Tuesday, August 9, 2011

An Alternate PMM Model

As mentioned in the last post, I'm looking at the code for implementing Danny Tarlow's PMM.  In Danny's (1 dimensional) model, each team has an Offense rating and a Defense rating and when Teami plays Teamj, the predicted score is:
Predi = Offensei * Defensej
In other words, we have a model where teams have a certain scoring potential (Offense) and the effect of defense is to reduce that potential proportionately.  So if a team has a Defense of 0.85, and plays a team with an Offense of 100, we expect the opponent to score 85 points -- 15 points below its potential.  On the other hand, if we play a team with an Offense of 60, we expect them to score 51 points -- 9 points below its potential.

That seems reasonable, but it's not the only possible model for the relationship between Offense and Defense.  For example, we might model defense as reducing the opposing team's offense by a fixed amount, e.g., Maryland holds opposing teams to 4 points less than they would score against an "average" defender.  With that model, our predicted score would look like this:
Predi = Offensei + Defensej
Which model is more reasonable?  My intuition says the first model, and it's certainly widely used, but I don't know of any work that has tried to determine the best model of the interaction between offense and defense in basketball.  (Although looking around did lead me to this interesting blog.)  Please correct my ignorance if you're aware of something relevant.  Easy enough, though, to test this model:

Predictor    % Correct    MOV Error
PMM71.7%11.23
PMM (alternate model) 64.5%13.65

So, not very good.  No real surprise, but it does raise the question of whether some different model for the interaction between Offense and Defense could out-perform the Offense*Defense model.