# A template for the _arxiv_ style

Shinpei Nakamura-Sakai
Department of Statistics and Data Science
Yale University
New Haven, CT
<EMAIL_ADDRESS>
&Laura Forastiere
Department of Biostatistics
Yale University
New Haven, CT
<EMAIL_ADDRESS>

&Brian Macdonald
Department of Statistics and Data Science
Yale University
New Haven, CT
<EMAIL_ADDRESS>

Corresponding author

# Estimating the age-conditioned average treatment effects curves: An
application for assessing load-management strategies in the NBA

Shinpei Nakamura-Sakai
Department of Statistics and Data Science
Yale University
New Haven, CT
<EMAIL_ADDRESS>
&Laura Forastiere
Department of Biostatistics
Yale University
New Haven, CT
<EMAIL_ADDRESS>

&Brian Macdonald
Department of Statistics and Data Science
Yale University
New Haven, CT
<EMAIL_ADDRESS>

Corresponding author

###### Abstract

In the realm of competitive sports, understanding the performance dynamics of
athletes, represented by the age curve (showing progression, peak, and
decline), is vital. Our research introduces a novel framework for quantifying
age-specific treatment effects, enhancing the granularity of performance
trajectory analysis. Firstly, we propose a methodology for estimating the age
curve using game-level data, diverging from traditional season-level data
approaches, and tackling its inherent complexities with a meta-learner
framework that leverages advanced machine learning models. This approach
uncovers intricate non-linear patterns missed by existing methods. Secondly,
our framework enables the identification of causal effects, allowing for a
detailed examination of age curves under various conditions. By defining the
Age-Conditioned Treatment Effect (ACTE), we facilitate the exploration of
causal relationships regarding treatment impacts at specific ages. Finally,
applying this methodology to study the effects of rest days on performance
metrics, particularly across different ages, offers valuable insights into
load management strategies’ effectiveness. Our findings underscore the
importance of tailored rest periods, highlighting their positive impact on
athlete performance and suggesting a reevaluation of current management
practices for optimizing athlete performance.

_Keywords_ Causal Inference $\cdot$ Age curve $\cdot$ Meta-learners $\cdot$
Load-management

##  1 Introduction

Athletic performance over an individual’s career is not a constant; it is a
dynamic interplay of improvement, peak performance, and eventual decline. This
performance trajectory, often visualized as an "age curve," encapsulates the
biological and career lifespan of athletes across various sports disciplines.
While it is widely recognized that these curves vary due to a myriad of
factors unique to each athlete, the scientific inquiry into the heterogeneity
of treatment effects across different ages remains underexplored. The nuanced
understanding of how specific interventions or treatments impact athletes at
various stages of their careers can be instrumental in enhancing their
performance and extending their peak years in the sport.

Schuckers et al. (2023)’ review focuses on regression-based models for age
curve analysis, specifically highlighting the predominance of fixed effects
models that incorporate quadratic and cubic age terms Brander et al. (2014),
and Lichtman’s delta method for age effect analysis Lichtman (2009).
Importantly, the review notes that all discussed models utilize season-level
data. It also points to advancements through semiparametric methods such as
spline techniques and generalized additive models Wakim and Jin (2014) Judge
(2020) Turtoro (2019), which offer more nuanced analytical capabilities.

The emphasis on season-level data in these models is critical; however,
transitioning to game-level data could significantly refine the analysis by
incorporating detailed variables like opponent strength, home advantage, and
team dynamics. While offering a richer dataset, this shift introduces
complexity due to the intricate interactions between these variables.
Traditional models, designed around season-level data using regression-based
models, face challenges in accurately capturing these dynamics, highlighting
the need for a new analytical framework. Such a framework would efficiently
handle non-linear interactions between game-level variables, representing a
significant methodological advancement in age curve analysis.

Integrating Conditional Average Treatment Effect (CATE) or Heterogeneous
Treatment Effect (HTE) estimation into the age-curve literature could
significantly enhance the precision of understanding how various treatments
impact athletes at different ages. HTE estimation is a statistical approach
that identifies subgroups of individuals who may experience varying responses
to a given treatment Athey and Imbens (2016). By applying HTE methods to the
age-curve literature, researchers can move beyond general trends to discover
how the age of an athlete interacts with the effectiveness of interventions.
Meta-learning techniques, such as those outlined in Künzel et al. (2019), can
improve the power and flexibility of HTE estimation in this context. For
example, HTE estimation could reveal varying benefits of injury prevention
strategies across different age groups, if particular nutritional adjustments
yield greater performance benefits for certain age groups, how mental health
interventions can be tailored to age-specific needs, or if recovery techniques
like cryotherapy demonstrate differential effects depending on an athlete’s
age. This precise analysis enables better resource allocation, targeting
interventions where they’ll be most beneficial to improve performance and
health outcomes cost-effectively. Beyond refining current age-curve analysis,
it paves the way for personalized athlete care strategies.

In the past, athletic management strategies regarding rest periods and "load
management" have largely relied on observed differences in how athletes of
various ages respond to training and recovery. However, this field lacks
rigorous quantitative analysis. As professional sports become increasingly
competitive, there’s a critical need to develop data-driven, individualized
strategies. Such strategies would integrate an athlete’s unique profile to
optimize performance outcomes while safeguarding their long-term health and
well-being. Recent research emphasizes a shift away from relying solely on the
concept of workload as a predictor of injury risk and performance. Instead,
there’s an increasing focus on understanding the complex relationship between
an athlete’s training load, internal responses, and external factors Gabbett
(2016) Drew and Finch (2016). This highlights the need for individualized load
management strategies that continuously monitor a variety of physiological and
subjective markers. The aim is to balance performance goals with injury
prevention by acutely measuring how athletes adapt to and recover from
training stress. This data-driven approach promises to enhance decision-making
for optimizing training schedules, and recovery protocols, and mitigating
athlete injury risk.

