Statistical model

Consider a game between team A and team B. In our model, the number of goals scored by country A are Poisson distributed with parameter (a number) L(A,B). This means that we expect that team A will score about L(A,B) goals in a match against team B. Here

L(A,B) = "Normal number of goals" x ("Strength of team A")/("Strength of team B").

"Normal number of goals" is a parameter (a number) which is interpreted as the average number of goals scored by one team in a match between to equally good teams. "Strength of team A" is a parameter (a number), which tells how good team A is, whereas "Strength of team B" tells how good team B is. The strenght of Germany is fixed to 100, and the strengths of all other counties are given relative to this.

Further, the number of goals scored by team B are Poisson distributed with parameter L(B,A), independent of the number of goals scored by team A.

This means that if team A has a higher strength than team B, we will expect that team A wins, because L(A,B) will be greater than L(B,A). However, it will still be a possibility for draw, or that team B wins.

Of course, our simple model is not able to cover all important aspects of a football match. However, in the World Cup, few match results will be available for parameter estimation, so a simple model is needed to avoid overfitting. However, in a national league, with many games over a season, one may consider several extensions to our model. These include

- Home-team advantage
- Offense strength and defense strength
- That the strengths of the teams vary over the season
- Number of goals to each team are correlated

In the later years, several articles about this theme have been published in the statistical literature. A good introduction is Lee, A. (1997), "Modeling Scores in the Premier League: Is Manchester United Really the Best?", Chance, Vol 10, s. 15-19.

Probabilities for the World Cup, based on similar ideas, is also presented by DonationCoder.com.

Parameter estimation

The parameters in the model are "Normal number of goals" and the strengths of each team. These parameters must be estimated before any probability calculations can be done. Before the start of the World Cup, the estimation is based on evaluations from sports jounalists in the Norwegian newspaper Aftenposten and six prominent Norwegian football experts (Egil "Drillo" Olsen, Lars Tjernås, Øyvind Alsaker, Vidar Davidsen, Nils Johan Semb og Karl-Petter Løken). The journalists and experts have guessed the results of several hundred hypothetical games, and these results are tranferred to number values of the parameters.

When the tournament starts, the real games are taken into account as well. The information value of the hypotethical games (the expert guesses) are weighted versus the real games, such that the hypothetical games are equally important as 24 real games. This means that when all teams have played two or more matches, the real games are the most important part in the estimation of the parameters.

Estimating the parameters means to find the parameter values that fit the data (the match results) as good as poosible. In our case, the parameters are estimated by maximising a modified Poisson likelihood. The difference from an ordinary Poisson likelihood is that it is robustified by downweighting large victories and by adding a penalty term that shrinks the individual strength parameters towards a common mean.

Estimated strength of all countries

The current estimate of "Normal number of goals" is 1.01.

The estimated strength values of each team is given in the table to the right (sorted), together with the FIFA ranking per May 17, 2006. The discrepancy between the strength and the FIFA ranking is due to the fact that the expert opinions differ somewhat from the FIFA-ranking. In addition, as the tournament makes progress, teams with good results will obtain high strength, even if they might have a low FIFA ranking.

Responsible for web-page: Magne Aldrin

The calculations are updated: juli 08, 2006 22:54

Method

Strength and ranking

Country	Strength	FIFA- ranking
Brazil	104	827
Germany	100	696
France	96	749
Italy	96	728
Argentina	87	746
England	78	741
Spain	73	756
Portugal	73	750
Switzerland	73	648
Netherlands	71	768
Mexico	61	758
Sweden	61	709
Czech-Republic	61	772
Ghana	58	600
Australia	57	612
Ecuador	56	631
Korea-Republic	55	677
Croatia	55	686
Ukraine	54	609
Ivory-Coast	53	669
Poland	51	677
USA	50	756
Paraguay	50	653
Angola	42	581
Japan	42	705
Serbia-Montenegro	39	610
Togo	39	569
Tunisia	37	693
Iran	37	686
Trinidad-Tobago	36	604
Costa-Rica	36	683
Saudi-Arabia	32	651