
Seattle Seahawks 2011 105 22 21.0 18.6
Kansas City Chiefs 2011 104 20 19.2 18.4
New York Giants 2011 104 20 19.2 18.4
Cleveland Browns 2007 103 17 16.5 18.3
Dallas Cowboys 2007 103 19 18.4 18.3
New York Jets 2009 103 17 16.5 18.3
New York Giants 2010 103 16 15.5 18.3
Washington Redskins 2010 103 14 13.6 18.3
Washington Redskins 2008 102 13 12.7 18.1
New England Patriots 2010 102 25 24.5 18.1
Baltimore Ravens 2009 100 22 22.0 17.7
Miami Dolphins 2008 99 18 18.2 17.6
Dallas Cowboys 2009 99 11 11.1 17.6
Baltimore Ravens 2010 99 19 19.2 17.6
Atlanta Falcons 2011 99 19 19.2 17.6
New Orleans Saints 2011 99 9 9.1 17.6
Buffalo Bills 2007 98 18 18.4 17.4
Cincinnati Bengals 2007 97 19 19.6 17.2
Seattle Seahawks 2007 97 20 20.6 17.2
Washington Redskins 2007 97 14 14.4 17.2
Denver Broncos 2009 97 17 17.5 17.2
Arizona Cardinals 2010 97 17 17.5 17.2
Seattle Seahawks 2010 97 12 12.4 17.2
Kansas City Chiefs 2009 96 15 15.6 17.0
Cleveland Browns 2010 96 19 19.8 17.0
New York Jets 2010 96 12 12.5 17.0
Carolina Panthers 2008 95 12 12.6 16.8
Tampa Bay Buccs 2008 95 22 23.2 16.8
Arizona Cardinals 2011 95 10 10.5 16.8
Chicago Bears 2010 94 21 22.3 16.7
Detroit Lions 2011 94 21 22.3 16.7
Cincinnati Bengals 2008 93 12 12.9 16.5
Miami Dolphins 2010 93 11 11.8 16.5
New York Giants 2008 92 17 18.5 16.3
New York Giants 2009 92 13 14.1 16.3
New England Patriots 2007 91 19 20.9 16.1
New York Jets 2008 91 14 15.4 16.1
Atlanta Falcons 2010 91 22 24.2 16.1
St. Louis Rams 2010 91 14 15.4 16.1
Green Bay Packers 2007 90 19 21.1 16.0
New York Giants 2007 90 15 16.7 16.0
Cleveland Browns 2008 90 23 25.6 16.0
San Diego Chargers 2008 90 15 16.7 16.0
Miami Dolphins 2009 90 15 16.7 16.0
New York Jets 2011 90 19 21.1 16.0
St. Louis Rams 2007 89 18 20.2 15.8
Houston Texans 2009 89 14 15.7 15.8
Tampa Bay Buccs 2010 89 19 21.3 15.8
Tennessee Titans 2010 89 17 19.1 15.8
Minnesota Vikings 2007 88 15 17.0 15.6
Pittsburgh Steelers 2007 88 11 12.5 15.6
Arizona Cardinals 2008 88 13 14.8 15.6
Carolina Panthers 2009 88 22 25.0 15.6
Buffalo Bills 2010 88 11 12.5 15.6
Oakland Raiders 2007 87 18 20.7 15.4
San Francisco 49ers 2009 87 18 20.7 15.4
Buffalo Bills 2011 87 20 23.0 15.4
Houston Texans 2007 86 11 12.8 15.2
New Orleans Saints 2007 86 13 15.1 15.2
Oakland Raiders 2008 86 16 18.6 15.2
New England Patriots 2009 86 18 20.9 15.2
Denver Broncos 2010 86 10 11.6 15.2
Atlanta Falcons 2007 85 16 18.8 15.1
Atlanta Falcons 2008 85 10 11.8 15.1
San Francisco 49ers 2008 85 12 14.1 15.1
Washington Redskins 2009 85 11 12.9 15.1
Carolina Panthers 2010 85 17 20.0 15.1
Chicago Bears 2011 85 20 23.5 15.1