In response to those needs, our work introduces a novel analytical framework
aimed at estimating age-specific treatment effects to construct a personalized
age curve for treatment impact. This framework is particularly focused on
decision-making processes regarding athlete rest and load management, a
concept at the forefront of sports science and athlete longevity. The
contributions of this study are threefold. Initially, we unveil a method for
estimating the age curve utilizing game-level data, a departure from
conventional approaches that depend on aggregated season-level data. This
shift to game-level data introduces complexities due to a larger array of
variables needing management. We address this by deploying a meta-learner
framework that allows for the application of sophisticated machine learning
models, adept at uncovering non-linear patterns that elude current leading
age-curve estimation techniques. Second, our framework offers a means to
discern causally identifiable treatment effects, facilitating a nuanced
comparison of age curves across various covariates and interventions. By
establishing the Age-Conditioned Treatment Effect (ACTE) for any given
intervention, our methodology supports the exploration of causal hypotheses
related to treatment impacts and outcomes for specific ages. Lastly, we apply
our methodology to investigate the influence of rest days between games on key
performance metrics, with a focus on athletes’ age. This analysis provides
insights into the effectiveness of load management strategies, highlighting
the performance benefits when athletes receive adequate rest. Through these
contributions, our study not only enhances our understanding of age-related
performance dynamics but also offers practical guidance for optimizing athlete
management strategies.

This introduction paves the way for the forthcoming sections, setting the
stage for a detailed exploration of our methodology in Section 2, a simulation
designed to offer an in-depth understanding of the framework in Section 3, an
examination of the outcomes and implications of our findings in Section 4, and
we will conclude with a discussion in Section 5. Our study proposes a
significant shift towards individualized athlete management strategies, moving
away from traditional, one-size-fits-all approaches. This potential shift aims
to redefine how athletic performance and longevity are optimized, marking a
pivotal change in the field.

##  2 Methodology

###  2.1 Framework

Let us consider a sample of N units, represented by the indicator
$i\in\\{1,...,N\\}$. For each unit, we observe the following variables:
$A\in\mathbb{Z}$, representing unit i’s age, $W\in\\{0,1\\}$, denoting a
binary treatment of interest, a vector of covariates ${X}_{i}$, and the
outcome of interest denoted by $Y_{i}^{obs}$. Under Rubin’s potential outcome
framework Rubin (1974), we make the Stable Unit Treatment Value Assumption
(SUTVA; see appendix A) and we denote by $Y(w)$ the potential outcome of unit
i under treatment $w$. We assume that for each unit i the tuple
$(Y_{i}(0),Y_{i}(1),A_{i},W_{i},X_{i})$ is independently drawn from a
distribution $\mathcal{P}$, such that
$(Y_{i}(0),Y_{i}(1),A_{i},W_{i},X_{i})\sim\mathcal{P}$.

For each potential outcome, we define the Conditional Expectation Function
(CFE), denoted by $\mu_{w}(a,x)$, as the expectation of the potential outcome
as a function of age and covariates, that is:

$\mu_{0}(a,x):=\mathop{{}\mathbb{E}}[Y_{i}(0)|A_{i}=a,X_{i}=x]\text{ and
}\mu_{1}(a,x):=\mathop{{}\mathbb{E}}[Y_{i}(1)|A_{i}=a,X_{i}=x]$ (1)

.

Under this framework, our main causal estimand of interest is the Age-
Conditioned Treatment Effect (ACTE), that is:

$\displaystyle\tau(a)$
$\displaystyle:=\mathop{{}\mathbb{E}}[Y_{i}(1)-Y_{i}(0)|A=a]$ (2)
$\displaystyle=\mu_{1}(a)-\mu_{0}(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mu_{1}(a,x)-\mu_{0}(a,x)]$

where $\mu_{w}(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mu_{1}(a,x)]$
is referred to as the Age-Conditioned Expectation Function (ACEF). Following
Schuckers et al. (2023), we can decompose the potential outcome as follows:

$Y_{i}(w)=g(a,w)+f_{i}(x,a,w)+\epsilon_{i}$ (3)

where $g(a,w)$ is the average performance at age $a$ for all players with
treatment $w$, $f(x,a,w)$ represents a performance adjustment at age $a$ for
player with covariates $x$, and treatment $w$, and $\epsilon$ is the mean zero
model error. Note that under this decomposition, $g$ is the representation of
all players, and hence $f$ represents the variation for all players at age
$a$. Namely, $\mathop{{}\mathbb{E}}_{\mathcal{X}}[f(x,a,w)|A=a]=0$

Note that using the decomposition in Equation 3, we have

$\displaystyle\tau(a)=\mathop{{}\mathbb{E}}[Y_{i}(1)-Y_{i}(0)|A=a]$
$\displaystyle=g(a,1)-g(a,0)+\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[f(x,a,1)-f(x,a,0)|A=a]$
$\displaystyle=g(a,1)-g(a,0)$

In order to properly identify this estimand we need the following assumptions:

######  Assumption 1 (Unconfoundedness).

There are no unmeasured factors affecting the probability of treatment
allocation and the outcome. Formally,

$(Y_{i}(0),Y_{i}(1))\perp\\!\\!\\!\perp W_{i}|X_{i}$

Under these assumptions, we can identify our estimand of interest using the
following theorem.

######  Theorem 1 (Identification of ACTE).

Under SUTVA and Assumption 1 we have

$\tau(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[Y_{i}^{obs}|W_{i}=1,A_{i}=a,X_{i}=x]]-\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[Y_{i}^{obs}|W_{i}=0,A_{i}=a,X_{i}=x]]$

###### Proof.

See appendix A ∎

###  2.2 Estimation of $\tau(a)$

To estimate $\tau(a)$, we implement a causal machine learning method,
particularly meta-learners, as introduced by Künzel et al. (2019). This
approach is well-suited for our observational study, especially considering
the complexities in estimating $g(a,x)$. Meta-learners offer notable
flexibility, allowing for the integration of various predictive models to suit
different goals. Künzel et al. (2019) proposes three distinct types of meta-
learners: S-learners, T-learners, and X-learners, each designed for specific
causal inference challenges.

