A Predictive Pythagorean For Football.

If baseball is the sport to which all other flavours of analytics can trace back their origins, then the most widely recognisable product of the sabermetrics movement is the Pythagorean Expectation. Elegantly simple and possessing a format that is instantly recalled by anyone who has taken maths at even the most rudimentary of levels, Bill James' equation allows a team's season long achievements to be seen with some of the unrepeatable, luck driven outcomes removed from the table.

In it's rawest form a baseball team's runs scored and allowed record is examined and mindful of the part that luck may play in scoring or conceding over various timescales, an expected win/loss record is produced that may differ from reality especially in cases where a team has managed to record an unsustainably large number of narrow victories. The aim is to produce an expected record that may be more indicative of a team's true ability level, and therefore future expected performances, than their actual record that may by partly a product of good or bad fortune.

As with many such developments the initial insight was immensely valuable and through refinement and input from the wider sabermetrics community Pythagorean expectation has become an extremely useful tool in the evaluation of team ability.

Other sports inevitably developed their own version of James' contribution and where basketball and American Football led, Association Football eventually followed. James initially suggested 2 as his exponent of choice (hence the Pythagorean name), but just has his initial attempt has undergone much change, a straight conversion from the diamond to the football pitch wasn't really possible. The most obvious problem is that draws aren't a feature of baseball, but football positively revels in producing them. So an approach based around success rate, where draws account for half a win was required in football.

Other potential problems for the formula existed in both sports. Scoring environment is a subtle, but important factor in producing results in both baseball and football. A football team which plays in contests where a below average number of goals are scored compared to the league average will see more draws than a team which plays with a more expansive approach. So inclusion of the goal environment for individual sides somewhere in the exponent is desirable for any sport choosing to develop a Pythagorean approach.

Finally, aspects of a particular sport that are well understood, but transient, may alter scoring or conceding rates in one season, but may be absent or substantially different in subsequent ones. Unusually large numbers of red cards, for example may result in a team's seasonal goal scoring records being correctly interpreted by a Pythag approach, but the team may improve next season through nothing more than better behaviour or a more evenly distribution of fouls.

Using Pythag in football is perfectly possible. It's really duplicating the goal difference approach where teams who have inflated points totals compared to those typically expected for a similar level of goal difference are labelled as over achievers who have probably got lucky in a large number of close games and will fall to earth sooner rather than later.

As a tool it also appears to make intuitive sense by downgrading those who appear to benefit by winning more than their fare share of close matches, while inflating the prospects of teams who appear capable of better if they had received a little more luck in how their scoring and conceding sequencing occurred. That alone, however, merely gives us an alternative opinion regarding the quality of teams. Our next step is to see if these opinions help to give an improved view of the future compared to some other measurement.

The aim of the Pythagorean expectation  for a team is to reduce the effect of non reproduceable, luck driven events and other sports routinely use a team's Pythag from one season to better predict their actual record in an upcoming campaign. Therefore to test the value to football I calculated the Pythag using an exponent that incorporates goal environment for every EPL team during the continuous "38 games era", comprising  the last 17 completed seasons. I then plotted these figures against the number of points gained by the teams in the EPL during the following season. As a comparison I repeated the process, but used just actual points gained in both seasons.

In total 340 teams recorded a Pythagorean expectation in the EPL and survived to compete in the division in the next season, so there is a survivor bias in the sample, although movement is less pronounced than in lower leagues where the best are also removed from the sample in subsequent seasons. 166 teams had Pythagorean points expectations that were in excess of their actual total for that season and 174 teams overperformed in reality. The average over or under performance in each case was around 3.5 points.

The plot using actual points in both seasons was very similar to the Pythag plot (shown above), but Pythag had a stronger correlation, with respective r^2 of 0.64 and 0.57 suggesting that "luck" corrected points totals are the more valuable predictor of future performance levels than mere raw results.

Individual cases are always interesting, if rarely indicative of a trend as a whole and the "unluckiest" team in the sample was Manchester City in 2003/04. They amassed just 41 points in finishing 16th against a Pythagorean expectation of 53 and in 2004/05 they lived up to their Pythag expectation from the previous year by gained 52 points in finishing 8th.

Neighbours, United were the sample's highest flying overachievers in 1999/00 when they amassed 91 actual points against a goals for and against derived expectation of just 80 and like City in the next season they gravitated towards their previous year's "deserved" points total by gathering exactly 80 points in retaining their title.

Team Performance Against Pythagorean Expectation Over The Last 17 EPL Seasons.

Team.	Number Of  Over Performing Seasons.	Number Of Under Performing Seasons.
Arsenal.	9	8
Aston Villa.	5	12
Birmingham.	4	3
Blackburn.	7	8
Bolton.	6	7
Charlton.	5	3
Chelsea.	8	9
Derby.	4	3
Everton.	4	13
Fulham.	4	7
Leeds.	6	3
Leicester.	3	4
Liverpool.	3	14
Manchester City.	6	6
Manchester United.	13	4
Middlesbrough.	5	8
Newcastle.	7	9
Stoke.	3	1
Sunderland.	6	5
Spurs.	9	8
WBA.	1	5
WHU.	10	4
Wigan.	6	1
Wolves.	3	1

The temptation is great to use Pythagorean expectation to try to identify persistently over or under achieving sides and to try to identify factors such as particularly astute managers or well balanced teams as crucial factors in producing more points for your scoring record. The narrative is appealing and baseball particularly has gone done this route. So for those interested I've tabulated the number of over and under performing seasons recorded by teams with four or more EPL seasons to their name since the league went exclusively to 20 teams.

It's also tempting to be immediately struck by Manchester United's over populated positive column. However, the majority of sides are within touching distance of parity between good and bad and many such as Wigan have achieved a lopsided record under multiple managers. Also survivor bias exists within the sample. Many "good/lucky" managers may have dropped of the radar if their tenure began with a "bad/unlucky" initial run of seasons. A fate that, anecdotally very nearly befell Sir Alec had Mark Robins not grabbed a winning FA Cup goal against Forest well before the start of the Premiership adventure. Overall evidence of a persistent ability to cheat your goal scoring records is fairly weak.

Pythagorean estimates are a useful tools, especially as a means to include goal environments into the equation and their versatility can be extended to a match by match seasonal analysis, but genuine and persistent over or under achievers are likely to be a very rare beast. Team's which are out performing their goal scoring and conceding records are probably just enjoying a slice of good fortune.