\section{Introduction}\label{sec:intro}}

Athlete's career trajectory is a popular topic of discussion in the
media nowadays. 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 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 \emph{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 \emph{survival bias}, as pointed out by Lichtman (2009),
Turtoro (2019), Judge (2020a), and Schuckers et al. (2021). In
particular, the aging effects are not often determined from a full
population of athletes in a given league. That is, only players that are
good enough to remain are observed; whereas those might be involved, but
do not actually participate or not talented enough to compete, are being
completely disregarded. This very likely results in an overestimation in
the construction of aging curves.

As such, player survivorship and dropout can be viewed in the context of
missing data. There are different cases of player absence from
professional sport at different points in their careers. At the
beginning, 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 one'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 might head for
retirement because they cannot perform at a level like they used to.

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
one research question: What would the aging curve look like if players
competed in every season from 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 \ref{sec:lit}. Next, we describe
our data and methods used to perform our analyses in Section
\ref{sec:meth}. After that, our approach is implemented through
simulation and analyses of real baseball data in Sections \ref{sec:sim}
and \ref{sec:app}. Finally, in Section \ref{sec:discuss}, we conclude
with a discussion of the results, limitations, and directions for future
work.

\hypertarget{sec:lit}{%
\section{Literature Review}\label{sec:lit}}

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.

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) looks at Ty Cobb's batting average
trajectory using parametric empirical Bayes and uses shrinkage methods
to obtain a parabolic curve for Cobb's career performance. Albert (1992)
proposes a quadratic random effects log-linear model for smoothing a
batter's home run rates throughout their career. A nonparametric method
is implemented to estimate the age effect on performance in baseball,
hockey, and golf by Berry et al. (1999). However, Albert (1999) weighs
in on this nonparametric approach and questions the assumptions that the
peak age and periods of growth and decline are the same for all players.
Albert (1999) ultimately prefers 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)
proposes a Bayesian exchangeable model for modeling hitting performance.
This approach combines quadratic regression estimates and assumes
similar careers for players born in the same decade. Fair (2008) and
Bradbury (2009) both use a fixed-effects regression to examine age
effects in the MLB, also assuming a quadratic aging curve form.

In addition to baseball, studies on aging curves have also been
conducted for other sports. Early on, Moore (1975) looks at the
association between age and running speed in track and field and
produces aging curves for different running distances using an
exponential model. Fair (1994) and Fair (2007) study 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) assume a quadratic relationship between performance and
age, as many have previously considered. As for basketball, Page et al.
(2013) use a Gaussian process regression in a hierarchical Bayesian
framework to model age effect in the NBA. Additionally, Lailvaux et al.
(2014) use NBA and WNBA data to investigate and test for potential sex
differences in the aging of comparable performance indicators. Vaci et
al. (2019) apply Bayesian cognitive latent variable modeling to explore
aging and career performance in the NBA, accounting for player position
and activity levels. In tennis, Kovalchik (2014) studies 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 \& Jin (2014) take a different route and consider 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.

A subset of the literature on aging and performance in sports provides
answers to the question: At what age do athletes peak? Schulz \& Curnow
(1988) look 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 focus on baseball and
find that the average peak age for baseball players is between 27 and
30, considering several performance measures. Later findings on baseball
peak age also show consistency with the results in Schulz et al. (1994).
Fair (2008) determines the peak-performance age in baseball to be 28,
whereas Bradbury (2009) determines that baseball hitters and pitchers
reach their career apex at about 29 years old. In soccer, Dendir (2016)
determines that the peak age for footballers in the top leagues falls
within the range of 25 to 27.

Last and most importantly, the idea of player survivorship is only
mentioned in a small number of articles. To our knowledge, not many
researchers have incorporated missing data methods into the estimation
of aging curves to account for missing but observable athletes. Schulz
et al. (1994) and Schell (2005) note the selection bias problem with
estimating performance averages by age in baseball, as better players
tend to have longer career longevity. Schall \& Smith (2000) predict
survival probabilities of baseball players using a logit model, and
examine the link between first-year performance and career length.
Lichtman (2009) studies different aging curves for different eras and
groups of players after correcting for survival bias, and shows that
survival bias results in an overestimation of the age effects.
Alternatively, Judge (2020a) concludes that survivorship bias leads to
an underestimation, not overestimation, of the aging curves. In
analyzing NHL player aging, Brander et al. (2014) apply their quadratic
and cubic fixed-effects regression models to predict performance for
unobserved players in the data.