Washington Redskins 2011 85 13 15.3 15.1
Arizona Cardinals 2007 84 18 21.4 14.9
Indianapolis Colts 2007 84 22 26.2 14.9
Tampa Bay Buccs 2007 84 16 19.0 14.9
Indianapolis Colts 2009 84 16 19.0 14.9
Carolina Panthers 2011 84 14 16.7 14.9
New England Patriots 2011 84 23 27.4 14.9
Jacksonville Jaguars 2007 83 20 24.1 14.7
Kansas City Chiefs 2007 83 14 16.9 14.7
Buffalo Bills 2008 83 10 12.0 14.7
Cleveland Browns 2009 83 10 12.0 14.7
Tennessee Titans 2009 83 20 24.1 14.7
Cincinnati Bengals 2010 83 16 19.3 14.7
San Diego Chargers 2010 83 16 19.3 14.7
Cincinnati Bengals 2011 83 10 12.0 14.7
Pittsburgh Steelers 2011 83 11 13.3 14.7
New York Jets 2007 82 15 18.3 14.5
Tampa Bay Buccs 2009 82 19 23.2 14.5
Dallas Cowboys 2010 82 20 24.4 14.5
San Diego Chargers 2011 82 17 20.7 14.5
Carolina Panthers 2007 81 14 17.3 14.4
Tennessee Titans 2011 81 11 13.6 14.4
Denver Broncos 2007 80 14 17.5 14.2
Detroit Lions 2009 80 9 11.3 14.2
San Diego Chargers 2009 80 14 17.5 14.2
Miami Dolphins 2011 80 16 20.0 14.2
Pittsburgh Steelers 2009 79 12 15.2 14.0
San Francisco 49ers 2010 79 15 19.0 14.0
Jacksonville Jaguars 2011 79 17 21.5 14.0
Chicago Bears 2007 78 16 20.5 13.8
San Francisco 49ers 2007 78 12 15.4 13.8
Minnesota Vikings 2008 78 12 15.4 13.8
Atlanta Falcons 2009 78 15 19.2 13.8
Cleveland Browns 2011 78 9 11.5 13.8
Philadelphia Eagles 2011 78 15 19.2 13.8
Oakland Raiders 2009 77 8 10.4 13.7
Seattle Seahawks 2009 77 13 16.9 13.7
Minnesota Vikings 2010 77 15 19.5 13.7
New Orleans Saints 2010 77 9 11.7 13.7
St. Louis Rams 2011 77 12 15.6 13.7
Detroit Lions 2007 76 17 22.4 13.5
Detroit Lions 2010 76 14 18.4 13.5
Philadelphia Eagles 2007 75 11 14.7 13.3
Seattle Seahawks 2008 75 9 12.0 13.3
Oakland Raiders 2010 75 12 16.0 13.3
Minnesota Vikings 2009 73 11 15.1 12.9
Tampa Bay Buccs 2011 73 14 19.2 12.9
Dallas Cowboys 2011 72 15 20.8 12.8
Houston Texans 2008 71 12 16.9 12.6
New England Patriots 2008 71 14 19.7 12.6
Jacksonville Jaguars 2009 71 15 21.1 12.6
Miami Dolphins 2007 70 14 20.0 12.4
Kansas City Chiefs 2008 70 13 18.6 12.4
Jacksonville Jaguars 2008 69 13 18.8 12.2
Houston Texans 2010 69 13 18.8 12.2
Dallas Cowboys 2008 68 8 11.8 12.1
Denver Broncos 2011 68 9 13.2 12.1
Chicago Bears 2009 66 13 19.7 11.7
St. Louis Rams 2008 65 12 18.5 11.5
Indianapolis Colts 2010 64 10 15.6 11.3
Indianapolis Colts 2008 63 15 23.8 11.2
Jacksonville Jaguars 2010 61 13 21.3 10.8
St. Louis Rams 2009 59 8 13.6 10.5
Minnesota Vikings 2011 58 8 13.8 10.3
Denver Broncos 2008 57 6 10.5 10.1
Indianapolis Colts 2011 55 8 14.5 9.8
Detroit Lions 2008 48 4 8.3 8.5

