Thursday, October 22, 2009

Under the Hood

There are three key concepts behind the statistics on this site:
The offensive and defensive efficiencies of a team simply represent the number of points a team will score against and have scored against them per 100 possessions.  In basketball the concept of a possession is simple: one team has a chance to score a basket and continues until they either succeed or somehow turn the ball over (turnover, foul, defensive rebound).

In football the concept of a possession is slightly less clear.  Is it a single snap from scrimage?  An entire drive?  Where does a kickoff fall in this definition?  If we were to use a single play from scrimage as our definition of a possession, we find that our symmetric relationship between possessions doesn't hold.  One team could easily have 100 possessions in a game, whereas their opponent may only muster 50.  In this scenario we would have to account for the amount of offense and defense in each game in addition to the efficiency of each squad.  This seems needlessly complicated.

On the flip side, if we were to use a drive as the definition of a possession then the game becomes significantly more symmetric and familiar.  One team takes the ball, attempts to score, and then returns the ball to their opponent.  Unfortunately this further reduces our already-small sample size.  In college basketball there are 340 teams playing a 30-game slate with roughly 70 possessions per game.  In football there are 120 teams playing an 10-game schedule with roughly 20 possessions per game. (All numbers approximate)  Furthermore, how do we account for the scenario in which a pass is intercepted and returned for a touchdown?  A kickoff returned for a touchdown (or the return is dropped and the kicking team returns it for a touchdown)?

I finally settled on the philosophy that every play is a possession.  Every snap, kickoff, punt ... it's all a single possession because each team has the potential to score on every play.  When one team is on offense and has the ball, they are in essense playing both offense and defense on each snap; the same goes for the "defense".  They are also trying to block the other team from scoring and -- if possible -- score themselves.

I may revisit this decision later, but for now it seems to serve me well and allows me to "fit" everything into the other two portions of the model.

This brings us to part two: Pythagorean expectation, a formula developed by Bill James that uses the average number of points scored by and against a team to determine their expected winning percentage.  If S is the number of points scored by a team and A is the number of points allowed, then the expected winning percentage of that team is

WinPct = 1 / (1 + (A/S)^E)

where E is an exponent that varies from sport-to-sport.  In baseball the most accurate value appears to be 1.81.  In college basketball the estimation of the exponent has varied from 14.0 (by Dean Oliver) to 16.5 (John Hollinger) to 11.5 (Pomeroy).  After analyzing data from several years of college football, I settled on a value of 3.0 for E, although many values in the range [2.5,3.2] produced reasonable results.

Now that we have a definition of a possession and a value of E for our Pythagorean winning percentage, that brings us to the log5 formula.  This formula states that if two teams were to play each other, and one team has a winning percentage of A and the other a winning percentage of B, the odds that the first team would win are

            A - A * B
  WPct = -----------------
         A + B - 2 * A * B

There are two other small factors my system takes into account: home field advantage and the primacy of more recent games.  In college football the home team wins approximately 62% of all games.  This effect can be seen to varying degrees in other sports as well.  It is interesting to note that the magnitude of home field advantage is larger during the early stages of the season but diminishes as the season progresses.  From 2006-2009, the home team won 68% of all games played in the first half of September, compared with 57% of all games played in the first half of November.  Even taking into account the tendency of Powerhouse U. to schedule early-season games against Sister Mary's School of the Blind, the effect deteriorates as the year progresses.  This may be caused by teams becoming more comfortable playing on the road as they mature during the year, but for now it's simply an observation.

There is also the matter of weighting more recent games more heavily.  Unlike many other systems, I do not completely discard games from the previous years.  I use an exponential decaying factor that causes games from the start of the year to count as roughly 3/5th of a full game, and games from a full year ago as roughly 1/6th of a game.  This allows early-season predictions to be "in the ballpark" for most teams.  Let's be honest; USC is going to remain USC from year-to-year, and Duke is going to  ... well, let's just say Durham becomes a much cheerier place starting in mid-November.

Now that you've had the whirlwind tour of the statistics behind the site, let's get on with the good stuff: rankings.