The S-learner is termed so due to its use of a ’single’ estimator. This
learner includes the treatment as one of the independent variables in the
outcome model, i.e.,

$\mu^{obs}(a,x,w):=\mathop{{}\mathbb{E}}[Y_{i}^{obs}|A=a,W=w,X=x],$

which is estimated by a base learner, which could be any supervised learning
or regression estimator. Consequently, the ACEF is given by

$\hat{\mu}_{w}(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\hat{\mu}^{obs}(a,x,w)],$

and the ACTE is then

$\hat{\tau}(a)=\hat{\mu}_{1}(a)-\hat{\mu}_{0}(a).$

The pseudocode of S-learner is provided in Appendix B

The T-learner uses ’two’ separate learners to estimate $\tau$. It
independently learns $\hat{\mu}_{0}(a,x)$ and $\hat{\mu}_{1}(a,x)$ for the
treatment and control groups, respectively, following the setup:

$\mu^{obs}_{0}(a,x):=\mathop{{}\mathbb{E}}[Y_{i}^{obs}|A_{i}=a,W_{i}=0,X_{i}=x]\text{
and
}\mu_{1}^{obs}(a,x):=\mathop{{}\mathbb{E}}[Y_{i}^{obs}|A_{i}=a,W_{i}=1,X_{i}=x]$

The first expression, the control response function, is estimated by any base
learner that could be any model of choice using the observation in the control
group, namely $\\{(A_{i},X_{i},Y_{i})\\}_{\\{i:W_{i}=0\\}}$ and we denote the
estimated function as $\hat{\mu}_{0}^{obs}(a,x)$. Similarly,
$\mu_{1}^{obs}(a,x)$ is learned using
$\\{(A_{i},X_{i},Y_{i})\\}_{\\{i:W_{i}=1\\}}$ with a potentially different
base learner of choice and denoting the estimator by
$\hat{\mu}^{obs}_{1}(x,a)$.

Then, the ACEF is given by

$\hat{\mu}_{w}(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\hat{\mu}_{w}^{obs}(a,x)],$

and the ACTE is given by $\hat{\tau}(a)=\hat{\mu}_{1}(a)-\hat{\mu}_{0}(a)$ as
before. The pseudocode of the T-learner is provided in Appendix B.

X-learner involves a two-step process. Initially, it estimates
$\hat{\mu}_{1}^{obs}(a,x)$ and $\hat{\mu}_{0}^{obs}(a,x)$ akin to the
T-learner either using the same of different base-learners. In the second
step, using the model trained in step one, it ’imputes’ the treatment effect
for the treated units, denoted by $D_{i}^{1}$, using
$\hat{\mu}_{0}^{obs}(a,x)$, that is,

$\tilde{D}^{1}_{i}=Y^{obs}_{i}-\hat{\mu}_{0}^{obs}(A_{i},X_{i})\qquad\forall
i:W_{i}=1,$

and it ’imputes’ the treatment effect for the control units, denoted by
$D_{i}^{0}$, using $\hat{\mu}_{1}^{obs}(a,x)$, that is,

$\tilde{D}^{0}_{i}=\hat{\mu}^{obs}_{1}(A_{i},X_{i})-Y^{obs}_{i}\qquad\forall
i:W_{i}=0.$

Then we could use any predictive model including regression or supervised
machine learning models to estimate

$\tau^{0}(a):=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[\tilde{D}_{i}^{1}|A_{i}=a,X_{i}=x]]\text{
and
}\tau^{1}(a):=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[\tilde{D}_{i}^{0}|A_{i}=a,X_{i}=x]],$

resulting in $\hat{\tau}^{0}(a)$ and $\hat{\tau}^{1}(a)$. The final
conditional average treatment effect for each age is estimated by

$\hat{\tau}(a)=e(a)\hat{\tau}_{0}(a)+(1-e(a))\hat{\tau}_{1}(a),$

with $e(a)\in[0,1]$ typically being the propensity score, i.e.,
$e(a):=Pr(W_{i}=1|A_{i}=a)$. Given the complex nature of the X-learner, it
does not directly yield the Conditional Expectation Function (CEF). However,
by analyzing the modified treatment variable $\hat{\tau}^{w}(a)$, we can
discern the differences between the treated and control groups. This approach
allows for a deeper understanding of the treatment effects within these
distinct groups. The pseudocode of the X-learner is provided in Appendix B.

Choosing between S, T, and X learners hinges on the data structure and the
specific objectives of causal inference analysis. The S-learner is most
effective when the dataset is relatively small or when the treatment effect is
hypothesized to be uniform across observations, making it a streamlined choice
for simpler causal relationships. In contrast, the T-learner excels with
larger datasets that can support separate models for treated and control
groups, offering a more detailed comparison of treatment effects and is
preferable when the data allows for a clear segregation of treatment groups.
The X-learner, however, stands out in scenarios where treatment effects are
expected to be highly heterogeneous or when there is an imbalance in the
number of observations between treatment groups. It is specifically tailored
for complex causal inquiries where the treatment effect varies significantly
across individuals or subgroups. Thus, the decision to use a specific learner
model should be informed by the dataset’s characteristics, including size,
balance, and the anticipated variability in treatment effects. This decision-
making process is underscored by a simulation study we conducted to assess the
comparative effectiveness of these models in estimating ACTE.

###  2.3 Base Learners

The selection of a base learner offers considerable flexibility, allowing for
the use of a simple regression algorithm or the adoption of more sophisticated
supervised machine learning models. This versatility ensures that the chosen
learner can be tailored to the specific needs and complexities of the data at
hand. Leveraging meta-learners’ versatility, we employ here two learners. The
primary model, drawn from Schuckers et al. (2023), is the spline with a fixed
effect model which achieved minimal average RMSE in their simulation study.
Their study evaluates estimated curves using different regression-based models
by comparing their average Root Mean Squared Error (RMSE) across different
ages.

To enhance the predictability of our models, we also integrate more
sophisticated methodologies, namely tree-based approaches. These, particularly
for tabular data, have demonstrated superior predictive capabilities over deep
neural networks, as evidenced by Grinsztajn et al. , Borisov et al. , and
Shwartz-Ziv and Armon . Among these, we specifically employ the Rforesty
package Künzel et al.  to implement a fast and honest random forest model. The
concept of ’honesty’ in this context signifies the utilization of distinct
data subsets for model training and prediction phases, thereby effectively
minimizing the overfitting issue commonly associated with standard random
forest approaches. This methodology ensures that our model’s predictions are
not only swift but also reliably accurate, setting a new standard for
predictive analysis in our field.