Perhaps the most closely related approach to our work is that by
Schuckers et al. (2021), which considers different regression and
imputation frameworks for estimating the aging curves in the National
Hockey League (NHL). First, they investigate different regression
approaches including spline, quadratic, quantile, and a delta plus
method, which is an extension to the delta method previously studied by
Lichtman (2009), Turtoro (2019), and Judge (2020b). This paper also
proposes an imputation approach for aging curve estimation, and
ultimately concludes that the estimation becomes stronger when
accounting for unobserved data, which addresses a major shortcoming in
the estimation of aging curves. However, it appears that the aging
curves are constructed without taking into account the variability as a
result of imputing missing data. This could be improved by applying
multiple imputation rather than considering only one imputed dataset. As
pointed out by Gelman \& Hill (2006) (Chapter 25), conducting only a
single imputation essentially assumes that the filled-in values
correctly estimate the true values of the missing observations. Yet,
there is uncertainty associated with the missingness, and multiple
imputation can incorporate the missing data uncertainty and provide
estimates for the different sources of variability.

\hypertarget{sec:meth}{%
\section{Methods}\label{sec:meth}}

\hypertarget{sec:data}{%
\subsection{Data Collection}\label{sec:data}}

In the forthcoming analyses, we rely on one primary source of publicly
available baseball data: the Lahman baseball database (Lahman, 1996 --
2021). Created and maintained by Sean Lahman, this database contains
pitching, hitting, and fielding information for Major League Baseball
players and teams dating back to 1871. The data are available in many
different formats, and the \texttt{Lahman} package in \texttt{R}
(Friendly et al., 2021; R Core Team, 2022) is utilized for our
investigation.

Due to our specific purpose of examining the aging curves for baseball
offensive players, the following datasets from the \texttt{Lahman}
library are considered: \texttt{Batting}, which provides
season-by-season batting statistics for baseball players; and
\texttt{People}, to obtain the date of birth of each player and
calculate their age for each season played. In each table, an athlete is
identified with their own \texttt{playerID}, hence this attribute is
used as a joining key to merge the two tables together. A player's age
for a season is determined as their age on June 30, and the formula
suggested by Marchi et al. (2018) for age adjustment based on one's
birth month is applied.

Throughout this paper, we consider on-base plus slugging (OPS), which
combines a hitter's ability to reach base on a swing and power-hitting,
as the baseball offensive performance measure. We scale the OPS for all
players and then apply an arcsine transformation to ensure a reasonable
range for the OPS values when conducting simulation and imputation. We
also assume a fixed length for a player's career, ranging from age 21 to
39. In terms of sample restriction, we observe all player-seasons with
at least 100 plate appearances, which means a season is determined as
missing if one's plate appearances is below that threshold.

\hypertarget{multiple-imputation}{%
\subsection{Multiple Imputation}\label{multiple-imputation}}

Multiple imputation (Rubin, 1987) is a popular statistical procedure for
addressing the presence of incomplete data. The goal of this approach is
to replace the missing data with plausible values to create multiple
completed datasets. These datasets can each be analyzed and results are
combined across the imputed versions. Multiple imputation consists of
three steps. First, based on an appropriate imputation model, \(m\)
copies of the dataset are created by filling the missing values. Next,
\(m\) analyses are performed on each of the \(m\) completed datasets.
Finally, the results from each of the \(m\) datasets are pooled together
to create a combined estimate and standard errors are estimated that
account for the between and within imputation variability. This last
step can be accomplished using asymptotic theory with Rubin's combining
rules (Little \& Rubin, 1987), which are as follows.