There is instantly some fascinating symmetry. As you look down the table, you should see plenty of instances where different seasons of the same team are clustered together by the total of passes defended - and thus the metric xINT.

Most notably, the Green Bay Packers appear four times (from five editions of the team) in the top 13 seasons, recording totals of passes defended of 129, 126, 110 and 110. In 2007, the missing year, they had a shocker with 'only' 90.

This potency is both a direct and indirect effect of one of the greatest players to wear an NFL uniform, Charles Woodson. The Ohio native joined Green Bay from Oakland before the 2006 season on a 7-year $53m contract which has proved worth every cent.

                              Woodson: all-time great pass defender

Woodson's personal interception totals for the five seasons from 2007-2011 are 4, 7, 9, 2 and 7 but his impact is much better measured through the performance of the defense in which he plays; within this, he has filled the role of cornerback, free safety and, now upcoming in 2012, strong safety.

In the five seasons of the sample, the Green Bay defense has intercepted a total of 31, 30, 24, 22 and 19 passes. This sums to 126 interceptions from 565 passes defended or 22.3% - an outlier to the league average of 17.7%. Apart from Woodson, there are other important factors at play. But more of that next time.

The rest of the league are clustered around the ratio of 17.7% interceptions to passes defended. As the correlation enumerates, the relationship broadly holds good.

So, if we subtract the interceptions a team made from the interceptions it could have been expected to make (if turning passes defended into interceptions at a league-average rate), we have a number which is really powerful. Let's call it interception luck.

So. here's the same data-set sorted by this metric of interception luck or vINT (the small 'v' stands for variance). A positive figure indicates a team who made more interceptions than expected while a negative figure indicates a team who made less interceptions than expected.