The choice between the spline and random forest models is context-dependent.
While the spline model is simpler and easier to interpret, the random forest
boasts higher predictive power. Selecting the appropriate model hinges on the
specific data and problem at hand.

##  3 Simulation Study

###  3.1 Simulation Design

In causal inference, simulation studies are crucial because only one of the
potential outcomes, either $Y(0)$ or $Y(1)$, can be observed. To discern the
most suitable meta-learner and estimation function for $\mu$, we have designed
three simulation scenarios that increase in complexity. The initial simulation
assumes a constant treatment effect, with the treatment and control groups
exhibiting the same quadratic shape. In the second simulation, the treatment
effect grows with age, introducing different shapes for the treatment and
control groups. The third and most complex simulation models a sophisticated
treatment effect, resulting in distinct shapes for each group.

Building on the ideas shared in Schuckers et al. (2023), our method sets
itself apart by focusing on the causal effect, $\tau(a)$, rather than just
trying to map out the age curve. This shift in focus allows us to explore the
underlying causal relationships. We simulate outcomes from

$Y_{i}(w)=g(a,w)+f_{i}(x,a,w)+\epsilon_{i}$

and

$g(a,w)=\omega+\beta_{1}(a-a_{max})^{2}+\beta_{2}(a-a_{max})^{2}\mathbbm{1}(a>a_{max})+\beta_{3}(a-a_{max})^{3}\mathbbm{1}(y>t_{max})+\tau(a)w$

.

where $i=1,...N$ and $a=\\{a_{1},...a_{max}\\}$ and for each scenario, we vary
$\tau(a)$ and $f(x,a,w)$ as follows

  1. 1.

Simulation 1 (Constant Treatment Effect):

$\tau(a)=2$ $f(x,a,w)=\gamma_{i}+b_{i}(a-a_{max})^{2}\mathbbm{1}(a>a_{max})$

  2. 2.

Simulation 2 (Linear Treatment Effect):

$\tau(a)=0.1(a-a_{min})$
$f(x,a,w)=\gamma_{i}+b_{i}(a-a_{max})^{2}\mathbbm{1}(a>a_{max})$

  3. 3.

Simulation 3 (Non-linear Treatment Effect):

$\tau(a)=2(a-16)+0.0005*\mathbbm{1}(a>20)*(a-a_{max})^{3}-0.0005*\mathbbm{1}(a>a_{max})*(a-a_{max})^{4}$
$f(x,a,w)=\gamma_{i}+b_{i}(a-a_{max})^{2}\mathbbm{1}(a>a_{max})+(2+3w)x_{i}$

In the simulation, individual random effects $\gamma_{i}$ and slope
coefficients $b_{i}$ are normally distributed with mean zero and standard
deviations $\sigma_{\gamma}$ and $\sigma_{\beta}$, respectively. The quadratic
and cubic terms are assigned specific values: $\beta_{1}=-\frac{1}{9}$,
$\beta_{2}=-\frac{6}{1000}$, and $\beta_{3}=\frac{45}{10000}$, reflecting the
empirical shape of the age-performance relationship observed in hockey data,
as discussed in Schuckers et al. (2023). The standard deviations
$\sigma_{\beta}=0.02$, $\sigma_{\epsilon}=1$, and $\sigma_{\gamma}=0.4$ are
set to capture the variability inherent in the data. Treatment assignment is
modeled as a Bernoulli distributed variable with a probability of $0.151$,
representing the proportion of back-to-back games played across ten seasons of
our cleaned data (2011-2021). Additionally, $x_{i}$ in scenario 3 is a
covariate from a uniform distribution on the interval $(-1,1)$

Figure 1: Simulation Scenario 1: Analysis of Constant Treatment Effect -
Potential Outcome Functions for Control and Treated Groups (Left), MSE in ACTE
Estimation (Right).

###  3.2 Simulation Results

We compute the $\tau(a)$ for a total of six combinations. Three from meta-
learning algorithm, S, T, and X learner and two models ols-spline and random
forest. Then we calculated the MSE to assess the performance of each
combination.

Figures 1, 2, 3 present the MSE by age, highlighting that in Simulation 1,
S-learner with OLS yields a minimal error, aligning with model assumptions due
to both groups’ constant shape derived from a spline family. In Simulation 2,
the T-learner with RF emerges as the top performer, capturing the distinct
age-curve shapes. For Simulation 3, the intricate X-learner with RF attains
the lowest MSE. Notably, S-learner with OLS is less effective for younger and
older ages, as it assumes the same shape for controlled and treated age
curves. While S-learner with RF improves upon this by differentiating
treatment in its tree splits especially for younger players, it still falls
short in encapsulating the full complexity of the group shapes.

In Table 1, it is evident that the optimal method varies by scenario. For
complex treatment effects, the X-learner paired with a random forest is
superior, whereas, for simpler effects with parallel trends in treatment and
control groups, the S-learner with ols-spline is preferable. Researchers must
establish well-informed assumptions and choose their models judiciously. In
practice, due to the fundamental problem of causal inference, the
unobservability of counterfactuals, actual MSE cannot be computed.

Figure 2: Simulation Scenario 2: Linear Treatment Effect - Potential Outcome
Functions for Control and Treated Groups (Left), MSE in ACTE Estimation
(Right).

Figure 3: Simulation Scenario 3: Non-linear Treatment Effect - Potential Outcome Functions for Control and Treated Groups (Left), MSE in ACTE Estimation (Right). Model | simulation1 | simulation2 | simulation3
---|---|---|---
s.ols | 0.00 | 0.44 | 89.23
t.ols | 1.53 | 1.44 | 9.40
x.ols | 1.53 | 1.44 | 9.40
s.rf | 0.35 | 0.29 | 9.90
t.rf | 0.05 | 0.07 | 4.17
x.rf | 0.07 | 0.08 | 3.74
Table 1: MSE for ACTE estimation

##  4 Application to National Basketball Association data

Load management in the NBA, a strategy increasingly adopted for injury
prevention and career longevity, also serves to optimize player performance.
Kawhi Leonard’s 2018-2019 season with the Toronto Raptors is a prime example.
By participating in only 60 of the 82 regular-season games, Leonard was in top
form for the playoffs, leading the Raptors to their inaugural championship.
Similarly, Gregg Popovich of the San Antonio Spurs masterfully applied this
approach with veteran Tim Duncan, preserving Duncan’s performance levels and
contributing to the Spurs’ repeated championship successes. This strategic
balance between player health and team achievement exemplifies the dual
benefits of load management.