Let \(Q\) be a parameter of interest and \(\widehat Q_i\) where
\(i=1,2,\dots,m\) are estimates of \(Q\) obtained from \(m\) imputed
datasets, with sampling variance \(U\) estimated by \(\widehat U_i\).
Then the point estimate for \(Q\) is the average of the \(m\) estimates
\[
\overline Q = \frac{1}{m} \sum_{i=1}^m \widehat Q_i
\,.
\] The variance for \(\overline Q\) is defined as \[
T=\overline U + \left(1 + \frac{1}{m}\right)B
\,,
\] where \[
\overline U = \frac{1}{m} \sum_{i=1}^m \widehat U_i
\] and \[
B=\frac{1}{m-1} \sum_{i=1}^m (\widehat Q_i - \overline Q)^2
\] are the estimated within and between variances, respectively.

Inferences for \(Q\) are based on the approximation \[
\frac{Q - \overline Q}{\sqrt{T}} \sim t_\nu
\,,
\] where \(t_\nu\) is the Student's \(t\)-distribution with
\(\displaystyle \nu = (m-1)\left(1+\frac{1}{r}\right)^2\) degrees of
freedom, with
\(\displaystyle r=\left(1+\frac{1}{m}\right)\frac{B}{\overline U}\)
representing the relative increase in variance due to missing data.

Accordingly, a \(100(1-\alpha)\)\% Wald confidence interval for \(Q\) is
computed as \[
\overline Q \ \pm \ t_{\nu,1-\alpha/2}\sqrt{T}
\,,
\] where \(t_{\nu,1-\alpha/2}\) is the \(1-\alpha/2\) quantile of
\(t_\nu\).

It is important to understand the reasons behind the missingness when
applying multiple imputation to handle incomplete data. Based on the
degree of bias attributable to the missingness, three types of missing
data are defined: missing completely at random (MCAR), missing at random
(MAR), and missing not at random (MNAR) (Rubin, 1976). MCAR occurs when
a missing observation is statistically independent of both the observed
and unobserved data. In the case of MAR, the missingness is associated
with the observed but not with the unobserved data. When data are MNAR,
there is a link between the missingness and the unobserved values in the
dataset.

Among the tools for performing multiple imputation, multivariate
imputations by chained equation (MICE) (van Buuren \&
Groothuis-Oudshoorn, 1999) is a flexible, robust, and widely used
method. This algorithm imputes missing data via an iterative series of
conditional models. In each iteration, each incomplete variable is
filled in by a separate model of all the remaining variables. The
iterations continue until apparent convergence is reached.

In this paper, we implement the MICE framework in \texttt{R} via the
popular \texttt{mice} package (van Buuren \& Groothuis-Oudshoorn, 2011).
Moreover, we focus on multilevel multiple imputation, due to the
hierarchical structure of our data. Specifically, we consider multiple
imputation by a two-level normal linear mixed model with heterogeneous
within-group variance (Kasim \& Raudenbush, 1998). In context, our data
consist of baseball seasons (ages) which are nested within the class
variable, player; and the season-by-season performance is considered to
be correlated for each athlete. The described imputation model can be
specified as the \texttt{2l.norm} method available in the \texttt{mice}
library.

\hypertarget{sec:sim}{%
\section{Simulation}\label{sec:sim}}

In this simulation, we demonstrate our aging curve estimation approach
with multiple imputation, and evaluate how different types of player
dropouts affect the curve. There are three steps to our simulation.
First, we fit a model for the performance-age relationship and utilize
its output to generate fictional careers for baseball players. Next, we
generate missing data by dropping players from the full dataset based on
different criteria, and examine how the missingness affects the original
aging curve obtained from fully observed data. Finally, we apply
multiple imputation to obtain completed datasets and assess how close
the imputed aging curves are to the true curve.

\hypertarget{generating-player-careers}{%
\subsection{Generating Player Careers}\label{generating-player-careers}}

