Estimating player contribution in hockey with regularized logistic regression

Robert B. Gramacy 1 , Shane T. Jensen 2  and Matt Taddy 1
  • 1 Booth School of Business, The University of Chicago, 5807 S Woodlawn Ave, Chicago, IL 60637, USA
  • 2 The Wharton School, University of Pennsylvania 3730 Walnut St., Philadelphia, PA 19102, USA
Robert B. Gramacy, Shane T. Jensen and Matt Taddy


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.

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JQAS, an official journal of the American Statistical Association, publishes research on the quantitative aspects of professional and collegiate sports. Articles deal with subjects as measurements of player performance, tournament structure, and the frequency and occurrence of records. Additionally, the journal serves as an outlet for professionals in the sports world to raise issues and ask questions that relate to quantitative sports analysis.