Critiques of load management point to its impact on the regular season’s
significance, suggesting that teams might not prioritize these games, thereby
diminishing fan engagement. This critique led to the NBA’s implementation of
the Player Participation Policy for the 2023-2024 season, requiring star
players to participate more frequently, with restrictions on resting multiple
star players simultaneously. This highlights the need for quantifying the
impact of rest on both player performance and the overall competitiveness of
the season.

Despite its growing acceptance, quantitative analysis of load management’s
precise impact remains scarce, particularly in comparison to the effects of
playing in back-to-back games. In this study, we apply the ACTE framework to
measure the impact of rest on players, segmented by age. While a randomized
controlled trial (RCT) would be the ideal method for such analysis, the
practical challenges and high costs make it infeasible in the context of an
NBA season. Therefore, our approach utilizes observational studies within this
framework. While acknowledging the inherent limitations and assumptions of
observational studies, we believe this approach offers a valuable starting
point for applying causal machine learning to age-related performance curves,
providing insightful results into the effects of load management.

###  4.1 Data

Leveraging ten seasons (2011-2022) of NBA data obtained through the hoopR
package Gilani , we utilize the discussed framework to analyze how days of
rest and age influence various box score statistics for NBA players. To ensure
the relevance of our data, we only consider instances where players were
active for at least 25 minutes in the previous game, thereby excluding minimal
play times. After the data processing, we have total of 827 players. In our
analysis, the treatment variable $w=0$ signifies games played on consecutive
days (back-to-back, b2b), while $w=1$ represents having at least one day of
rest. The potential outcome under treatment $w$ is denoted as $Y(w)$, with $X$
representing the covariate matrix. We define $\mu_{w}(a,x)$ as the conditional
expectation of the potential outcome $Y(w)$, given the player’s age $a$. This
calculation facilitates the construction of the age curve. Through
$\tau(a,x)$, we ascertain the ACTE, quantifying the impact of rest versus
participating in b2b games at specific ages. Given the observational nature of
our study, completely eliminating confounders might not be feasible. However,
we aim to mitigate their influence by adjusting for variables such as player,
team, opponent team, home or away games, and season-fixed effects. Our
analysis primarily focuses on players aged between 18 and 39, as the treatment
group’s sample size outside this range is significantly limited, as shown in
table 2.

Table 2: Number of games played across age and treatment status Age | b2b | non-b2b
---|---|---
18 | 0 (0.00%) | 5 (100.00%)
19 | 266 (14.22%) | 1604 (85.78%)
20 | 679 (14.66%) | 3952 (85.34%)
21 | 917 (14.49%) | 5412 (85.51%)
22 | 1312 (15.47%) | 7168 (84.53%)
23 | 1734 (16.32%) | 8892 (83.68%)
24 | 1795 (15.97%) | 9446 (84.03%)
25 | 1715 (15.14%) | 9616 (84.86%)
26 | 1850 (15.66%) | 9964 (84.34%)
27 | 1785 (16.04%) | 9344 (83.96%)
28 | 1529 (14.94%) | 8706 (85.06%)
29 | 1221 (14.98%) | 6930 (85.02%)
30 | 984 (14.25%) | 5921 (85.75%)
31 | 867 (14.46%) | 5129 (85.54%)
32 | 633 (13.71%) | 3984 (86.29%)
33 | 464 (13.98%) | 2855 (86.02%)
34 | 316 (13.98%) | 1944 (86.02%)
35 | 218 (13.53%) | 1393 (86.47%)
36 | 128 (12.04%) | 935 (87.96%)
37 | 69 (11.54%) | 529 (88.46%)
38 | 39 (13.18%) | 257 (86.82%)
39 | 22 (10.38%) | 190 (89.62%)
40 | 3 (5.66%) | 50 (94.34%)
41 | 0 (0.00%) | 5 (100.00%)
42 | 0 (0.00%) | 10 (100.00%)

###  4.2 Average Conditional Expectation Function (ACEF)

In this section, we will analyze the ACEF for S and T learners using ols-
spline and random forest.

In all ACEF plots, a finer line denotes the $\hat{\mu}_{w}(a)$, which
represents the estimated expectation given age. We also employ a smoothing
spline function, fitted through a generalized additive model (GAM) with a
smoothness degree of six, as suggested by Schuckers et al. (2023). This
approach helps us understand the general age-curve trend for both the
treatment and control groups, represented by a solid line. The shaded regions
show the 90% bootstrap confidence intervals for each age group. Non-
overlapping shaded areas provide statistical evidence of the impact of rest on
subsequent game performance. For a detailed discussion on the confidence
interval calculation, refer to Appendix B.

First, we analyze the net, offensive, and defensive ratings. Net, Offensive,
and Defensive Ratings are crucial statistics in basketball analytics.
Offensive Rating measures a player’s or team’s efficiency in scoring,
calculated as the number of points produced per 100 possessions; a higher
offensive rating indicates better offensive performance. Conversely, a
defensive rating assesses the ability to prevent opponents from scoring, with
points allowed per 100 opponent possessions; a lower defensive rating means
stronger defense. Net Rating is the difference between offensive and defensive
rating, offering a holistic view of overall impact. A positive net rating
implies a player or team contributes more offensively than they concede
defensively, while a negative value suggests a need for improvement in scoring
efficiency, defensive effectiveness, or both.

In Figures 4, 5, 6, and 7, the Conditional Expectation Function (CEF) for net,
offensive, and defensive ratings is displayed. Across all three metrics, the
effects are significantly pronounced with age except for older players where
we lack sample size.

With the S-learner using OLS-Spline, most effects show significance. Yet, this
model is the most constrained, potentially hiding strong intrinsic bias. It
operates under the assumption of uniform treatment effects across ages, which
can be mitigated only by incorporating two-way interaction terms between
covariates and treatment. The S-learner with Random Forest (RF) addresses this
limitation, as trees adeptly capture nonlinear relationships between treatment
and covariates. However, this model reveals a non-significant effect on
offensive ratings for several ages. One hypothesis for this observation could
be that older players, in back-to-back games, emphasize offense over defense.
Defensively, rotating younger players to cover key offensive opponents might
mitigate physical fatigue. Another contributing factor could be the limited
data on older and younger players. As Table 2 indicates, maintaining a career
in the competitive NBA is challenging, leading to less precise inferences for
this demographic.

