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Estimation of player aging curves using regression and imputation

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Abstract

The impact of age on performance is a fundamental component to models of player valuation and prediction across sport. Age effects are typically measured using age curves, which reflect the expected average performance at each age among all players that are eligible to participate. Most age curve methods, however, ignore the reality that age likewise influences which players receive opportunities to perform. In this paper we begin by highlighting how selection bias is linked to the ages in which we observe players perform. Next, using underlying distributions of how players move in and out of sport organizations, we assess the performance of various methods for age curve estimation under the selection bias of player entry and issues of small samples at younger and older ages. We propose several methods for player age curve estimation, introduce a missing data framework, and compare these new methods to more familiar approaches including both parametric and semi-parametric modeling. We then use simulations to compare several approaches for estimating aging curves. Imputation-based methods, as well as models that account for individual player skill, tend to generate lower root mean squared error (RMSE) and age curve shapes that better match the truth. We implement our approach using data from the National Hockey League. All of the data and code for this paper are available in a Github repository.

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Notes

  1. MLB stands for Major League Baseball and NBA stands for National Basketball League

  2. Imputation with truncation limits the range of values that generated realizations can take. For example, if \(X \sim N(0,1)\) is truncated below at \(-2\) then the realized values of X could only take values in \((-2,\infty )\) whereas untruncated realizations could take values in \((-\infty , \infty )\).

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Acknowledgements

We would like to thank CJ Turturo for making his code available.

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Authors and Affiliations

  1. St. Lawrence University, Canton, NY, USA

    Michael Schuckers

  2. National Football League, New York, USA

    Michael Lopez

  3. Yale University, New Haven, USA

    Brian Macdonald

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  1. Michael Schuckers
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  2. Michael Lopez
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Correspondence to Michael Schuckers.

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This material is based upon work supported by the U.S. National Science Foundation under Grant No. CNS-1919554.

Appendix

Appendix

Below is a table highlighting average RMSE for different simulation settings, by age.

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figure 10

Average Root Mean Squared Error by Age (column), Method (row), and number of players (sets of rows) across simulations

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Schuckers, M., Lopez, M. & Macdonald, B. Estimation of player aging curves using regression and imputation. Ann Oper Res 325, 681–699 (2023). https://doi.org/10.1007/s10479-022-05127-y

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