Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

Journal of Quantitative Analysis in Sports

An official journal of the American Statistical Association

Editor-in-Chief: Glickman, PhD, Mark

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
See all formats and pricing
In This Section

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.

  • 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.

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

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

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

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

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

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

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

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

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

  • 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.

  • 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.

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

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

  • 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.

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

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

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. Export Citation

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Jack Davis, Harsha Perera, and Tim B. Swartz
Journal of Sports Analytics, 2015, Volume 1, Number 1, Page 19

Comments (0)

Please log in or register to comment.
Log in