We fit a mixed-effects model using the player data described in Section
\ref{sec:data}. Our goal is to obtain the variance components of the
fitted model to simulate baseball player careers. The model of
consideration is of the form \[
\displaylines{
Y_{pq} = (\beta_0 + b_{0p}) + \beta_1X_q + \beta_2X_q^2 + \beta_3X_q^3 + \epsilon_{pq} 
\cr
b_{0p} \sim N(0, \tau^2) 
\cr 
\epsilon_{pq} \sim N(0, \sigma^2).
}
\] In detail, this model relates the performance metric \(Y_{pq}\) (in
our case, transformed OPS) for player \(p\) at age (season) \(q\) to a
baseline level via the fixed effect \(\beta_0\). The only covariate
\(X\) in the model is age, which is assumed to have a cubic relationship
with the response variable, transformed OPS. Another component is the
observational-level error \(\epsilon_{pq}\) with variance \(\sigma^2\)
for player \(p\) at age \(q\). We also introduce the random effects
\(b_{0p}\), which represents the deviation from the grand mean
\(\beta_0\) for player \(p\), allowing a fixed amount of shift to the
performance prediction for each player. In addition, to incorporate the
variability in production across the season \(q\), a random effect
parameter \(\tau^2\) is included. Our modeling approach is implemented
using the \texttt{lme4} package in \texttt{R} (Bates et al., 2015). We
utilize the sources of variance from the fitted model to simulate 1000
careers for baseball players of age 21 to 39.

\hypertarget{sec:drop}{%
\subsection{Generating Missing Data}\label{sec:drop}}

After obtaining reasonable simulated careers for baseball players, we
create different types of dropouts and examine how they cause deviations
from the fully observed aging curve. We consider the following cases of
players' retirement from the league:

\begin{enumerate}
\def(\arabic{enumi}){(\arabic{enumi})}
\tightlist
\item
  Dropout players with 4-year OPS average below a certain threshold, say
  0.55.
\item
  Dropout players with OPS average from age 21 (start of career) to 25
  of less than 0.55.
\item
  25\% of the players randomly retire at age 30.
\end{enumerate}

For the first two scenarios, the missingness mechanism is MAR, since
players get removed due to low previously observed performance. Dropout
case (3) falls under MCAR, since athletes are selected at random to
retire without considering past or future offensive production.

Figure \ref{fig:drop-compare} displays the average OPS aging curves for
all baseball players obtained from the original data with no missingness
and data with only the surviving players associated with the dropout
mechanisms mentioned above. These are smoothed curves obtained from
loess model fits, and we use mean absolute error (MAE) to evaluate the
discrepancy between the dropout and true aging curves. It is clear that
randomly removing players have minimal effect on the aging curve, as the
curve obtained from (3) and the original curve essentially overlap (MAE
\(= 7.42 \times 10^{-4}\)). On the other hand, a positive shift from the
fully observed curve occurs for the remaining two cases of dropout based
on OPS average (MAE \(= 0.031\) for (1) and MAE \(=0.019\) for (2)).
This means the aging curves with only the surviving players are
overestimated in the presence of missing data due to past performance.
More specifically, the player performance drops off faster as they age
than when it is estimated with only complete case analysis.

\begin{figure}

{\centering \includegraphics{drop-compare-1} 

}

\caption{Comparison of the average OPS aging curve constructed with the fully observed data and different cases of dropouts (obtained only for the surviving players and without imputation).}\label{fig:drop-compare}
\end{figure}

\hypertarget{sec:imp}{%
\subsection{Imputation}\label{sec:imp}}

Next, we implement the multiple imputation with a hierarchical structure
procedure described in Section \ref{sec:meth} to the cases of dropout
that shifts the aging effect on performance. We perform \(m=5\)
imputations with each made up of 30 iterations, and apply Rubin's rules
for combining the imputation estimates. The following results are
illustrated for dropout mechanism (2), where players with a low OPS
average at the start of their careers (ages 21--25) are forced out of
the league.

Figure \ref{fig:drop-imp} (left) shows smoothed fitting loess aging
curves for all 5 imputations and a combined version of them, in addition
to the curves constructed with fully observed and only surviving players
data. The 95\% confidence interval for the mean OPS at each age point in
the combined curve obtained from Rubin's rules is further illustrated in
Figure \ref{fig:drop-imp} (right). It appears that the combined imputed
curve follows the same shape as the true, known curve. Moreover,
imputation seems to capture the rate of change for the beginning and end
career periods quite well, whereas the middle of career looks to be
slightly underestimated. The resulting MAE of \(0.0039\) confirms that
there is little deviation of the combined curve from the true one.