The T-learner results offer more insights, particularly regarding older
players, due to its complexity. Being more intricate than the S-learner, the
T-learner demands a larger sample size. While the general conclusions for both
OLS-Spline and RF mirror those from the S-learner with RF, it becomes more
apparent that rest impacts the defensive aspect more than the offensive end.

Figure 4: CEF using S-learner and OLS-Spline for net, offensive, and defensive
rating

Figure 5: CEF using S-learner and RF for net, offensive, and defensive rating

Figure 6: CEF using T-learner and OLS-Spline for net, offensive, and defensive
rating

Figure 7: CEF using T-learner and RF for net, offensive, and defensive rating

We now turn our attention to analyzing box score statistics in the NBA, a
detailed set of metrics that measure a player’s game performance. These
include points, assists, rebounds (divided into offensive and defensive),
steals, blocks, turnovers, and fouls. Assists tally a player’s contribution to
teammates’ scoring, while steals and blocks are key defensive indicators of
disrupting the opponent’s play. Turnovers and fouls, conversely, denote player
errors and infractions. Such statistics are vital for assessing both
individual and team play, offering a quantitative recap of crucial game
elements. Our analysis particularly focuses on points, assists, total
rebounds, steals, blocks, and turnovers.

To ensure equitable comparisons among players with varying game minutes, we
examine these statistics normalized per 100 possessions. This approach aids in
more accurately interpreting the respective trends and patterns in player
performance across different game scenarios.

In Figures 8, 9, 10, and 11, we display the Conditional Expectation Function
(CEF) for the previously mentioned box score statistics. Notably, steals are
the only statistic showing a stronger effect than other metrics.

In the analysis using the S-learner, it becomes clear that steals are the only
statistic consistently demonstrating a positive effect. This pattern suggests
that rest could significantly influence steals, likely due to the physical
nature required for successful execution. Although the effect on turnovers is
not marked by statistical significance, the notable U-shaped curve indicates
that prime-aged players generally incur fewer turnovers compared to their
counterparts. Blocks exhibit a similar U-shaped trend. However, this might be
skewed by Tim Duncan’s extraordinary blocking in his later career years. As
shown in Table 3, Duncan’s performance predominantly drives this trend, with
the smaller sample size in his age bracket possibly leading to the unusual
pattern in blocks. For assists, the older the players get they improve their
assist skills. One hypothesis for this is that when the players get old,
instead of being physical and attacking the rim, veteran players prefer to
dish the ball taking advantage of their experience consistent players
rendering high assists are mostly Steve Nash and Chris Paul as we can observe
in the table 4. Moreover, the assist category is an area where treatment has a
minimal impact, primarily because it involves less physical contact compared
to other statistics.

The T-learner largely aligns with the conclusions drawn from other models, but
it uniquely captures the rest effect on blocks when we use random forest,
particularly among younger players. This model highlights that rest has a
significant impact on this demographic. Across all four combinations analyzed,
the conclusions are similar, indicating consistent trends. This consistency is
especially notable in most box score statistics, where the T-learner
reinforces the patterns observed in other estimators, emphasizing the
influence of rest on certain aspects of player performance.

Table 3: Top 10 blocks per 100 possessions by players above 35 years old Player | age | b100 | date
---|---|---|---
Tim Duncan | 36.00 | 12.07 | 2013-01-13
Tim Duncan | 39.00 | 10.91 | 2015-11-18
Tim Duncan | 37.00 | 10.53 | 2014-02-03
Marcus Camby | 37.00 | 10.20 | 2012-02-01
Tim Duncan | 36.00 | 9.80 | 2012-11-28
Tim Duncan | 36.00 | 9.43 | 2013-03-03
Tim Duncan | 36.00 | 9.09 | 2013-02-13
Tim Duncan | 36.00 | 8.96 | 2012-12-12
Marcus Camby | 37.00 | 8.96 | 2012-03-30
Tim Duncan | 36.00 | 8.93 | 2013-01-16
Table 4: Top 10 assists per 100 possessions by players above 35 years old Player | age | a100 | date
---|---|---|---
Steve Nash | 37.00 | 30.91 | 2012-01-24
Chris Paul | 36.00 | 27.59 | 2012-02-04
Kobe Bryant | 36.00 | 27.42 | 2012-03-05
Steve Nash | 38.00 | 26.15 | 2012-12-12
Steve Nash | 38.00 | 25.86 | 2015-11-28
Chris Paul | 36.00 | 25.42 | 2013-12-02
Steve Nash | 37.00 | 25.42 | 2014-01-24
Chris Paul | 36.00 | 25.00 | 2022-02-25
Steve Nash | 38.00 | 25.00 | 2012-04-13
Steve Nash | 38.00 | 25.00 | 2013-11-15

Figure 8: CEF using S-learner and OLS-Spline for box score statistics per 100
possessions

Figure 9: CEF using S-learner and RF for box score statistics per 100
possessions

Figure 10: CEF using T-learner and OLS for box score statistics per 100
possessions

Figure 11: CEF using T-learner and RF for box score statistics per 100
possessions

Finally, we analyze the shooting statistics. Field Goal Percentage, Three-
Point Field Goal Percentage, and True Shooting Percentage are essential
metrics in basketball for assessing shooting efficiency. Field Goal Percentage
is calculated by dividing the number of successful field goals (any shot made
from within the three-point line) by the total number of field goal attempts.
Three-point field Goal Percentage focuses on shots made from beyond the three-
point line, calculated similarly by dividing successful three-point field
goals by three-point attempts. True Shooting Percentage is a more
comprehensive measure, accounting for field goals, three-point field goals,
and free throws, offering a complete picture of a player’s shooting efficiency
by considering the value of each shot type. It’s calculated using a formula
that adjusts for the fact that three-point field goals are worth more than
other field goals and include free throws.

