Regularized Adjusted Plus-Minus (RAPM) in Hockey

Adjusted plus-minus started in basketball. Dan Rosenbaum laid it out at 82games.com in April 2004, building on the WINVAL ratings Jeff Sagarin and Wayne Winston had designed. Regress the scoring margin of every stretch of play on who was on the floor, and each player's coefficient is his impact with his teammates and opponents held constant.

The first versions were noisy, because players who share most of their minutes are hard to pull apart. Joseph Sill's fix, presented at the MIT Sloan Sports Analytics Conference in March 2010, was regularization (ridge regression, in statistics terms), which he said nearly doubled the method's accuracy on games it hadn't seen.

Brian Macdonald brought it to hockey the same year. His first version, posted in June 2010 and published in the Journal of Quantitative Analysis in Sports in 2011, used weighted least squares on even-strength goals, and his 2012 follow-up switched to ridge regression, ran it on goals, shots, Fenwick and Corsi, and produced estimates he described as independent of teammates, opponents and the zone where a shift began. Robert Gramacy, Matthew Taddy and Shane Jensen published a regularized logistic version the next year.

Evolving-Hockey revived it for the public at Hockey-Graphs on January 14, 2019, and its version sits under the even-strength components of its WAR model. That's the RAPM this site plots, taken from Evolving-Hockey's single-season, even-strength export for every skater with 20 games.

The regression treats every stint, a stretch of play with no substitutions, as one observation, weighted by how long it lasted. The target is the team's rate of shot attempts, expected goals or goals per 60 minutes during that stint, with a separate regression for each. Evolving-Hockey's 2017-18 demonstration ran 293,009 shifts averaging 12.94 seconds apiece, entered as 586,018 rows (one for each team's offense). These are the predictors:

• Each skater on the ice for the offense
• Each skater on the ice for the defense
• The score state
• The strength state (five-on-five, four-on-four and so on)
• Whether the stint included a faceoff in either team's offensive zone
• Whether either team was on the second night of a back-to-back
• Home ice

Here's a worked example with invented round numbers. A defenseman's team generates 2.9 expected goals per 60 with him on the ice, against a league rate of 2.5. Raw on-ice numbers give him all of that plus-0.4. His usual partner and the top line he plays behind were out there for most of those minutes, though, and the regression has seen how they do without him. Say it hands them plus-0.3 of the credit. He's left with plus-0.1, and the ridge penalty trims that a little further toward zero.

Ridge regularization pulls every rating toward zero, which is league average, and the strength of the pull is chosen by cross-validation. A player with very little ice time gets pulled almost all the way (Evolving-Hockey's write-up puts the line at roughly 100 to 150 even-strength minutes), so you'll see plenty of fringe players sitting a hair from zero. Each season is run on its own with players split by team. The multi-season versions combine single seasons weighted by ice time.

RAPM measures isolated impact against an average player. Zero is league average and the unit is goals (or expected goals, or attempts) per 60 minutes. If you see a net rating of plus-0.20, it means a team's expected-goal margin ran a fifth of a goal per 60 better with that player on the ice than his teammates, opponents and usage can explain.

Detroit had the best number in the league last season. It wasn't a forward. In Evolving-Hockey's 2025-26 even-strength file, net expected-goal impact among skaters with 20 or more games ran from minus-0.58 to plus-0.53 per 60. The middle 80% sat between minus-0.19 and plus-0.21. Moritz Seider was the plus-0.53, ahead of Adam Fox and Jordan Spence, with plus-0.24 on offense and 0.30 fewer expected goals against per 60 on defense. Over his 1,640 even-strength minutes that's about 14.6 expected goals of impact.

The rest of the roster spread out behind him. Alex DeBrincat ranked 50th at plus-0.26. Dylan Larkin was 542nd at minus-0.08, a little below average at both ends, and Ben Chiarot was 774th of 795 at minus-0.30, most of it on defense.

Why trust a number that's been deliberately pulled toward zero? Because the unpulled version predicts worse. Sill's out-of-sample tests are the case for the penalty, and Evolving-Hockey sizes its own penalty the same way, on data the model hasn't seen.

The isolated-impact chart

Every player page on this site plots offense against defense RAPM for every NHL skater with 20 games, Red Wings highlighted, up and right good at both ends. It's the quickest way you can check whether a player's on-ice numbers are his own or borrowed from the people around him. Only four skaters cleared plus-0.20 at both ends last season, and Seider was one of them.

The engine under WAR

Evolving-Hockey runs long-term RAPM for each component, then trains a statistical plus-minus model to predict it and uses that model's output for each season of its WAR. It goes the long way because single-season RAPM still struggles with linemates who always play together. Patrick Bacon's WAR at HockeyStats uses a weighted ridge regression for the same job, so both of the main public value models rest on it.

Seeing past the box score

Marco Kasper scored 0.93 five-on-five points per 60 last season, 354th of 385 forwards with 500 minutes. RAPM rated him plus-0.17 net, 115th in the league, and most of it came on offense (plus-0.12 expected goals for per 60). His line's chances went up with him on the ice even when the points didn't come, which is the case for patience.

Linemates are hard to separate

What happens when two players share almost every shift? The regression has little to tell them apart with, and Evolving-Hockey says multicollinearity is still an issue within a single season. That's why its WAR uses long-term RAPM.

Shrinkage hides extremes

Ridge pulls every rating toward zero, so a strong player on limited minutes looks closer to average than he is, and anyone under roughly 100 to 150 even-strength minutes is pulled almost all the way.

The controls are coarse

Zone starts enter only as whether a stint included a faceoff in a given zone, and score as a handful of states. Anything that isn't on the list, from shift length to who's in goal for the expected-goal versions, ends up folded into somebody's rating, and you can't tell whose.

It assumes impacts add up

Because the model is linear, a player's rating stays the same whoever he's with, so chemistry between two linemates who are better together than apart has nowhere to go but into their individual numbers.

Is RAPM the same as relative Corsi?

Not quite. Relative Corsi is the difference between a team's numbers with a player on the ice and with him off it, so it still carries whoever he played with and against. RAPM puts all ten skaters into one regression together, so each rating is net of the other nine.