# Least squares rating of football teams

The Wikipedia article Statistical association football predictions mentions a method for least squares rating of football teams. The article does not give any source for this, but I found what I think may be the origin of this method. It appears to be from an undergrad thesis titled Statistical Models Applied to the Rating of Sports Teams by Kenneth Massey. It is not on football in particular, but on sports in general where two teams compete for points. A link to the thesis can be found here.

The basic method as described in Massey’s paper and the Wikipedia article is to use a n*k design matrix A where each of the k columns represents one team, and each of the n rows represents a match. In each match (or row) the home team is indicated by 1, and the away team by -1. Then we have a vector y indicating goal differences in each match, with respect to the home team (i.e. positive values for home wins, negative for away wins). Then the least squares solution to the system Ax = y is found, with the x vector now containing the rating values for each team.

When it comes to interpretation, the difference in least squares estimate for the rating of two teams can be seen as the expected goal difference between the teams in a game. The individual rating can be seen as how many goals a teams scores compared to the overall average.

Massey’s paper also discusses some extensions to this simple model that is not mentioned in the Wikipedia article. The most obvious is incorporation of home field advantage, but there is also a section on splitting the teams’ performances into offensive and defensive components. I am not going to go into these extensions here, you can read more about them i Massey’s paper, along with some other rating systems that are also discussed. What I will do, is to take a closer look at the simple least squares rating and compare it to the ordinary three points for a win rating used to determine the league winner.

I used the function I made earlier to compute the points for the 2011-2012 Premier League season, then I computed the least squares rating. Here you can see the result:

PTS LSR LSRrank RankDiff
Man City 89 1.600 1 0
Man United 89 1.400 2 0
Arsenal 70 0.625 3 0
Tottenham 69 0.625 4 0
Newcastle 65 0.125 8 3
Chelsea 64 0.475 5 -1
Everton 56 0.250 6 -1
Liverpool 52 0.175 7 -1
Fulham 52 -0.075 10 1
West Brom 47 -0.175 12 2
Swansea 47 -0.175 11 0
Norwich 47 -0.350 13 1
Sunderland 45 -0.025 9 -4
Stoke 45 -0.425 15 1
Wigan 43 -0.500 16 1
Aston Villa 38 -0.400 14 -2
QPR 37 -0.575 17 0
Bolton 36 -0.775 19 1
Blackburn 31 -0.750 18 -1
Wolves 25 -1.050 20 0

It looks like the Least squares approach gives similar results as the standard points system. It differentiates between the two top teams, Manchester City and Manchester United, even if they have the same number of points. This is perhaps not so surprising since City won the league because of greater goal difference than United, and this is what the least squares rating is based on. Another, perhaps more surprising thing is how relatively low least squares rating Newcastle has, compared to the other teams with approximately same number of points. If ranked according to the least squares rating, Newcastle should have been below Liverpool, instead they are three places above. This hints at Newcastle being better at winning, but with few goals, and Liverpool winning fewer times, but when they win, they win with more goals. We can also see that Sunderland comes poor out in the least squares rating, dropping four places.

If we now plot the number of points to the least squares rating we see that the two methods generally gives similar results. This is perhaps not so surprising, and despite some disparities like the ones I pointed out, there are no obvious outliers. I also calculated the correlation coefficient, 0.978, and I was actually a bit surprised of how big it was.