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Journal of Quantitative Analysis in Sports

An official journal of the American Statistical Association

Editor-in-Chief: Glickman, PhD, Mark / Rigdon, Steve

4 Issues per year


CiteScore 2016: 0.44

SCImago Journal Rank (SJR) 2015: 0.288
Source Normalized Impact per Paper (SNIP) 2015: 0.358

Online
ISSN
1559-0410
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Estimating player contribution in hockey with regularized logistic regression

Robert B. Gramacy
  • Corresponding author
  • Booth School of Business, The University of Chicago, 5807 S Woodlawn Ave, Chicago, IL 60637, USA
  • Email:
/ Shane T. Jensen
  • The Wharton School, University of Pennsylvania 3730 Walnut St., Philadelphia, PA 19102, USA
/ Matt Taddy
  • Booth School of Business, The University of Chicago, 5807 S Woodlawn Ave, Chicago, IL 60637, USA
Published Online: 2013-03-30 | DOI: https://doi.org/10.1515/jqas-2012-0001

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.

Keywords: Bayesian shrinkage; lasso; logistic regression; regularization; sports analytics

References

  • Awad, T. 2009. “Numbers On Ice: Fixing Plus/Minus.” Hockey Prospectus. See www.puckprospectus.com

  • Friedman, J. H., T. Hastie, and R. Tibshirani. 2010. “Regularization Paths for Generalized Linear Models via Coordinate Descent.” Journal of Statistical Software 33(1): 1–22.Google Scholar

  • Geman, S. and D. Geman. 1984. “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images.” IEEE Transaction on Pattern Analysis and Machine Intelligence 6: 721–741.Google Scholar

  • Gramacy, R. 2012a. Reglogit: Simulation-based Regularized Logistic Regression. R package version 1.1.Google Scholar

  • Gramacy, R. B. 2012b. Monomvn: Estimation for multivariate normal and Student-t data with monotone missingness. R package version 1.8-9.Google Scholar

  • Gramacy, R. and N. Polson. 2012. “Simulation-Based Regularized Logistic Regression.” Bayesian Analysis 7: 1–24.Google Scholar

  • Hoerl, A. E. and R. W. Kennard. (1970). “Ridge Regression: Biased Estimation for Nonorthogonal Problems.” Technometrics 12: 55–67.CrossrefGoogle Scholar

  • Holmes, C. and K. Held. 2006. “Bayesian Auxilliary Variable Models for Binary and Multinomial Regression.” Bayesian Analysis 1(1): 145–168.Google Scholar

  • Hornik, K., D. Meyer, and C. Buchta. (2011). slam: Sparse Lightweight Arrays and Matrices. R package version 0.1-23.Google Scholar

  • Ilardi, S. and A. Barzilai. 2004. “Adjusted Plus-Minus Ratings: New and Improved for 2007–2008.” 82games.com.Google Scholar

  • Macdonald, B. 2010. “A Regression-based Adjusted Plus-Minus Statistic for NHL Players.” Tech. rep., arXiv: 1006.4310.Google Scholar

  • Rosenbaum, D. T. 2004. “Measuring How NBA Players Help Their Teams Win.” 82games.com.Google Scholar

  • Schuckers, M. E., D. F. Lock, C. Wells, C. J. Knickerbocker, and R. H. Lock. 2010. “National Hockey League Skater Ratings Based upon All On-Ice Events: An Adjusted Minus/Plus Probability (AMPP) Approach.” Tech. rep., St. Lawrence University.Google Scholar

  • R Development Core Team 2010. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.Google Scholar

  • Taddy, M. 2012a. textir: Inverse Regression for Text. R package version 1.8-6.Google Scholar

  • Taddy, M. 2012b. “Multinomial Inverse Regression for Text Analysis.” Journal of the American Statistical Association, accepted for publication.Web of ScienceGoogle Scholar

  • Thomas, A. C., S. L. Ventura, S. Jensen, and S. Ma. 2012. “Competing Process Hazard Function Models for Player Ratings in Ice Hockey.” Tech. rep., ArXiv:1208.0799.Google Scholar

  • Tibshirani, R. 1996. “Regression shrinkage and selection via the lasso.” J. R. Statist. Soc. B, 58: 267–288.Google Scholar

  • Vollman, R. 2010. “Howe and Why: Ten Ways to Measure Defensive Contributions.” Hockey Prospectus.Google Scholar

About the article

Corresponding author: Robert B. Gramacy, Booth School of Business, The University of Chicago, 5807 S Woodlawn Ave, Chicago, IL 60637, USA, Tel.: +773-702-0739


Published Online: 2013-03-30


1Note that we include goalies in our analysis.

2Fitted in R using the command fit>-glm(goals~XP, family=“binomial”).

3We used forward step-wise regression with the Bayes information criterion (BIC).

4This is the lowest possible budget from which lines can be formed satisfying (4).

5Sweater sales is another matter.

6We omitted goalie-skater and goalie-goalie interaction terms.


Citation Information: Journal of Quantitative Analysis in Sports, ISSN (Online) 1559-0410, ISSN (Print) 2194-6388, DOI: https://doi.org/10.1515/jqas-2012-0001.

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