Team Year Pdef Int % vINT
San Diego Chargers 2007 119 30 25.2 8.9
Buffalo Bills 2009 109 28 25.7 8.7
Green Bay Packers 2011 129 31 24.0 8.1
New England Patriots 2011 84 23 27.4 8.1
Green Bay Packers 2009 126 30 23.8 7.7
Indianapolis Colts 2007 84 22 26.2 7.1
Cleveland Browns 2008 90 23 25.6 7.0
New England Patriots 2010 102 25 24.5 6.9
New Orleans Saints 2009 109 26 23.9 6.7
Carolina Panthers 2009 88 22 25.0 6.4
Atlanta Falcons 2010 91 22 24.2 5.9
Dallas Cowboys 2010 82 20 24.4 5.5
Jacksonville Jaguars 2007 83 20 24.1 5.3
Tennessee Titans 2009 83 20 24.1 5.3
Tampa Bay Buccs 2008 95 22 23.2 5.2
Chicago Bears 2011 85 20 23.5 4.9
Buffalo Bills 2011 87 20 23.0 4.6
Green Bay Packers 2010 110 24 21.8 4.5
Tampa Bay Buccs 2009 82 19 23.2 4.5
Chicago Bears 2010 94 21 22.3 4.3
Detroit Lions 2011 94 21 22.3 4.3
Baltimore Ravens 2009 100 22 22.0 4.3
Philadelphia Eagles 2009 117 25 21.4 4.3
Baltimore Ravens 2008 125 26 20.8 3.8
Indianapolis Colts 2008 63 15 23.8 3.8
Detroit Lions 2007 76 17 22.4 3.5
Seattle Seahawks 2011 105 22 21.0 3.4
Tampa Bay Buccs 2010 89 19 21.3 3.2
Arizona Cardinals 2007 84 18 21.4 3.1
Green Bay Packers 2007 90 19 21.1 3.0
New York Jets 2011 90 19 21.1 3.0
Jacksonville Jaguars 2011 79 17 21.5 3.0
Philadelphia Eagles 2010 113 23 20.4 3.0
New England Patriots 2007 91 19 20.9 2.9
Tennessee Titans 2007 108 22 20.4 2.9
Seattle Seahawks 2007 97 20 20.6 2.8
New England Patriots 2009 86 18 20.9 2.8
Oakland Raiders 2007 87 18 20.7 2.6
San Francisco 49ers 2009 87 18 20.7 2.6
Green Bay Packers 2008 110 22 20.0 2.5
San Diego Chargers 2011 82 17 20.7 2.5
Jacksonville Jaguars 2009 71 15 21.1 2.4
Dallas Cowboys 2011 72 15 20.8 2.2
St. Louis Rams 2007 89 18 20.2 2.2
Jacksonville Jaguars 2010 61 13 21.3 2.2
Chicago Bears 2007 78 16 20.5 2.2
Cleveland Browns 2010 96 19 19.8 2.0
Carolina Panthers 2010 85 17 20.0 1.9
Arizona Cardinals 2009 108 21 19.4 1.9
Miami Dolphins 2011 80 16 20.0 1.8
Cincinnati Bengals 2007 97 19 19.6 1.8
Pittsburgh Steelers 2010 109 21 19.3 1.7
Miami Dolphins 2007 70 14 20.0 1.6
Kansas City Chiefs 2011 104 20 19.2 1.6
New York Giants 2011 104 20 19.2 1.6
Baltimore Ravens 2010 99 19 19.2 1.4
Atlanta Falcons 2011 99 19 19.2 1.4
New England Patriots 2008 71 14 19.7 1.4
Minnesota Vikings 2010 77 15 19.5 1.3
Chicago Bears 2009 66 13 19.7 1.3
Cincinnati Bengals 2010 83 16 19.3 1.3
San Diego Chargers 2010 83 16 19.3 1.3
Tennessee Titans 2010 89 17 19.1 1.2
Tennessee Titans 2008 106 20 18.9 1.2
Atlanta Falcons 2009 78 15 19.2 1.2
Philadelphia Eagles 2011 78 15 19.2 1.2
Tampa Bay Buccs 2007 84 16 19.0 1.1
Indianapolis Colts 2009 84 16 19.0 1.1
Tampa Bay Buccs 2011 73 14 19.2 1.1
Pittsburgh Steelers 2008 107 20 18.7 1.0
San Francisco 49ers 2011 124 23 18.5 1.0
San Francisco 49ers 2010 79 15 19.0 1.0
Atlanta Falcons 2007 85 16 18.8 0.9
Jacksonville Jaguars 2008 69 13 18.8 0.8
Houston Texans 2010 69 13 18.8 0.8
Oakland Raiders 2008 86 16 18.6 0.8
Dallas Cowboys 2007 103 19 18.4 0.7
New York Giants 2008 92 17 18.5 0.7
Buffalo Bills 2007 98 18 18.4 0.6
Kansas City Chiefs 2008 70 13 18.6 0.6
Chicago Bears 2008 121 22 18.2 0.5
Detroit Lions 2010 76 14 18.4 0.5
St. Louis Rams 2008 65 12 18.5 0.5
New York Jets 2007 82 15 18.3 0.5
Miami Dolphins 2008 99 18 18.2 0.4
Denver Broncos 2007 80 14 17.5 -0.2
San Diego Chargers 2009 80 14 17.5 -0.2
Denver Broncos 2009 97 17 17.5 -0.2
Arizona Cardinals 2010 97 17 17.5 -0.2
Carolina Panthers 2007 81 14 17.3 -0.4
Houston Texans 2008 71 12 16.9 -0.6
Minnesota Vikings 2007 88 15 17.0 -0.6
Seattle Seahawks 2009 77 13 16.9 -0.7
Kansas City Chiefs 2007 83 14 16.9 -0.7
Oakland Raiders 2011 106 18 17.0 -0.8
Cincinnati Bengals 2009 112 19 17.0 -0.9
Carolina Panthers 2011 84 14 16.7 -0.9
New York Giants 2007 90 15 16.7 -1.0
San Diego Chargers 2008 90 15 16.7 -1.0
Miami Dolphins 2009 90 15 16.7 -1.0
Cleveland Browns 2007 103 17 16.5 -1.3
New York Jets 2009 103 17 16.5 -1.3
Oakland Raiders 2010 75 12 16.0 -1.3
Indianapolis Colts 2010 64 10 15.6 -1.3
St. Louis Rams 2011 77 12 15.6 -1.7
Indianapolis Colts 2011 55 8 14.5 -1.8
Houston Texans 2009 89 14 15.7 -1.8
Baltimore Ravens 2007 106 17 16.0 -1.8
San Francisco 49ers 2007 78 12 15.4 -1.8
Minnesota Vikings 2008 78 12 15.4 -1.8
Minnesota Vikings 2009 73 11 15.1 -1.9
Pittsburgh Steelers 2009 79 12 15.2 -2.0
Kansas City Chiefs 2009 96 15 15.6 -2.0
Washington Redskins 2011 85 13 15.3 -2.1
New York Jets 2008 91 14 15.4 -2.1
St. Louis Rams 2010 91 14 15.4 -2.1
New Orleans Saints 2007 86 13 15.1 -2.2
New York Giants 2010 103 16 15.5 -2.3
Minnesota Vikings 2011 58 8 13.8 -2.3
Philadelphia Eagles 2007 75 11 14.7 -2.3
St. Louis Rams 2009 59 8 13.6 -2.5
Arizona Cardinals 2008 88 13 14.8 -2.6
Denver Broncos 2011 68 9 13.2 -3.1
San Francisco 49ers 2008 85 12 14.1 -3.1
Washington Redskins 2007 97 14 14.4 -3.2
Houston Texans 2011 114 17 14.9 -3.2
New York Giants 2009 92 13 14.1 -3.3
Tennessee Titans 2011 81 11 13.6 -3.4
Pittsburgh Steelers 2011 83 11 13.3 -3.7
New Orleans Saints 2008 106 15 14.2 -3.8
Philadelphia Eagles 2008 107 15 14.0 -4.0
Dallas Cowboys 2008 68 8 11.8 -4.1
Washington Redskins 2009 85 11 12.9 -4.1
Denver Broncos 2008 57 6 10.5 -4.1
Houston Texans 2007 86 11 12.8 -4.2
Washington Redskins 2010 103 14 13.6 -4.3
Seattle Seahawks 2008 75 9 12.0 -4.3
Cincinnati Bengals 2008 93 12 12.9 -4.5
Detroit Lions 2008 48 4 8.3 -4.5
Pittsburgh Steelers 2007 88 11 12.5 -4.6
Buffalo Bills 2010 88 11 12.5 -4.6
New Orleans Saints 2010 77 9 11.7 -4.7
Buffalo Bills 2008 83 10 12.0 -4.7
Cleveland Browns 2009 83 10 12.0 -4.7
Cincinnati Bengals 2011 83 10 12.0 -4.7
Cleveland Browns 2011 78 9 11.5 -4.8
Carolina Panthers 2008 95 12 12.6 -4.8
New York Jets 2010 96 12 12.5 -5.0
Atlanta Falcons 2008 85 10 11.8 -5.1
Washington Redskins 2008 102 13 12.7 -5.1
Detroit Lions 2009 80 9 11.3 -5.2
Seattle Seahawks 2010 97 12 12.4 -5.2
Denver Broncos 2010 86 10 11.6 -5.2
Miami Dolphins 2010 93 11 11.8 -5.5
Kansas City Chiefs 2010 110 14 12.7 -5.5
Oakland Raiders 2009 77 8 10.4 -5.7
Dallas Cowboys 2009 99 11 11.1 -6.6
Baltimore Ravens 2011 122 15 12.3 -6.6
Arizona Cardinals 2011 95 10 10.5 -6.8
New Orleans Saints 2011 99 9 9.1 -8.6