In Figures 12, 13, 14, and 15, we analyze the impact of rest on various
shooting metrics, represented by four distinct curves. Generally, rest
influences shooting percentages; however, for three-point attempts, this
effect is less pronounced compared to the other two metrics. Specifically, for
regular Field Goal Percentage and True Shooting Percentage, the influence of
rest appears to be significant predominantly among younger players. However,
for older players, the data presents a degree of uncertainty and erratic
patterns. This inconsistency can be attributed to the limited sample size for
this age group, leading to imprecise curve estimations that are vulnerable to
skewing by outliers. The observed trend suggests that scoring two-point field
goals, which often involve more physical exertion due to closer proximity to
the basket and defensive pressure, is more affected by rest than shooting
three-pointers, which are typically less physically demanding. This
distinction could explain why rest has a more pronounced effect on FG and TS
percentage compared to three-point shooting.

Figure 12: CEF using S-learner and OLS-Spline for the field goal, three-point,
and true shooting percentage

Figure 13: CEF using S-learner and RF for the field goal, three-point, and
true shooting percentage

Figure 14: CEF using T-learner and OLS-Spline for the field goal, three-point,
and true shooting percentage

Figure 15: CEF using T-learner and RF for the field goal, three-point, and
true shooting percentage

###  4.3 Age-Conditioned Treatment Effect (ACTE)

In this section, we present the results obtained from the X-learner. Due to
the algorithm’s design, which ’crosses’ inferential information between
groups, it’s not feasible to plot the two Conditional Expectation Functions
(CEFs) for this model. Instead, we’ll focus on each metric’s Average Treatment
Effect (ATE). It’s important to note that in our specific covariate context,
the X-learner’s results using an OLS-spline approach would mirror those of the
T-learner with an OLS-spline. Therefore, we will limit our discussion to the
outcomes of employing a random forest with honesty. A significant benefit of
this approach is that it incorporates information from both the treatment and
control groups in the estimation of treatment effects.

In all the figures presented, various elements are strategically employed to
illustrate the ACTE and its context. The finer line in each graph represents
the ACTE itself, providing a direct visual representation of the treatment
effect. Accompanying the ACTE line is its associated bandwidth, which
delineates the 90% bootstrap confidence interval. This interval is crucial for
understanding the statistical significance of the observed effects.
Additionally, a dashed horizontal line is plotted at zero. The relationship of
the confidence interval with this dashed line is key: if the interval does not
intersect the dashed line, it indicates that the treatment effect is
statistically significant. Such visual demarcations in the figures are
essential for a nuanced interpretation of the data, allowing for a more
informed understanding of the significance and implications of the ACTE in
various contexts.

In Figure 16, the analysis upholds the main findings observed in the ACEFs.
The net rating emerges as significantly impactful across various age groups,
except older players where the limited sample size poses a challenge to
conclusive analysis. A noteworthy aspect is the differential impact on the
defensive and offensive ends; the ACTE in defense is notably greater in
absolute value than in offense. This disparity suggests a more pronounced
effect of rest on defensive capabilities. Consequently, we can infer that rest
influences defensive performance more substantially than offensive output.

Upon analyzing Figure 17, it is evident that the effects of rest on most box
score statistics are minimal. However, notable exceptions can be observed in
the domains of steals, blocks, and turnovers, particularly among younger
athletes. This trend tends to be masked in the visualization due to the high
variability present in older players’ performance. For the older cohort, a
clear adverse effect on scoring is observed, which could be attributed to
older athletes possibly focusing more on offensive play during times of
fatigue, as previously mentioned, or it may stem from a smaller sample size
that renders the findings for this age group less definitive. Nevertheless,
the reliability of these observations is limited by the overall small sample
size, preventing a firm conclusion on the hypothesis.

In Figure 18, our findings are in concordance with those observed by the S and
T learners, indicating that the influence of rest is particularly pronounced
in the areas of field goal and true shooting percentages. This trend can be
attributed to the physically demanding nature of two-point plays, as discussed
in previous sections. The agreement among various learning models strengthens
the evidence that rest plays a crucial role in enhancing shooting efficiency,
especially in situations that require higher physical effort. This thorough
analysis highlights the complex impact of rest on different facets of player
performance, providing critical insights for the development of effective
player management strategies and rest schedules. To better illustrate the
impact of rest, we focused on plotting data for ages between 20 to 38, as the
effects were less discernible across all ages in the initial plot, allowing
for a clearer observation of these effects.

Figure 16: ACTE using X-learner and RF for net, offensive, and defensive
ratings

Figure 17: ACTE using X-learner and RF for box score statistics per 100
possessions

Figure 18: ACTE using X-learner and RF for field goal, three-point, and true
shooting percentage for age 20 to 40 (up) and 20 to 38 (bottom)

##  5 Discussion

This study has analyzed the impact of load management strategies on player
performance over time, with a particular focus on the NBA. By employing
statistical methodologies, including meta-learners and various machine
learning techniques, the research provides a nuanced understanding of how rest
days influence player output, especially when considering different age
groups. The findings suggest that strategic rest can enhance performance for
several metrics, offering crucial insights for team management, coaches, and
sports scientists aiming to optimize athlete longevity and effectiveness. This
aligns with and expands upon existing literature that advocates for the
careful management of player workload to prevent injury and maintain peak
performance levels throughout a season.

Crucially, our work advances the age-curve literature by establishing a novel
framework to estimate these curves using game-level data, as opposed to the
traditional season-level approach. This innovation allows for a more granular
analysis of how performance evolves with age and under different conditions,
significantly enhancing our ability to understand and predict player
trajectories. Additionally, by introducing a method to measure different
treatment effects conditional on age, our study opens new avenues for testing
multiple causal hypotheses. This represents a methodological leap forward,
providing researchers, team analysts, and sports scientists with a powerful
tool to explore a wide range of questions about athlete performance and
management.

Finally, the research acknowledges its limitations, including the
observational nature of the data and potential biases in performance
measurement. These constraints underline the need for further studies,
possibly incorporating controlled experiments or longitudinal studies across
different sports, to validate and refine the findings. Future research could
also explore the psychological impacts of load management, including athlete
motivation and team dynamics, to provide a more comprehensive understanding of
its effects. Such endeavors would not only enhance our grasp of optimal load
management strategies but also contribute to developing healthier, more
sustainable careers for professional athletes.

