Thursday, July 05, 2007
Building a Better Mousetrap: Adjusted Pythagorean Winning Percentage
Time and again I've espoused on this blog the virtues of using a team's Pythagorean winning percentage to project more accurately how they will perform in the upcoming season. For the uninitiated the formula for the Pythagorean winning percentage is as follows:
Points Scored^2.37)/(Points Scored^2.37 + Points Allowed ^2.37)
The resulting number is a team's Pythagorean winning percentage. Multiplying that number by the number of games played will give a reasonable estimation of how many games a team should have won. However, the the formula is not without its flaws. For starters, blowouts, especially extreme blowouts can artificially inflate or deflate a team's Pythagorean record depending on whether or not they received or doled out the beating. The solution? Compute the Pythagorean winning percentage on a game by game basis, add up the totals, and divide by games played. This way each game is counted the same and the effect of blowouts is lessened. Here is a hypothetical example of the adjusted theorem in action for Eponymous State University.
ESU went 8-3 while scoring 312 points and allowing 198. Their Pythagorean winning percentage is .746. Their expected record is then 8.21-2.89. Their actual record is aligned pretty well with their Pythagorean record. However, one game sticks out like a sore thumb and is unjustly influencing the ratings. In the fourth game ESU dropped 70 on their opponent. Perhaps they were a Division III school, maybe they were a Division I school with a slew of injuries, maybe they turned the ball over nine times, maybe ESU ran up the score. Either way, we need to find a way to lessen that game's impact. Say for example, ESU stopped scoring after 30 points. They still win the game rather easily, but their seasonal Pythagorean winning percentage drops to .680 (7.48-3.52). That's almost 3/4 of a decrease in expected wins. If we determine the Pythagorean winning percentage of each game, add them up, and divide by 11 we get an adjusted Pythagorean winning percentage of .669 (7.36-3.64).
When we compute the Pythagorean winning percentage on a per game basis the difference between beating a team 30-3 and 70-3 is only about 3/100 of a point in winning percentage.
70 points is just piling on. Each extra score above a certain point negligibly increases the odds of winning the ball game. This in effect puts the proverbial 'cap' on margin of victory.
Now the important part. Is the adjusted Pythagorean winning percentage a decent predictor of a team's fortunes. Here are the r squared values for how three 2005 statistics predicted a teams 2006 winning percentage (BCS schools and Notre dame only--sample size 66).
2005 winning percentage: .352
2005 Pythagorean winning percentage: .3653
2005 adjusted winning percentage: .39
All three measures were reasonable predictors with adjusted Pythagorean winning percentage being the best predictor. It should be noted that in a post last off season, click here to view it, we found a team's (again BCS schools and Notre dame only) 2004 Pythagorean winning percentage to be a much better predictor than its 2004 winning percentage of its 2005 winning percentage. Part of the reason for the lowered predictive powers for both measures could be the added 12th game in 2006. The majority of the time the 12th game features a BCS school taking on a low-level Division IA or non-Division IA school for a guaranteed victory, thus boosting a team's winning percentage.