Regularized Adjusted Plus-Minus (RAPM)
A regression-based metric that isolates a player's individual impact on team goal or expected goal rates by controlling for teammates, opponents, score state, zone starts, and venue using ridge regression.
Regularized Adjusted Plus-Minus (RAPM) is the gold standard for isolating individual player impact in hockey. Traditional plus-minus conflates a player's own ability with the quality of their linemates, opponents, and goaltenders. RAPM solves this by running a ridge regression on every shift or stint of play: the dependent variable is the goal rate (or xG rate), and the independent variables are indicator columns for every player on the ice, plus controls for score state, zone starts, venue, and rest.
The "regularized" part refers to the ridge (L2) penalty applied to the regression coefficients. Without regularization, the model would overfit to small samples and produce wildly unstable estimates -- especially for players who rarely play apart from their usual linemates. The ridge penalty shrinks coefficients toward zero, trading a small amount of bias for a large reduction in variance.
Hockey Alchemy uses a daisy-chain RAPM approach where each season's output serves as the Bayesian prior for the next season. This allows the model to accumulate information over time while naturally decaying older data. Separate RAPM regressions are run for even-strength, power play (5v4), and penalty kill (4v5) situations. The resulting coefficients feed into the GAR model as a key ingredient for the team-adjustment and defensive evaluation stages.
Formula
RAPM solves via ridge regression: y (GF/60 or xGF/60) = X * beta + controls + epsilon X = player indicator matrix (1 if on ice, -1 if opponent) Controls: score state, zone starts, venue, back-to-back Regularization: L2 penalty (ridge) to stabilize estimates Daisy-chain: prior season output -> Bayesian prior for next season
Frequently Asked Questions
Why use ridge regression instead of ordinary least squares?
NHL players frequently share ice time with the same linemates, creating severe multicollinearity. Ordinary least squares would produce wildly unstable coefficients. Ridge regression adds a penalty that shrinks estimates toward zero, producing more stable and reliable results at the cost of slightly compressing the range.
What is the difference between RAPM and GAR?
RAPM produces rate-based coefficients (e.g., impact on goals per 60 minutes). GAR is a cumulative value metric that incorporates RAPM along with counting stats, replacement levels, and team adjustments to produce a total value number. Think of RAPM as a key ingredient in the GAR recipe.
How many seasons of data does RAPM use?
Hockey Alchemy uses a daisy-chain approach where each season's RAPM output becomes the prior for the next. Decay rates (0.6-0.7) control how quickly old information fades. This provides the stability of multi-year data while remaining responsive to current-season performance.
Can RAPM separate a player from their linemates?
That is exactly what RAPM is designed to do. By including every on-ice player as a variable and applying regularization, the model attributes results to individual players rather than line combinations. Players who always play together will have less certain individual estimates, but the model still produces reasonable splits.