##  Appendix A Proofs

The Stable Unit Treatment Value Assumption (SUTVA) consists of the following
two sub-assumptions.

######  Assumption 2 (No interference).

The potential outcomes for any unit do not vary with the treatments assigned
to other units. Formally,

$\forall\mathbf{w,w^{\prime}}\in\textstyle\mathop{{}\mathbb{R}}^{N}\text{ such
that }w_{i}=w^{\prime}_{i}\Rightarrow
Y_{i}(\mathbf{w})=Y_{i}(\mathbf{w^{\prime}})$

under this assumption we have $Y_{i}(\mathbf{w})=Y_{i}(w_{i})$, namely we
index the potential outcome by the individual treatment $w_{i}$.

######  Assumption 3 (Consistency).

There is only one version of each treatment $w$ and thus,

$Y_{i}^{obs}=Y_{i}(\mathbf{w})$

where if we combine with assumtion 2 we get $Y_{i}^{obs}=Y_{i}(w_{i})$.

###  A.1 Proof of Theorem 1

Under Assumptions 1, 2 , and 3 we have

$\displaystyle\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[Y_{i}^{obs}|W_{i}=1,A_{i}=a,X_{i}=x]]-\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[Y_{i}^{obs}|W_{i}=0,A_{i}=a,X_{i}=x]]$
$\displaystyle=$
$\displaystyle\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[Y_{i}(1)|W_{i}=1,A=a,X=x]-\mathop{{}\mathbb{E}}[Y_{i}(0)|W_{i}=0,A_{i}=a,X_{i}=x]]$
$\displaystyle=$
$\displaystyle\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\mathop{{}\mathbb{E}}[Y_{i}(1)|A_{i}=a,X_{i}=x]-\mathop{{}\mathbb{E}}[Y_{i}(0)|A_{i}=a,X_{i}=x]]$
$\displaystyle=$
$\displaystyle\textstyle\mathop{{}\mathbb{E}}[Y_{i}(1)|A_{i}=a]-\mathop{{}\mathbb{E}}[Y_{i}(0)|A_{i}=a]$
$\displaystyle=$ $\displaystyle g(a,1)-g(a,0)=\tau(a)$

where the second line follows from Assumption 2 and 3, and the third line
follows from assumption 1,

##  Appendix B Algorithms

In the subsequent text, we detail the algorithmic frameworks utilized in this
study, in the form of pseudocode. The symbols $Y_{0}$ and $Y_{1}$ are employed
to denote the observed outcomes for the control and treatment groups,
respectively. For instance, $Y_{1i}$ represents the observed outcome for the
$i$th subject within the treatment group. Similarly, $A_{0},X_{0}$ and
$A_{1},X_{1}$ represent the attributes of the control and treated subjects,
respectively, with $A^{1}_{i},X^{1}_{i}$ being the age and attribute vector
for the $i$th treated subject. We utilize the expression $M_{k}(Y\sim X)$ to
denote a regression estimator that models regressing outcome $Y$ on $X$. This
estimator may be any regression or machine learning technique. Following the
format given by Künzel et al. (2019).

procedure S-learner($A,X,Y,W$)

$\hat{\mu}_{w}=M_{0}(Y^{obs}\sim(A,X,W))$

$\hat{\tau}(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\hat{\mu}_{1}(a,x)-\hat{\mu}_{0}(a,x)]$

Algorithm 1 S-learner

procedure T-learner($A,X,Y,W$)

$\hat{\mu}_{0}=M_{0}(Y^{0}\sim(A^{0},X^{0}))$

$\hat{\mu}_{1}=M_{1}(Y^{1}\sim(A^{1},X^{1}))$

$\hat{\tau}(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[\hat{\mu}_{1}(a,x)-\hat{\mu}_{0}(a,x)]$

Algorithm 2 T-learner

1:procedure X-learner($A,X,Y,W,g$)

2: $\hat{\mu}_{0}=M_{1}(Y^{0}\sim(A^{0},X^{0}))$

3: $\hat{\mu}_{1}=M_{2}(Y^{1}\sim(A^{1},X^{1}))$

4: $D_{i}^{1}=Y_{i}^{1}-\hat{\mu}_{0}(A_{i}^{1},X_{i}^{1})$

5: $D_{i}^{0}=\hat{\mu}_{1}(A_{i}^{0},X_{i}^{0})-Y_{i}^{0}$

6: $\hat{\tau}_{1}=M_{3}(\bar{D}^{1}\sim(A^{1},X^{1}))$

7: $\hat{\tau}_{0}=M_{4}(\bar{D}^{0}\sim(A^{0},X^{0}))$

8:
$\hat{\tau}(a)=\textstyle\mathop{{}\mathbb{E}}_{\mathcal{X}}[g(a)\hat{\tau}_{0}(a,x)+(1-g(a))\hat{\tau}_{1}(a,x)]$

The function $g(x)$, whose values range from 0 to 1, serves as a weighting
mechanism selected to reduce the variance in the estimator $\hat{\tau}(x)$.

Algorithm 3 X-learner

Input: $a$:age column of the training data , $x$: features of the training
data, $w$: treatment assignments of the training data, $y$: observed outcomes
of the training data, $p$: point of interest, $N$: number of observation

Output: Bootstrap Confidence Interval

1 S={1,…,N}

2 for  _$b\in\\{1,\dots,B\\}$_ do

3
$s_{b}^{*}\leftarrow\text{sample}(S,\text{replace}=\text{True},\text{size}=N)$

4 $x_{b}^{*}\leftarrow x[s_{b}^{*}]$

5 $w_{b}^{*}\leftarrow w[s_{b}^{*}]$

6 $y_{b}^{*}\leftarrow y[s_{b}^{*}]$

7
$\hat{\tau}_{b}^{*}(p)\leftarrow\text{learner}(x_{b}^{*},w_{b}^{*},y_{b}^{*})(p)$

8

9 end for

10return _(
$Q_{\alpha/2}(\\{\hat{\tau}_{b}^{*}(p)\\}_{b=1}^{B}),Q_{1-\alpha/2}(\\{\hat{\tau}_{b}^{*}(p)\\}_{b=1}^{B})$)_

Algorithm 4 Procedure to compute bootstrap confidence intervals

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