Additionally, we perform diagnostics to assess the plausibility of the
imputations, and also examine whether the algorithm converges. We first
check for distributional discrepancy by comparing the distributions of
the fully observed and imputed data. Figure \ref{fig:diag} (left)
presents the density curves of the OPS values for each imputed dataset
and the fully simulated data. It is obvious that the imputation
distributions are well-matched with the observed data. To confirm
convergence of the MICE algorithm, we inspect trace plots for the mean
and standard deviation of the imputed OPS values. As shown in Figure
\ref{fig:diag} (right), convergence is apparent, since no definite trend
is revealed and the imputation chains are intermingled with one another.

\begin{figure}

{\centering \includegraphics{drop-imp-1} 

}

\caption{At left, comparison of the average OPS aging curve constructed with the fully observed data, only surviving players, and imputation. At right, combined imputed curve with 95\% confidence intervals obtained from Rubin's rules. Results shown here are for dropout case of players having OPS average from age 21 to 25 below 0.55.}\label{fig:drop-imp}
\end{figure}

\begin{figure}

{\centering \includegraphics{diag-1} \includegraphics{diag-2} 

}

\caption{At left, kernel density estimates for the fully observed and imputed OPS values. At right, trace plots for the mean and standard deviation of the imputed OPS values against the iteration number for the imputed data. Results shown here are for dropout case of players having OPS average from age 21 to 25 below 0.55.}\label{fig:diag}
\end{figure}

\hypertarget{sec:app}{%
\section{Application: MLB Data}\label{sec:app}}

Lastly, we apply the multilevel multiple imputation model to estimate
the average OPS aging curve for MLB players. For this investigation,
besides the data pre-processing tasks mentioned in Section
\ref{sec:data}, our sample is limited to all players who made their
major league debut no sooner than 1985, resulting in a total of 2323
players. To perform imputation, we pass in the parameters similar to our
simulation study (\(m=5\) with 30 iterations for each imputation).

Figure \ref{fig:mlb-imp} shows the OPS aging curves for MLB players
estimated with and without imputation. The plot illustrates a similar
result as from simulation, as the combined imputed curve is lower than
the curve obtained when ignoring the missing data. It is clear that the
aging effect is overestimated without considering the unobserved
player-seasons that are observable. In other words, the actual
performance declines with age more rapidly than estimates based on only
the observed data.

\begin{figure}

{\centering \includegraphics{mlb-imp-1} 

}

\caption{Comparison of the average OPS aging curve constructed with only observed players and imputation for MLB data.}\label{fig:mlb-imp}
\end{figure}

\hypertarget{sec:discuss}{%
\section{Discussion}\label{sec:discuss}}

The concept of survivorship bias is frequently seen in professional
sports, and our paper approaches the topic of aging curves and player
dropout in baseball as a missing data problem. We utilize multiple
imputation with a multilevel structure to improve estimates for the
baseball aging curves. Through simulation, we highlight that ignoring
the missing seasons leads to an overestimation of the age effect on
baseball offensive performance. With imputation, we achieve a better
aging curve which shows that players actually decline faster as they get
older than previously estimated.

There are many limitations in our study which leave room for improvement
in future work. In our current imputation model, age is the only
predictor for estimating performance. It is possible to include more
covariates in the algorithm and determine whether a better aging curve
estimate is achieved. In particular, we could factor in other baseball
offensive statistics (e.g.~home run rate, strikeout rate, WOBA, walk
rate,\ldots) in building an imputation model for OPS.

Furthermore, the aging curve estimation problem can be investigated in a
completely different statistical setting. As noted in Section
\ref{sec:lit}, rather than considering discrete observations, another
way of studying aging curves is through a continuous approach, assuming
a smooth curve for career performance. As pointed out by Wakim \& Jin
(2014), methods such as functional data analysis (FDA) and principal
components analysis through conditional expectation (PACE) possess many
modeling advantages, in regard to flexibility and robustness. There
exists a number of proposed multiple imputation algorithms for
functional data (Ciarleglio et al., 2021; He et al., 2011; Rao \&
Reimherr, 2021), which all can be applied in future studies on aging
curves in sports.

\hypertarget{references}{%
\section*{References}\label{references}}
\addcontentsline{toc}{section}{References}

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\end{document}
