Filling the Gaps A Multiple Imputation Approach to Estimating Aging Curves in Baseball Quang NguyenGregory J. Matthews

2025-04-27 0 0 428.99KB 19 页 10玖币

侵权投诉

Filling the Gaps: A Multiple Imputation Approach to

Estimating Aging Curves in Baseball

Quang Nguyen∗Gregory J. Matthews†

March 11, 2024

Abstract

In sports, an aging curve depicts the relationship between average performance and

age in athletes’ careers. This paper investigates the aging curves for oﬀensive players in

Major League Baseball. We study this problem in a missing data context and account

for diﬀerent types of dropouts of baseball players during their careers. We employ a

multiple imputation framework for multilevel data to impute the player performance

associated with the missing seasons, and estimate the aging curves based on the imputed

datasets. We then evaluate the eﬀects of diﬀerent dropout mechanisms on the aging

curves through simulation, before applying our method to analyze MLB player data

from past seasons. Results suggest an overestimation of the aging curves constructed

without considering the unobserved seasons, whereas estimates obtained from multiple

imputation address this shortcoming.

Keywords: aging curve; baseball; multiple imputation; survival bias

∗Department of Statistics & Data Science, Carnegie Mellon University

†Department of Mathematics and Statistics, Loyola University Chicago

arXiv:2210.02383v3 [stat.AP] 11 Mar 2024

1 Introduction

The rise and fall of an athlete is a popular topic of discussion in the sports media today.

Questions regarding whether a player has reached their peak, is past their prime, or is good

enough to remain in their respective professional league are often seen in diﬀerent media

outlets such as news articles, television debate shows, and podcasts. The average performance

of players by age throughout their careers is visually represented by an aging curve. This

graph typically consists of a horizontal axis representing a time variable (usually age or

season) and a vertical axis showing a performance metric at each time point in a player’s

career.

One signiﬁcant challenge associated with the study of aging curves in sports is survival

bias, as pointed out by Lichtman (2009), Turtoro (2019), Judge (2020a), and Schuckers et al.

(2023). In particular, the aging eﬀects are not often determined from a full population of

athletes (i.e., all players who have ever played) in a given league. That is, only players that

are good enough to remain are observed; whereas those who might be involved, but do not

actually participate or are not talented enough to compete, are being completely disregarded.

This very likely results in an overestimation of the aging curves.

As such, player survivorship and dropout can be viewed as a missing data problem. There

are several distinct cases of player absence from professional sport at diﬀerent points in their

careers. Early on, teams may elect to assign their young prospects to their minor/development

league aﬃliates for several years of nurture. Many of those players would end up receiving a

call-up to join the senior squad, when the team believes they are ready. During the middle of

a player’s career, a nonappearance could occur due to various reasons. Injury is unavoidable

in sports, and this could cost a player at least one year of their playing time. Personal reasons

such as contract situation and more recently, concerns regarding a global pandemic, could

also lead to athletes sitting out a season. Later on, a player, by either their choice or their

team’s choice, might head for retirement because they cannot perform at a level like they

used to, leading to unobserved seasons that could have been played.

The primary aim of this paper is to apply missing data techniques to the estimation of

aging curves. In doing so, we focus on baseball and pose the following research question:

What would the aging curve look like if all players competed in every season within a ﬁxed

range of age? In other words, what would have happened if a player who was forced to retire

from their league at a certain age had played a full career? The manuscript continues with a

review of existing literature on aging curves in baseball and other sports in Section 2. Next,

we describe our data and methods used to perform our analyses in Section 3. After that, our

approach is implemented through simulation and analyses of real baseball data in Sections 4

and 5. Finally, in Section 6, we conclude with a discussion of the results, limitations, and

directions for future work.

2 Literature Review

To date, we ﬁnd a considerable amount of previous work related to aging curves and career

trajectory of athletes. This body of work consists of several key themes, a wide array of

statistical methods, and applications in many sports besides baseball such as basketball,

hockey, and track and ﬁeld, to name a few.

