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# NHL Adjusted Plus-Minus, Part 1

Problems with plus-minus have been well-documented by hockey analysts.  For example, the topic of plus-minus was Frequently Asked Question #1 in Gabe's 10-part series on FAQs about Statistical Analysis in the NHL.  Various analysts have different methods for correcting the issues associated with plus-minus, and have come up their own statistics that attempt to do what plus-minus attempts to do.  (Some of those are also described in that 10-part series).  Examples of a few of the most well-known advanced statistics are Alan Ryder's Player Contribution, Gabe's relative +/- and Corsi stuff, and Tom Awad's Goals Versus Threshold.

The purpose of the next few posts that I'll make is to describe another statistic with the same goal. It is an Adjusted Plus-Minus (APM) statistic that is similar to the APM for basketball developed by Rosenbaum, and Ilardi and Barzilai.  In these posts, I'll summarize an NHL version of APM, attempting to describe the models in plain language, and keeping the technical details to a minimum.  But if you're interested in the details, you can check out these two papers:

1. A Regression-Based Adjusted Plus-Minus Statistic for NHL Players (should be free to sign up and download, and there are other interesting articles there, including these four articles)

2. An Improved Adjusted Plus-Minus Statistic for NHL Players (while at that site, you can also check out the DIGR paper by Schuckers)

The first paper, A Regression-Based Adjusted Plus-Minus Statistic for NHL Players, has the most details, the most results, and the most discussion, and is the best one to read first.

Here is the jist of those papers.  The basic goal of the APM statistics, like the other player performance metrics mentioned above, is to measure a player's individual contribution to his team.  The APM model does this by using a huge multiple linear regression.  The explanatory variables are players, and indicate whether or not they were on the ice during each shift.  The outcome variable is goals per 60 minutes during each shift.  Each line of data corresponds to a "shift", by which we mean a period of time on the ice when no substitutions are made.  So when one player changes, that is a different group of players on the ice, and is considered a new "shift".

The estimates obtained in any multiple linear regression have the interpretation that they estimate the effect that each variable has on the outcome, independent of all other variables.   That's the key... independent of all other variables.  (There are assumptions being made here, and the models are of course not perfect.  I'll discuss that stuff in a later post.)

This means that for this model, the results are interpreted as an estimate of the goals per 60 minutes that a player contributed to his team, independent of all the other players in the league.  In other words, it estimates the goals per 60 minutes contributed by a player, independent of (or adjusted for) the strength of his teammates and opponents.  I also included zone start variables in the model, so the ratings are independent of zone starts too.

There are two different models, one for even strength situations, and one for power play and shorthanded situations.  For each player, there are separate ratings for offense and defense, and separate ratings for even strength, power play, and short handed situations.  Also, the goals per 60 minutes ratings can be used as is, or playing time can be used to make the ratings goals per season.  For these posts, I'll choose to show goals per season, mostly because then it makes sense to add even strength and special teams ratings to get overall ratings.

Here's an example of the results, the top 10 players in OPM (Offensive component of Adjusted Plus-Minus):

 Player Team Pos GP MinG OPM DPM APM AErr EVO EVD EVA PPO PPD PPA SHO SHD SHA S. Crosby PIT C 63 20.3 46.2 -5.3 40.9 15.1 34.0 -3.0 31.1 12.2 -2.4 9.8 J. Toews CHI C 76 19.1 43.6 13.7 57.4 15.0 26.6 8.6 35.2 16.2 1.1 17.4 0.8 4.0 4.8 J. Thornton S.J C 81 19.1 42.5 9.5 52.0 16.2 25.8 8.7 34.6 16.7 0.7 17.4 A. Ovechkin WSH LW 78 21.5 41.9 7.1 49.0 19.8 29.5 9.0 38.4 12.4 -1.8 10.6 R. Getzlaf ANA C 73 19.6 40.5 13.2 53.7 17.8 19.6 12.8 32.4 20.9 0.4 21.3 M. Richards PHI C 79 19.9 37.1 11.1 48.3 15.3 17.5 15.5 33.0 15.9 -2.6 13.3 3.7 -1.8 2.0 D. Sedin VAN LW 77 17.9 37.1 19.0 56.1 23.8 26.7 17.8 44.5 10.4 1.2 11.6 J. Spezza OTT C 70 18.8 36.1 0.8 36.9 14.2 18.4 3.0 21.3 17.8 -2.2 15.6 E. Staal CAR C 79 20.6 35.4 1.7 37.2 15.1 21.4 2.6 24.1 14.7 0.0 14.8 -0.7 -0.9 -1.7 N. Backstrom WSH C 81 18.7 34.4 10.1 44.5 16.8 13.0 10.2 23.2 21.4 -0.1 21.3

(Glossary: The columns OPM and DPM are the offensive and defensive components of APM, and the prefixes EV, PP, SH denote that those results are for even strength, power play and short handed situations only.  For example, EVO is even strength offensive APM, EVD is even strength defensive APM, and EVA is even strength APM (offense+defense). AErr is the standard error in the APM estimates, GP is average games played during the last 4 seasons, and MinG is average minutes per season.)

No huge surprises here.  Despite the fact that Crosby has only played on average 63 games a season during the last 4 seasons, he still comes out on top.  Toews ranking was a little surprising to me.  Not that I don't believe it, just that it surprised me... I figured Ovechkin would be second.  Of course, we should always keep in mind that these rankings are noisy, even with 4 years of data.  The fact that Toews has one of the highest defensive ratings in this list, on the other hand, did not surprise me at all.

I liked seeing that Richards shows up with a high offensive rating in short handed situations.  That's actually the highest in the league.  I also thought it was kinda cool that Daniel's AErr is so high.  The model isn't really sure about his rating or Henrik's rating because it can't tell them apart.... not (just) because they are twins, but because they are almost always playing together.

Some notable players that are just outside the top 10:
Malkin (33.8, 12th) - injuries kept him off the list
Datsyuk (28.8, 16th) - don't worry, Datsyuk will be on plenty of other lists
H. Sedin (28.1, 17th) - the Sedin ratings are too noisy to be trusted.  I should probably just make them one person.

I should also point out that a player like Stamkos isn't on the list because the four seasons of data used in the model include his rookie year, when he didn't put up the superstar numbers that he does now.  That's one of the downsides of using 4 years of data.

I think that's good for now.  I'll finish by listing some of the things I'd like to talk about in future posts:

Even strength ratings
Power play and short handed ratings
Overall ratings and the 4-yr trophy finalists
General Limitations
Limitations with using only goals
Some remedies
Goals/60 versus Goals/season
Future work

Also, I'm always open to suggestions.

-bmac
(Brian Macdonald)