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\hypersetup{
  pdftitle={Supplementary material for the manuscript ``Development of vertical jump performance in young male football players: a longitudinal meta-analysis''},
  pdfauthor={Ahlan B. Lima, Federal University of Santa Catarina; Ricardo T. Quinaud, University of the Extreme South of Santa Catarina; Fábio C Karasiak, Federal University of Santa Catarina; Luciano G. Galvão, Federal University of Santa Catarina; Carlo Castagna, University of Southern Denmark, Carlo Bo Urbino University; Carlos E. Gonçalves, University of Coimbra; Humberto M. Carvalho, Federal University of Santa Catarina},
  pdfkeywords={Youth sports, Adolescent development, Soccer, Athletic
Performance, Bayesian approach, Multilevel analysis},
  hidelinks,
  pdfcreator={LaTeX via pandoc}}

\title{\textbf{Supplementary material for the manuscript ``Development
of vertical jump performance in young male football players: a
longitudinal meta-analysis''}}
\author{Ahlan B. Lima, Federal University of Santa Catarina \and Ricardo
T. Quinaud, University of the Extreme South of Santa Catarina \and Fábio
C Karasiak, Federal University of Santa Catarina \and Luciano G. Galvão,
Federal University of Santa Catarina \and Carlo Castagna, University of
Southern Denmark, Carlo Bo Urbino University \and Carlos E. Gonçalves,
University of Coimbra \and Humberto M. Carvalho, Federal University of
Santa Catarina\footnote{Corresponding author:Campus Trindade, 88040-900,
  Florianópolis, Santa Catarina, Brasil}}
\date{}

\begin{document}
\maketitle

\newpage

This supplementary material file details the model specification in our
analysis and provides supplementary results about posterior predictive
checks. The data and model codes are in the Open Science Foundation
repository at \url{https://osf.io/yb57c/}.

\section{Model specification}\label{model-specification}

We fitted a longitudinal meta-analysis model to describe the development
of vertical jump in young male football, using a fully Bayesian approach
{[}1{]}. We initially considered a basic Bayesian meta-analysis model:

\[ y_j  \sim  Normal(\theta_j,\sigma_j= se_j)\]

where \(y_j\) is the point estimate for the stature of a single study
\(j\), drawn from a normal distribution that is centered on that study's
``true'' estimate, \(\theta_j\), and with a standard deviation
\(\sigma_j\) given by the study's standard error \(se_j\). These are
taken directly from each study \(j\). We assumed that each study \(j\)
is drawn from a population of related studies, each with its point
estimate for stature and respective standard error. Hence, it is assumed
that \(\theta_j\) has a normal distribution, where \(\mu_j\) is the
population mean (meta-analytic) estimate, and \(\tau\) is the variation
around the population mean, i.e., between-study variation:

\[ \theta_j  \sim Normal(\mu_j,\tau)\]

The longitudinal meta-analysis model describing the development of
vertical jump of young football players was specified by considering
each study's outcome \(y_j\), i.e., veritical jump performance, at time
point \(t\) (i.e., chronological age) in study \(j\), with an error
variance of the within-study sampling errors, assumed to be normally
distributed. We explored the possibility of a nonlinear development by
considering a polynomial structure in the multilevel model to model each
study's maximum vertical jump velocity developmental growth, by
considering at least growth coefficients up to the cubic to capture the
known shape of pubertal growth. Hence, \(\mu_j\) is given by the
intercept \(\alpha_{j[t]}^{age}\), the linear, \(\beta_{j[t]}^{age}\),
and nonlinear terms,\(\beta_{j[t]}^{age^2}\) and \(\beta_{j}^{age^3}\).
The notation for the \(\mu_j\), \(\beta_{j[t]}^{age}\), and
\(\beta_{j[t]}^{age^2}\) indicates that each study \(j\) is given a
unique intercept, linear and the second-order slopes are issued from a
normal distribution centered, respectively on \(\mu\), \(\beta^{age}\),
and \(\beta^{age^2}\), the grand intercept and slopes, meaning that
there might be different mean scores for each study. The correlation
between the varying intercepts and the varying slopes was estimated by
modeling \(\alpha_{j[t]}^{age}\), \(\beta_{j[t]}^{age}\), and
\(\beta_{j[t]}^{age^2}\) as issued from the same multivariate normal
distribution (a multivariate normal distribution is a generalization of
the usual normal distribution to more than one dimension), centered on 0
and with some covariance matrix S, as specified in the following form:

\[\mu_j = \alpha_{j[t]}^{age} + \beta_{j[t]}^{age}+ \beta_{j[t]}^{age^2} + \beta_{j}^{age^3} \]

\[\begin{bmatrix}
\alpha_{j[t]}^{age} \\
\beta_{j[t]}^{age} \\
\beta_{j[t]}^{age^2} \\
\end{bmatrix} 
\sim MVNormal
\begin{pmatrix}
\begin{bmatrix}
0\\
0\\
0
\end{bmatrix}, S 
\end{pmatrix} \]

\[ S=\begin{bmatrix}
\sigma_{\alpha}^2 & 0 & 0 \\
0 & \sigma_{\beta^{age}}^2 & 0 \\
0 & 0 & \sigma_{\beta^{age^2}}^2 & 
\end{bmatrix}  \]

For computational efficiency, we set the correlation between intercepts
and slopes used for the computation of the covariance matrix to 0. Also,
we centered the outcomes on the aggregated sample means (sample mean for
CMJ = 31.12 cm; sample mean for age = 12.71 year).

We also fitted our model by aligning each study's outcome \(y_j\) by a
age peak height velocity (PHV). The age at PHV estimated was derived
from a longitudinal meta-analysis of young male athletes data {[}1{]}.
For football, the estimated age at PHV was 12.95 years.

We specified weakly regularizing priors in our models. For the
population-level parameters we used a normal prior:
\(\mu \sim Normal(0,10),(\beta, \beta^{age^2}, \beta^{age^3}) \sim Normal(0,5)\);
and for group-level effects we used
\((\sigma_\alpha, \sigma_{\beta^{age}}, \sigma_{\beta^{age^2}}) \sim \text{exponential}(1)\).

\section{Supplementary Figures}\label{supplementary-figures}

We used the adjusted signaling questions for each domain considering
specifically our meta-analysis research question and design {[}1{]}.
Specifically, we considered the following signaling questions: ``Q1. Was
the design adequate to describe jump height performance for the
meta-analysis research question?''; ``Q2. Was the sample described in
detail?''; ``Q3. Was the method described in detail?''; ``Q4. Were any
drop-outs reported?''; ``Q5. Were the outcome measures reliable?'';
``Q6. Were results reported adequate for extraction?''; ``Q7. In case of
multiple reports, was the design, sample, and results consistent?'';
``Q8. Were the analysis methods appropriate?''. The traffic light plot
of the degree of judgment in each signaling question across the studies
and the distribution of the degree of judgment by signaling questions in
the studies considered in the meta-analysis are presented in Figure S1
and Figure S2, respectively.

\begin{figure}
\centering
\includegraphics{figures/Traffic lights plot jump footbal meta-analysis.png}
\caption{Traffic light plots of the degree of judgment in each signaling
question across the studies included in the meta-analysis.}
\end{figure}

\begin{figure}
\centering
\includegraphics{figures/Summary plot jump football meta-analysis.png}
\caption{Distribution of degree of judgment by signaling questions in
the studies included in the meta-analysis.}
\end{figure}

The distributions of the study's standard error (Figure S3) and inverse
sample sizes (both on reverse scales) against the Bayesian meta-analysis
model residuals (Figure S4) showed a trend of asymmetry the funnel
plots, mainly present in the studies wuth smaller samples.

\begin{figure}
\centering
\includegraphics{figures/supplementary figure 3.jpeg}
\caption{Distribution of study standard error (on a reverse scale)
against the Bayesian meta- analysis model residuals.}
\end{figure}

\begin{figure}
\centering
\includegraphics{figures/supplementary figure 4.jpeg}
\caption{Distribution of study inverse sample sizes (on a reverse scale)
against the Bayesian meta-analysis model residuals.}
\end{figure}

Based on the posterior predictive checks, our models generated
predictions compatible with the observed data, i.e., they could produce
data that resembled the observed data. The posterior prediction plots
are displayed in Figure S5 and Figure S6 for the nonlinear models of
vertical jump performance, aligned by chronological age and age at PHV,
respectively. Our models were able to generate predictions that matched
the observed data.