A typical notion in the baseball aging curves literature is the assumption of a quadratic

form for modeling the relationship between performance and age. Morris (1983) looked at

Ty Cobb’s batting average trajectory using parametric empirical Bayes and used shrinkage

methods to obtain a parabolic curve for Cobb’s career performance. Albert (1992) proposed a

quadratic random eﬀects log-linear model for smoothing a batter’s home run rates throughout

their career. Berry et al. (1999) implemented a nonparametric method to estimate the age

eﬀect on performance in baseball, hockey, and golf. However, Albert (1999) weighed in on

this nonparametric approach and questioned the assumptions that the peak age and periods

of growth and decline are the same for all players. Albert (1999) ultimately preferred a

second-degree polynomial function for estimating age eﬀect in baseball, which is a parametric

model. Continuing his series of work on aging trajectories, Albert (2002) proposed a Bayesian

exchangeable model for baseball hitting performance. This approach combined quadratic

regression estimates and assumes similar careers for players born in the same decade. Fair

(2008) and Bradbury (2009) both used a ﬁxed-eﬀects regression to examine age eﬀects in the

MLB, also assuming a quadratic aging curve form. A major drawback of Bradbury (2009)’s

study is that the analysis only considered players with longer baseball careers.

In addition to baseball, studies on aging curves have also been conducted for other sports.

Early on, Moore (1975) looked at the association between age and running speed in track

and ﬁeld and produced aging curves for diﬀerent running distances using an exponential

model. Fair (1994) and Fair (2007) studied the age eﬀects in track and ﬁeld, swimming, chess,

and running, in addition to their latter work in baseball, as mentioned earlier. In triathlon,

Villaroel et al. (2011) assumed a quadratic relationship between performance and age, as

many have previously considered. As for basketball, Page et al. (2013) used a Gaussian

process regression in a hierarchical Bayesian framework to model age eﬀect in the NBA.

Additionally, Lailvaux et al. (2014) used NBA and WNBA data to investigate and test for

potential sex diﬀerences in the aging of comparable performance indicators. Vaci et al. (2019)

applied Bayesian cognitive latent variable modeling to explore aging and career performance

in the NBA, accounting for player position and activity level. In tennis, Kovalchik (2014)

studied age and performance trends in men’s tennis using change point analysis.

Another convention in the aging curve modeling literature is the assumption of discrete

observations. Speciﬁcally, most researchers use regression modeling and consider a data

measurement for each season played throughout a player’s career. In contrast to previous

approaches, Wakim and Jin (2014) took a diﬀerent route and considered functional data

analysis as the primary tool for modeling MLB and NBA aging curves. This is a continuous

framework which treats the entire career performance of an athlete as a smooth function. In

a similar functional data setting, Leroy et al. (2018) investigated the performance progression

curves in swimming.

A subset of the literature on aging and performance in sports speciﬁcally studies the

question: At what age do athletes peak? Schulz and Curnow (1988) looked at the age of

peak performance for track and ﬁeld, swimming, baseball, tennis, and golf. A follow-up study

to this work was done by Schulz et al. (1994), where the authors focused on baseball and

found that the average peak age for baseball players is between 27 and 30, considering several

performance measures. Later ﬁndings on baseball peak age also showed consistency with the

results in Schulz et al. (1994). Fair (2008) determined the peak-performance age in baseball

to be 28, whereas Bradbury (2009) determined that baseball hitters and pitchers reach the

top of their careers at about 29 years old. In soccer, Dendir (2016) determined that the peak

age for footballers in the top leagues falls within the range of 25 to 27.

The idea of player survivorship is only mentioned in a small number of articles. To our

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

FillingtheGaps:AMultipleImputationApproachtoEstimatingAgingCurvesinBaseballQuangNguyen∗GregoryJ.Matthews†March11,2024AbstractInsports,anagingcurvedepictstherelationshipbetweenaverageperformanceandageinathletes’careers.ThispaperinvestigatestheagingcurvesforoffensiveplayersinMajorLeagueBaseball.Westud...

展开>> 收起<<

Filling the Gaps A Multiple Imputation Approach to Estimating Aging Curves in Baseball Quang NguyenGregory J. Matthews.pdf

共19页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Filling the Gaps A Multiple Imputation Approach to Estimating Aging Curves in Baseball Quang NguyenGregory J. Matthews

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: