Pythagoras solved?: An R-squared of 97.8 percent
November 5, 2007 2 Comments
The question that just won’t die: Why do teams out-perform their Pythagorean projections? What if I told you I know how (at least a first-order answer), although it’s not exactly earth-shattering. A little while ago, I had 2/3 of the answer. A regression incorporating a team’s actual winning percentage, the standard deviation of their runs scored distribution over the 162 games of the schedule, the SD of their runs allowed, and their one-run game winning percentage predicted to Pythagorean residuals with an R-square of .665.
As the headline screams, I’ve gotten something even better. My data set is all teams from 1980-2006, via Retrosheet‘s game logs. I calculated each team’s Pythagorean projection (using the David Smyth/Patriot formula) and how far off the formula’s projection was from the actual retail price. I saved those residuals and calculated to see what else might explain them statistically.
I calculated what each team’s average margin of victory was when they won, and their average margin of losing. I also took standard deviations of those two numbers. I dropped all four of those numbers into a regression with the four variables I had used in my original study (Actual winning percentage, SD of runs scored, SD of runs allowed, and winning percentage in one-run games) trying to predict the Pythagorean residuals. Three variables shook out as significant: a team’s actual winning percentage, average margin of victory, and average margin of losing. I re-ran the regression with only those three variables, and they combined to explain 97.8% of the variance.
The directions were what you might expect. Teams that had a small average margin of victory (a lot of close wins) out-performed their estimations, while teams that lost a lot of blowouts were also more likely to outperform. Those who lost a lot of close games, but won a bunch of blowouts were under-performers. This finding isn’t anything that we couldn’t have (and haven’t already) guessed from looking at and thinking about the formula, but the magnitude of how much of the variance is being explained is huge. Pythagorean residuals tell us how a team won and lost its games and that’s about it. Now, what factors influence a team’s ability to win or lose one-run games and what factors influence a team’s ability to win or lose in blowouts still could use some investigation.
I do have one clue. A team’s average margin of victory and average margin of defeat were only correlated at .197, which isn’t all that big. It did point in the direction that if a team won games by a lot, they also tended to lose games by a lot. However, since the correlation isn’t all that big, I would go so far as to say that the two concepts are largely un-related to one another. This means that in order to explain Pythagorean over (or under) performance, we need to explain two “skills”: Why teams are able to win a lot of close games (as opposed to losing them) and why they get blown out. It’s possible that it’s mostly luck that drives these “skills,” but then there might be something to them.