Football Ranking Model Details

 

The model consists of fitting N observations (games) involving division 1 teams. All non-division 1 teams are considered the same. Only the margin of victory and the home team is relevant for each game. Only games so far in a season are relevant. There is no other information used, i.e. previous year’s record, conference strength, etc. All games count the same, although a parameter can be changed to weight recent games more heavily. Margins above a certain threshold are ignored.

 

The model can be written generically as the following, where “score difference” refers to the margin:

 

            Score difference = S-Function(power difference, home team)                                  (1)

 

The “power difference” is the difference in “power” value for a given game. The “power” values are a vector of real numbers, which form the ranking system, going between 0 (worst) and arbitrary maximum (best). There is one power value for each division 1 team and one for a generic non-division 1 team. The “S-Function” referred to above can be any generic S-shaped function, which goes through 0,0 for games without home field advantage (i.e. teams with the same power value will tie at a neutral site). One form is given by:

 

                                                                      (2)

 

where “k” is a fitting parameter and the term Dscoremax is a limiting margin to which all actual margins are collapsed. Thus, if this term is set to 21 (the default value), margins beyond 21 have no impact on the rankings. Also in Equation 2, the parameter x0 is used to account for home field advantage. For neutral site games, x0 = 0, and the function in Equation 2 is fully symmetric. For games with a home team, x0 is shifted so that the home team has an advantage of h points. The value h is the natural fitting parameter for home field advantage and thus x0 is calculated based on h. For delta values referring to the home team minus the away team, this takes the form:

 

                                                                                                  (3)

 

An example is shown in Figure 1 for k = 2, h=5 and Dscoremax= 21.

 

Fitting the model involves the following steps:

 

  1. Choose Dscoremax (= 21 for example), and any weighting for recent games, etc.
  2. Choose values for k and h
  3. Invert Equation 2 for all of the games played so far. This yields Dpower values for each game.
  4. Solve the linear regression problem formed by having 121 (120 FBS teams plus one generic non-FBS team) power values satisfy the N inverted Dpower values determined in Step 2 above.
  5. Shift the 121 power values to a minimum of  0 (upper bound unbounded)
  6. Use Equation 2 with the new power values to solve for the score difference for each game
  7. Compute the lack of fit, and repeat step 1 as necessary

 

One difficulty in this procedure is inverting Equation 2 in Step 2 for games whose margin matches or exceeds the maximum value of Dscoremax. One solution is to assign a slightly lower margin in this case, anywhere from one-half to one-tenth of a point, which is arbitrary but necessary for the inversion done in Step 2. An alternative strategy is to solve the non-linear regression problem over all of the 121 + 2 parameters. While similar in degrees of freedom, and perhaps giving better goodness of fit to scores data for particular cases, this strategy tends to be less stable with regard to starting values for the power rankings (which are unnecessary in the present formulation) and gives greater variability week to week.

 

Figure 1. A plot of Equation 2, with DSmax=21, k=2 and h=5 (Eq. 3) for illustration,

showing the mapping between power difference and margin of score.