Tuesday, November 5, 2013


The rating model used here is based on multivariate Poisson regression and is an application of the work by Dixon and Coles (1997): Dixon, M. J. and Coles, S. G. (1997), Modelling Association Football Scores and Inefficiencies in the Football Betting Market. Journal of the Royal Statistical Society: Series C (Applied Statistics), 46: 265–280.

Although the method was used originally in the context of setting odds for betting (see wikipedia article on predicting association football matches), I am interested in testing the method for developing rankings for youth soccer teams that could be used for league formations, tournament seedings, etc. based on summer tournament and league match results. Elo rankings are used for the same purpose in youth chess and could be used for the same purpose for soccer, but Dixon's approach allows one to compute attack and defense strengths which can then be used to simulate number of goals in a match up (for example, check out the Match Predictor to to the left to predict your next match). This allows one to compare the characteristics of teams using their attack and defense strengths. Also Elo ratings do not use the goal differential information from games, and thus are not using a significant piece of data available from competitions involving accumulated points.

The strength ratings you see on this site are based on a statistical analysis of 13 months of games and ranks the attack and defense of the team based on the number of goals it scores and allows against other teams of different strengths. The model used here weights away and home games differently because home teams have an advantage and accounts for the surface on which the game was played (turf versus grass) because more goals tend to be scored on grass. More recent games are weighted more heavily thus team strength can evolve over time.

Unlike a points standing (as used in league standing), a win does not necessary increase a team's strength rating. If a team is expected to win 4-0 against a weaker team, but instead wins 2-0, its attack rating will go down. Similarly, if it is expected to allow no goals in a game against a weaker team, but instead allows a goal, its defense rating will go down. The expected goals for team A in a match-up with team B is the attack score of team A divided by the defense score of team B. However, the actual outcome of the game (win/tie/loss) is chance.

The only data that enters into the analysis are the match results---the opponents and the goals scored by each team. Cross-league tournament games are used to rank teams who play in different fall leagues. In order for all teams to be ranked, there must be a path from every team to every other team. Thus a sufficient number of cross-league games is required to ensure full connectivity and to ensure that the relative ratings of different leagues does not hinge on 1-2 matches.

One of the nice features of a probability model (like I am using) is that it can be used to predict the outcome of games. The following table shows how the probability of a win, tie, or loss depends on the difference between two team's strength ratings (those "total strength" numbers appearing in the 2nd column in my tables). The table is based on "balanced" teams, where an individual team's attack and defense values are similar. If, for example, the two teams involved were highly defense oriented with a much higher defense strength than attack strength, the results would be more skewed to a tie.difference in the total strength scores and probability of win (by the stronger team), tie, or loss. Note, Elo ratings in chess are also based on a probability model and the different in rating of chess players can similarly be translated into a prediction regarding win/loss/tie. There are also Elo ratings used in soccer but the update algorithm used for soccer does not (to my eye) come from a statistical model (unlike the chess Elo algorithm). Dixon and Cole's approach has a better statistical foundation for modeling soccer match data and makes testable predictions about the performance of the ratings.
0.00 25 50 75 100 125 150 175 200 225 250 275 300
win 35 39 44 49 54 59 64 68 73 77 81 85 88
tie 31 31 30 29 27 26 23 22 19 17 14 11 9
lose 34 30 26 22 19 15 13 10 8 6 5 4 3
As you can see a difference of 1.5 between teams indicates that the stronger team is heavily favored. A difference 3.0 between teams is very large.

If you want the expected (mean) goals scored by team A against team B. Divide team A's attack score (column 3) by team B's defense score (column 4).