Estimating Player Contribution in Hockey with Regularized Logistic Regression

TL;DR

This paper introduces a regularized logistic regression approach to evaluate hockey players' contributions, addressing limitations of traditional plus-minus metrics by accounting for sample size and providing more reliable estimates.

Contribution

The paper develops and applies a regularized logistic regression model with prior shrinkage techniques to assess individual player contributions in hockey, improving upon existing metrics.

Findings

Most players do not have significant measurable contributions.

Star players stand out as significant contributors.

Some highly paid players do not contribute proportionally to their salaries.

Abstract

We present a regularized logistic regression model for evaluating player contributions in hockey. The traditional metric for this purpose is the plus-minus statistic, which allocates a single unit of credit (for or against) to each player on the ice for a goal. However, plus-minus scores measure only the marginal effect of players, do not account for sample size, and provide a very noisy estimate of performance. We investigate a related regression problem: what does each player on the ice contribute, beyond aggregate team performance and other factors, to the odds that a given goal was scored by their team? Due to the large-p (number of players) and imbalanced design setting of hockey analysis, a major part of our contribution is a careful treatment of prior shrinkage in model estimation. We showcase two recently developed techniques -- for posterior maximization or simulation -- that make such analysis feasible. Each approach is accompanied with publicly available software and we include the simple commands used in our analysis. Our results show that most players do not stand out as measurably strong (positive or negative) contributors. This allows the stars to really shine, reveals diamonds in the rough overlooked by earlier analyses, and argues that some of the highest paid players in the league are not making contributions worth their expense.