Statistical Models
Statistical Modeling
Below is an exmaple of the type of statistical modeling made possible with the rating system. The dataset that the model below is based in is all FIFA international matches from 1872 to August 17, 2013 (in total 35,872 matches).

FIFA Internationals
The sport of football has evolved substantially since the first recognized FIFA match between England and Scotland on the 30th of November 1872. Consequently, a regression model based over the entire period that the dataset comprises would be inadvisable. The number of matches per game has varied considerably over time. Below is a chart of goals per match 1872-2013 (the line is a 5 year moving average): Goals per match
A series of simple regression models that cover a decade (or more) of data would provide a better fit and thus will be used in this example.

Simple Linear Regression
A simple linear regression model will suffice to provide a reasonable fit.

Goals per match
In the equation above, the y represents what we are trying to model (the amount of goals scored for a given country), the α represents the intercept of the regression, while the β is the value used to predict (the difference between the two country ratings).

Simple Linear Regression Models
After fitting the data, the following models can be used to model and predict FIFA international matches. First we model the goals scored for the Favorite (the team with the higher rating). Then we model for the Underdog (the country with the lower rating). Note that for the underdog there is a MIN column. This value is a minimum floor for goals scored for the underdog.

TIME Matches α β
Pre 1920 544 1.4610.664
1920-1945 2,022 1.5800.667
1946-1959 2,025 1.5860.676
1960-1969 2,564 1.1860.800
1970-1979 3,705 1.0090.773
1980-1989 5,033 0.8520.779
1990-1999 6,971 0.8940.808
2000-2009 9,440 0.8670.822
2010- 3,572 0.8990.753
TIME Matches α βMIN
Pre 1920 544 1.461-0.1300.68
1920-1945 2,022 1.580-0.1610.68
1946-1959 2,025 1.586-0.1540.47
1960-1969 2,564 1.186-0.1390.43
1970-1979 3,705 1.009-0.1270.25
1980-1989 5,033 0.852-0.1680.23
1990-1999 6,971 0.894-0.1250.33
2000-2009 9,440 0.867-0.1320.36
2010- 3,572 0.899-0.1240.33

Consider the friendly match between England and Scotland played at Wembley Stadium, London, on 14th of August, 2013. The annual ratings going into the match were England at 100.20 and Scotland 98.35. Empirically it was found that the home field advantage for FIFA matches dictates that the home team increase its rating by 0.27, while the away team decreases its rating by 0.27. This changes the ratings (including home field advantage) to England 100.47, Scotland 98.08, with the rating differential calculated as (100.47 - 98.08 = 2.39). Thus the predictor variable (x) is 2.39

To get the final scores, we now utilize the appropriate simple linear equations. Since this match occurred in 2013 we use the following equations:

Goals per match
y= 0.899 + (.753)* 2.39
y= 0.899 + 1.7997
y= 2.669

England would be expected to score 2.7 goals.

Goals per match
y= 0.899 + (-0.124)* 2.39
y= 0.899 -.2964
y= 0.6026

Scotland would be expected to score 0.6 goals.

The model score was: England 2.7 Scotland 0.6
The actual score was: England 3 Scotland 2

We can use empirical historical data to guage the error of the model. Below is a table detailing the actual goal scoring distributions for those countries that the model predicted 2.7 and 0.6 goals, respectively:


Below is the percent distribution:


From these empirical distributions we can derive an estimate of the probabilities of all scorelines. Below is the table of all scorelines ranked from most probable to least. The actual 3-2 scoreline should occur approximately 2% of the time. Given the differential in ratings, we would expect an England win 82.04% of the time, a draw 12.53% of the time, and a Scotland win 5.42% of the time.

EnglandScotlandPct EngPct ScoPROBCUM PROBRESULT
2026.85%60.81%16.329%16.329%ENGLAND WIN
3026.85%60.81%16.329%32.658%ENGLAND WIN
1016.67%60.81%10.135%42.793%ENGLAND WIN
4015.74%60.81%9.572%52.365%ENGLAND WIN
2126.85%29.73%7.983%60.348%ENGLAND WIN
3126.85%29.73%7.983%68.331%ENGLAND WIN
4115.74%29.73%4.680%83.033%ENGLAND WIN
504.63%60.81%2.815%85.848%ENGLAND WIN
018.33%29.73%2.477%88.326%SCOTLAND WIN
3226.85%7.57%2.032%92.390%ENGLAND WIN
514.63%29.73%1.376%93.766%ENGLAND WIN
1216.67%7.57%1.261%95.028%SCOTLAND WIN
4215.74%7.57%1.191%96.219%ENGLAND WIN
028.33%7.57%0.631%96.849%SCOTLAND WIN
600.93%60.81%0.563%97.412%ENGLAND WIN
2326.85%1.62%0.435%97.848%SCOTLAND WIN
524.63%7.57%0.350%98.634%ENGLAND WIN
610.93%29.73%0.275%98.909%ENGLAND WIN
1316.67%1.62%0.270%99.179%SCOTLAND WIN
4315.74%1.62%0.255%99.434%ENGLAND WIN
038.33%1.62%0.135%99.570%SCOTLAND WIN
534.63%1.62%0.075%99.645%ENGLAND WIN
2426.85%0.27%0.073%99.717%SCOTLAND WIN
3426.85%0.27%0.073%99.790%SCOTLAND WIN
620.93%7.57%0.070%99.860%ENGLAND WIN
1416.67%0.27%0.045%99.905%SCOTLAND WIN
048.33%0.27%0.023%99.970%SCOTLAND WIN
630.93%1.62%0.015%99.985%ENGLAND WIN
544.63%0.27%0.013%99.997%ENGLAND WIN
640.93%0.27%0.003%100.000%ENGLAND WIN

Copyright Crankshaw Sports Stats. All rights reserved.