\begin{figure}
\centering
\includegraphics{figures/supplementary figure 1.jpeg}
\caption{Distributions of replicated data sets drawn from the posterior
predictive distribution in our model alligned by chronological age
compared with the empirical distribution of the stature values extracted
from the literature.}
\end{figure}

\begin{figure}
\centering
\includegraphics{figures/supplementary figure 2.jpeg}
\caption{Distributions of replicated data sets drawn from the posterior
predictive distribution in our model alligned by age at peak height
velocity compared with the empirical distribution of the stature values
extracted from the literature.}
\end{figure}

\section{Full Search Strategy}\label{full-search-strategy}

Our search strategy used the search terms outlined in Table 1 across
multiple databases, including MEDLINE (PubMed), Web of Science, and
SCOPUS. The search terms were systematically organized into blocks
corresponding to the PIOS framework (adapted from the PICOS framework,
excluding the comparator item), covering population (two blocks),
intervention/sports, outcomes, and study type. We applied the search
terms to each database, with the strategy adapted to the specific
characteristics of each database. We then conducted a combined search to
ensure a comprehensive exploration of the literature. The search was
performed on March 15, 2024. We intended to identify longitudinal
studies encompassing a range of performance data (sprints, jumps,
anaerobic endurance, etc.) in various sports. However, the present
analysis focused on articles related to youth football and vertical jump
performance.

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.2603}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.2603}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.4795}}@{}}
\caption{Terms used during electronic databases search}\tabularnewline
\toprule\noalign{}
\endfirsthead
\endhead
\bottomrule\noalign{}
\endlastfoot
\textbf{Block 1} & Population AND & young OR youth OR adolescent OR teen
OR youths OR adolescents OR teens OR child OR children OR teenagers OR
teenager \\
\textbf{Block 2} & Population AND & players OR player OR athletes OR
athlete \\
\textbf{Block 3} & Intervention/ Sports AND & sports OR sport OR soccer
OR football OR basketball OR volleyball OR handball OR hockey OR cricket
OR baseball OR rugby OR lacrosse OR softball OR netball OR bicycling OR
boxing OR golf OR gymnastics OR ``martial arts'' OR tennis OR running OR
skating OR skiing OR ``track and field'' OR swimming OR mountaineering
OR ``racquet sports'' OR ``water sports'' OR walking OR ``weight
lifting'' OR wrestling OR athletics \\
\textbf{Block 4} & Outcomes AND & sprint OR jump OR ``aerobic
endurance'' OR ``physical endurance'' OR strength OR power OR agility OR
``medicine ball throw'' OR ``physical performance'' OR ``functional
performance'' OR ``physical fitness'' OR ``functional capacities'' OR
``functional capacity'' OR ``physiological responses'' OR
``physiological response'' OR ``physiological adaptations'' OR
``physiological adaptation'' OR height OR stature OR ``body mass'' OR
``body weight'' OR vo2 OR ``peak height velocity'' OR PHV OR maturation
OR ``short-term power output'' \\
\textbf{Block 5} & Study type & longitudinal OR ``repeated measures'' OR
``repeated measure'' \\
\end{longtable}

\subsection{MEDLINE (via PubMed,
1946-present)}\label{medline-via-pubmed-1946-present}

\textbf{Date of last search}: 2024-03-15.

\textbf{Field codes}: None.

\textbf{Limits and restrictions:} Language = ``English''.

\textbf{Search filters:} None.

\textbf{Total number of hits}: 3,216 hits.

\textbf{Full search strategy:} ((((young OR youth OR adolescent OR teen
OR youths OR adolescents OR teens OR child OR children OR teenagers OR
teenager) AND (players OR player OR athletes OR athlete)) AND (sports OR
sport OR soccer OR football OR basketball OR volleyball OR handball OR
hockey OR cricket OR baseball OR rugby OR lacrosse OR softball OR
netball OR bicycling OR boxing OR golf OR gymnastics OR ``martial arts''
OR tennis OR running OR skating OR skiing OR ``track and field'' OR
swimming OR mountaineering OR ``racquet sports'' OR ``water sports'' OR
walking OR ``weight lifting'' OR wrestling OR athletics)) AND (sprint OR
jump OR ``aerobic endurance'' OR ``physical endurance'' OR strength OR
power OR agility OR ``medicine ball throw'' OR ``physical performance''
OR ``functional performance'' OR ``physical fitness'' OR ``functional
capacities'' OR ``functional capacity'' OR vo2 OR vo2max OR CMJ OR
``countermovement jump'' OR yo-yo OR intermittent test OR ``shuttle
run'')) AND (longitudinal OR ``repeated measures'' OR ``repeated
measure'')

\subsection{WEB OF SCIENCE (Core Collection
1945-present)}\label{web-of-science-core-collection-1945-present}

\textbf{Date of last search}: 2024-03-15.

