Graves-Reese College Football Rankings
These ratings and rankings were developed by Todd Graves (a Ph.D. statistician working at Los Alamos National Laboratory) and Shane Reese (a Ph.D. statistician and professor at Brigham Young University). The ratings are based on who won each game of the current season, they include a home field advantage effect, and they collapse all teams not in Division 1-A into a single "Miscellaneous" team. We do not consider margin of victory, nor do we bias our ratings using results from past seasons, even very early in the year. Hence our rankings would be considered "Retrodictive" in the computer rankings parlance (i.e. they strive to rate the performance of the teams without bias, rather than using all possible information to get the best possible predictions for future games). Still, it is possible to construct predictions from our ratings.

Our ratings are based on the Bradley-Terry model. Suppose the rating of the visiting team is v, the home team has rating h, and the home field advantage parameter is a. Then the probability that the visiting team wins the game is exp(v+a)/{exp(h) + exp(v+a)}. One could estimate all teams' abilities using maximum likelihood, but then all undefeated teams would have infinite ratings. A solution is to use Bayesian hierarchical modeling; we assume that each team's rating has a normal prior distribution with mean zero and standard deviation sigma, we further put a prior distribution on sigma and on the home field advantage parameter. (The effect of using prior distributions is to multiply the likelihood function by additional terms). We then maximize this product and report the maximizing values (the "posterior mode") as each team's rating.