## Wednesday, April 3, 2013

### Lunchtime Sports Science: Fitting a Bradley-Terry Model

Power rankings are game rankings that also allow you to estimate the likely outcome if two opponents were to face each other. One of the simplest of these models is known as the Bradley-Terry-Luce model (or commonly, Bradley-Terry). The idea is that each player $$i$$ is assumed to have an unknown rating $$R_i$$. If players $$i$$ and $$j$$ compete, the probability that $$i$$ wins under this model is expected to be about $\frac{R_i}{R_i + R_j}.$ This model is very popular for hockey and other games; one commonly seen version is called KRACH.

Let's fit a Bradley-Terry model to the current season of NCAA D1 men's hockey. The Frozen Four starts on Thursday, April 11, so you'll get to see how well your predictions do.

You'll need to have R installed. Once R is installed, install the "BradleyTerry2" package that's freely available for R (thanks to Heather Turner and David Firth). To do this, start R and run the following command; you'll have to pick a source.
install.packages("BradleyTerry2")
Next, download two files from my hockey GitHub - R code that fits a basic Bradley-Terry model and a data file containing the NCAA D1 men's hockey game results going back to 1998.

https://github.com/octonion/hockey/blob/master/lunchtime/uscho_btl.R
https://github.com/octonion/hockey/blob/master/lunchtime/uscho_games.csv

Make sure both files are in the same directory and run the R code. That's it, you've built a power ranking using a Bradley-Terry model. You should get output that looks like this:

ability      s.e.
Quinnipiac              1.687042594 0.5678939
Massachusetts-Lowell    1.480098569 0.5872701
Minnesota               1.428503522 0.5638946
Yale                    1.115338226 0.5641414
Miami                   1.114307264 0.5479346
Notre Dame              1.109912670 0.5523657
Boston College          1.091836391 0.5815233
St. Cloud State         1.079314965 0.5573018

The order should be the same as USCHO's KRACH rankings.

How do we use these ability estimates to predict game outcomes? These values are the logarithms of the ratings I've mentioned above, so first apply the exponential to get the rating, then the estimated winning probability is the team's rating divided by the sum of the team and opponent ratings. For the teams in the Frozen Four we get a power rating of $$e^{1.687} = 5.40$$ for Quinnipiac and $$e^{1.079} = 2.94$$ for St. Cloud State, so we estimate the probability of Quinnipiac beating St. Cloud State to be about $\frac{5.40}{5.40+2.94} = 0.65.$ What's your estimate for Massachusetts-Lowell beating Yale?