[Paper Review] Hockey Player Performance via Regularized Logistic Regression

Robert B. Gramacy, Matt Taddy+1 more|arXiv (Cornell University)|Dec 21, 2016

Sports Analytics and Performance19 references3 citations

TL;DR

This paper proposes a regularized logistic regression model to estimate individual hockey players' partial contributions to goal scoring, accounting for teammates and opponents by modeling on-ice effects jointly. By applying penalized likelihood estimation to high-dimensional game data, it isolates each player's influence beyond team-level noise, offering a more accurate and stable measure of performance than traditional plus-minus.

ABSTRACT

A hockey player's plus-minus measures the difference between goals scored by and against that player's team while the player was on the ice. This measures only a marginal effect, failing to account for the influence of the others he is playing with and against. A better approach would be to jointly model the effects of all players, and any other confounding information, in order to infer a partial effect for this individual: his influence on the box score regardless of who else is on the ice. This chapter describes and illustrates a simple algorithm for recovering such partial effects. There are two main ingredients. First, we provide a logistic regression model that can predict which team has scored a given goal as a function of who was on the ice, what teams were playing, and details of the game situation (e.g. full-strength or power-play). Since the resulting model is so high dimensional that standard maximum likelihood estimation techniques fail, our second ingredient is a scheme for regularized estimation. This adds a penalty to the objective that favors parsimonious models and stabilizes estimation. Such techniques have proven useful in fields from genetics to finance over the past two decades, and have demonstrated an impressive ability to gracefully handle large and highly imbalanced data sets. The latest software packages accompanying this new methodology -- which exploit parallel computing environments, sparse matrices, and other features of modern data structures -- are widely available and make it straightforward for interested analysts to explore their own models of player contribution.

Motivation & Objective

• To address the limitations of traditional plus-minus in isolating individual player contributions in hockey.
• To model the joint influence of all players on goal outcomes, accounting for teammates, opponents, and game context.
• To develop a stable, parsimonious estimation method for high-dimensional, imbalanced hockey data.
• To provide a statistically rigorous alternative to marginal metrics like plus-minus using modern regularization techniques.
• To enable practical application of player evaluation models through accessible software and computational efficiency.

Proposed method

• Formulate a logistic regression model where the outcome is which team scored a goal, based on player on-ice presence, team identities, and game situation (e.g. power-play, even-strength).
• Use regularization (e.g. L1 or L2 penalty) to stabilize estimation in high-dimensional parameter space where standard maximum likelihood fails.
• Incorporate sparse matrix representations and parallel computing to handle large-scale data efficiently.
• Optimize the penalized log-likelihood objective to favor simpler models that generalize well to unseen data.
• Estimate partial effects for each player, representing their isolated contribution to goal scoring independent of team context.
• Leverage modern statistical software packages that support regularization and high-performance computing for model fitting.

Experimental results

Research questions

• RQ1 How can individual player contributions in hockey be isolated from confounding team and opponent effects?
• RQ2 Can a joint modeling approach using logistic regression better estimate player influence than marginal metrics like plus-minus?
• RQ3 How does regularization improve estimation stability in high-dimensional, sparse hockey data?
• RQ4 To what extent can game context (e.g. power-play, shorthanded) be meaningfully incorporated into player performance models?
• RQ5 Can scalable computational methods make advanced player evaluation models accessible to analysts without high-end resources?

Key findings

• The proposed model successfully isolates individual player effects by accounting for teammates and opponents, providing a more accurate measure of on-ice influence than traditional plus-minus.
• Regularization stabilizes estimation in high-dimensional settings where standard maximum likelihood would fail due to overfitting or non-identifiability.
• The method effectively handles imbalanced data common in hockey, such as rare events like shorthanded goals, without requiring data transformation.
• The use of sparse matrices and parallel computing enables efficient model fitting on large-scale datasets, making the approach practical for real-world analysis.
• The resulting partial effect estimates are interpretable and robust, offering a principled alternative to heuristic performance metrics.
• Modern software tools now make it feasible for researchers and analysts to implement and extend such models independently.