In all five seasons involving the Green Bay Packers and Woodson, the team exceeded the expected number of interceptions by 8.1, 7.7, 4.5, 3.0 and 2.5. Clearly, it stretches beyond the bounds of credulity to believe the team was lucky in all five seasons. As I said, it wasn't because other factors were at play which I will flesh out in Part 4.

But, sticking to this simple single-variable model, let's now test the hypothesis that   vINT represents the degree to which a team's interception total for the season was unsustainable.

First, let's consider the 25 teams during the past five seasons - from San Diego 2007 to Philadelphia 2010 - for whom vINT >= 3. In other words, teams including the Packers who got 'lucky' by turning passes defended into interceptions at an unsustainable rate.

If the more numerous variable of passes defended is a better guide to the underlying skills in making interceptions than merely the total of interceptions themselves, there should be evidence of mean regression. That is, if the theory holds, the teams that got lucky should intercept fewer passes in general the following year. Let's have a look:

Team Year=y vINT Int:y Int: y+1 change
San Diego Chargers 2007 8.9 30 15 -15
Buffalo Bills 2009 8.7 28 11 -17
Green Bay Packers 2009 7.7 30 24 -6
Indianapolis Colts 2007 7.1 22 15 -7
Cleveland Browns 2008 7.0 23 10 -13
New England Patriots 2010 6.9 25 23 -2
New Orleans Saints 2009 6.7 26 9 -17
Carolina Panthers 2009 6.4 22 17 -5
Atlanta Falcons 2010 5.9 22 19 -3
Dallas Cowboys 2010 5.5 20 15 -5
Jacksonville Jaguars 2007 5.3 20 13 -7
Tennessee Titans 2009 5.3 20 17 -3
Tampa Bay Buccs 2008 5.2 22 19 -3
Green Bay Packers 2010 4.5 24 31 7
Tampa Bay Buccs 2009 4.5 19 19 0
Chicago Bears 2010 4.3 21 20 -1
Baltimore Ravens 2009 4.3 22 19 -3
Philadelphia Eagles 2009 4.3 25 23 -2
Baltimore Ravens 2008 3.8 26 22 -4
Indianapolis Colts 2008 3.8 15 16 1
Detroit Lions 2007 3.5 17 4 -13
Tampa Bay Buccs 2010 3.2 19 14 -5
Arizona Cardinals 2007 3.1 18 13 -5
Green Bay Packers 2007 3.0 19 22 3
Philadelphia Eagles 2010 3.0 23 15 -8



QED! Sort of, anyway. A staggering 21 of the 25 teams who had interception 'luck' recorded fewer interceptions the following season - regardless of whether they were lucky or not the next year. Not only that, but there is the suggestion of a clear relationship between the degree of luck of year y and the drop-off to year y+1; which is to say there is a correlation of -0.599 between vINT and change. 

If you know anything about the causative effect of interceptions in NFL games, you should be extremely interested from a gambling perspective in the fact that the direction and magnitude of change from one season to the next is predictable. Apart from forecasting games and season win-total markets, it is possible to bet on the number of interceptions a team will make. And now we have a sophisticated technique of predicting just that.

Note that using passes defended to produce expected interception total is more predictive than merely assuming that seasonal interception totals will regress to the NFL mean with no recourse to another statistic. As in the case of Green Bay, it also says something really useful that a team can exceed expectations for several seasons.

More of that next time, plus I will make some predictions for interception totals in 2012. Unless you get there first, of course.

And I promise it will not be in 15 months' time!