Friday, August 18, 2006

Hockey: a shot quality formula

In this post from last week, I described a study by Alan Ryder that analyzed play-by-play data to determine the expected value of different types of shots on goal, based on distance and shot type.

Ryder’s data was in the form of graphs, rather than formulae – if you wanted to find out how many goals an even-strength 40-foot wrist shot was worth, you’d up some values on his graphs and do a quick calculation.

But
this study by Ken Krzywicki (also on hockeyanalytics.com), actually calculates a single formula for all shots.

Krzywicki uses a logistic regression, which is just a regular regression, but instead of trying to predict the probability of scoring, it predicts the logarithm of the odds of scoring. The regression results are on page 4 of the study, but I’ll just run through a quick example.

What is the chance of scoring on a 20 foot wrist shot on the power play that’s not a rebound? According to Krzywicki’s formula,

Add 0.3654 for a shot of 17-22 feet.
Add 0.0093 for a wrist shot.
Add 0.0000 because the shot is not a rebound.
Add 0.4007 for a shot that’s on the power play.
The total is –1.4615. Take the negative, which is 1.4615.
Take the natural antilog (e to the power) of 1.4615, giving 4.31.
Add 1, giving 5.31.
Finally, compute 1 divided by 5.31. That gives 0.19.
So the chance of scoring on the shot is 0.19.

Or, put another way, a 20 foot power-play wrist shot is worth 0.19 goals.

Krzywicki runs through a bunch of statistical tests to validate the model ... for instance, he presents a graph of predicted versus actual probabilities by decile, and the model appears to fit quite well. As far as I understand the tests, it’s still possible that the model is biased for certain types and distances of shots, but if so, the biases appear to balance out overall.