\textbf{Field codes}: Topic (``ALL'').

\textbf{Limits and restrictions:} Language = ``English''; Document Type
= ``Article'', ``Book Chapter''.

\textbf{Search filters:} None.

\textbf{Total number of hits}: 562 hits.

\textbf{Full search strategy:} ((((ALL=(young OR youth OR adolescent OR
teen OR youths OR adolescents OR teens OR child OR children OR teenagers
OR teenager)) AND ALL=(players OR player OR athletes OR athlete)) AND
ALL=(sports OR sport OR soccer OR football OR basketball OR volleyball
OR handball OR hockey OR cricket OR baseball OR rugby OR lacrosse OR
softball OR netball OR bicycling OR boxing OR golf OR gymnastics OR
``martial arts'' OR tennis OR running OR skating OR skiing OR ``track
and field'' OR swimming OR mountaineering OR ``racquet sports'' OR
``water sports'' OR walking OR ``weight lifting'' OR wrestling OR
athletics)) AND ALL=(sprint OR jump OR ``aerobic endurance'' OR
``physical endurance'' OR strength OR power OR agility OR ``medicine
ball throw'' OR ``physical performance'' OR ``functional performance''
OR ``physical fitness'' OR ``functional capacities'' OR ``functional
capacity'' OR vo2 OR vo2max OR CMJ OR ``countermovement jump'' OR yo-yo
OR intermittent test OR ``shuttle run'')) AND ALL=(longitudinal OR
``repeated measures'' OR ``repeated measure'')

\subsection{SCOPUS}\label{scopus}

\textbf{Date of last search:} 2024-03-15.

\textbf{Field codes:} Article title, abstract, keywords(TITLE-ABS-KEY)~

\textbf{Limits and restrictions:} Language = ``English''; Document Type
= ``Article''; Source Type = ``Journal Article''.

\textbf{Search filters:} None.

\textbf{Total number of hits:} 576 hits.

\textbf{Full search strategy:} (TITLE-ABS-KEY (young OR youth OR
adolescent OR teen OR youths OR adolescents OR teens OR child OR
children OR teenagers OR teenager) AND TITLE-ABS-KEY (players OR player
OR athletes OR athlete) AND TITLE-ABS-KEY (sports OR sport OR soccer OR
football OR basketball OR volleyball OR handball OR hockey OR cricket OR
baseball OR rugby OR lacrosse OR softball OR netball OR bicycling OR
boxing OR golf OR gymnastics OR ``martial arts'' OR tennis OR running OR
skating OR skiing OR ``track and field'' OR swimming OR mountaineering
OR ``racquet sports'' OR ``water sports'' OR walking OR ``weight
lifting'' OR wrestling OR athletics) AND TITLE-ABS-KEY (sprint OR jump
OR ``aerobic endurance'' OR ``physical endurance'' OR strength OR power
OR agility OR ``medicine ball throw'' OR ``physical performance'' OR
``functional performance'' OR ``physical fitness'' OR ``functional
capacities'' OR ``functional capacity'' OR vo2 OR vo2max OR cmj OR
``countermovement jump'' OR yo-yo OR intermittent AND test OR ``shuttle
run'') AND TITLE-ABS-KEY (longitudinal OR ``repeated measures'' OR
``repeated measure'')) AND (LIMIT-TO ( DOCTYPE , ``ar'')) AND ( LIMIT-TO
(LANGUAGE , ``English''))

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

\phantomsection\label{refs}
\begin{CSLReferences}{0}{1}
\bibitem[\citeproctext]{ref-Lima2023}
1. Lima AB, Quinaud RT, Gonçalves CE, Carvalho HM. Peak height velocity
in young athletes: A longitudinal meta-analysis. Journal of Sports
Sciences {[}Internet{]}. 2023;41:151--63. Available from:
\url{https://doi.org/10.1080/02640414.2023.2203484}

\end{CSLReferences}

\end{document}
