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Bias in estimated short sprints profiles using timing gates due to the flying start: Simulation study and proposed solutions

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Abstract

Short sprints have been modeled using the mono-exponential equation that involves two parameters: (1) maximum sprinting speed (MSS) and (2) relative acceleration (TAU), most often performed using the timing gates. I have named this the No correction model. Unfortunately, due to the often utilized flying start, a bias is introduced when estimating parameters. In this paper, I have (1) proposed two additional models (Estimated TC and Estimated FD) that aim to correct this bias, and (2) provided a theoretical simulation study that provides model performances in estimating parameters. In conclusion, both Estimated TC and Estimated FD models provided more precise parameter estimates, but surprisingly, the No correction model provided better estimates of some parameter changes.
Bias in estimated short sprint profiles using
timing gates due to the flying start:
simulation study and proposed solutions
Mladen Jovanovi´
c1
1Faculty of Sport and Physical Education, University of Belgrade, Serbia
Corresponding author:
Mladen Jovanovi´
c1
Email address: coach.mladen.jovanovic@gmail.com
ABSTRACT
Short sprints have been modeled using the mono-exponential equation that involves two parameters:
(1) maximum sprinting speed (MSS) and (2) relative acceleration (TAU), most often performed using the
timing gates. I have named this the No correction model. Unfortunately, due to the often utilized flying start,
a bias is introduced when estimating parameters. In this paper, I have (1) proposed two additional models
(Estimated TC and Estimated FD ) that aim to correct this bias, and (2) provided a theoretical simulation
study that provides model performances in estimating parameters. In conclusion, both Estimated TC and
Estimated FD models provided more precise parameter estimates, but surprisingly, the No correction
model provided better estimates of some parameter changes.
1 INTRODUCTION
Sprint speed is one of the most distinctive and admired physical characteristics in sports. In the majority
of team sports (e.g., soccer, field hockey, handball, etc.), short sprints are defined as maximal sprinting
from a standstill across a distance that does not result in deceleration at the finish. Peak anaerobic power
is reached during the first few seconds (
<
5
s
) of maximal efforts (Mangine et al., 2014), however the
capacity to attain maximal sprint speed is athlete- and sport-specific. For instance, track and field sprinters
are trained to achieve maximal speed later in a race (i.e., 50-60
m
) (Ward-Smith, 2001), whereas team
sport athletes have sport-specific attributes and reach maximal speed much earlier (i.e., 30-40
m
) (Brown,
Vescovi & Vanheest, 2004). The evaluation of short sprint performance is frequently included in a battery
of fitness tests for a wide variety of sports, regardless of the kinematic differences between athletes.
The use of force plates is regarded as the gold standard for analyzing the mechanical features of
sprinting; nevertheless collecting the profile of a whole sprint presents practical and cost problems
(Samozino et al., 2016; Morin et al., 2019). Radar and laser technology are frequently utilized laboratory-
grade methods (Buchheit et al., 2014; Jim
´
enez-Reyes et al., 2018; Marcote-Peque
˜
no et al., 2019; Edwards
et al., 2020) that are typically unavailable to sports practitioners. Timing gates are unquestionably the
most prevalent method available for evaluating sprint performance. Multiple gates are frequently placed
at different distances to capture split times (e.g., 10, 20, 30, and 40
m
), which can now be incorporated
into the method for determining sprint mechanical properties (Samozino et al., 2016; Morin et al., 2019).
Practitioners can utilize the outcomes to explain individual differences, quantify the effects of training
interventions, and gain a better knowledge of the limiting variables of performance, which is an advantage
of this method.
1.1 Mathematical model
The mono-exponential equation
(1)
has been used to model short sprints. It was first proposed by
Furusawa, Hill & Parkinson (1927) and made more popular by Clark et al. (2017) and Samozino et al.
(2016). Equation
(1)
is the function for instantaneous horizontal velocity
v
given time
t
and two model
parameters.
PREPRINT - NOT PEER REVIEWED
v(t) = MSS ×(1et
TAU )(1)
Maximum sprinting speed (MSS; expressed in
ms 1
) and relative acceleration (TAU; expressed
in
s
) are the parameters of the equation
(1)
. TAU represents the ratio of MSS to initial acceleration
(MAC; maximal acceleration, expressed in
ms 2
)
(2)
. Note that TAU, given the equation
(1)
, is the time
required to reach a velocity equal to 63.2% of MSS.
MAC =MSS
TAU (2)
Although TAU is utilized in the equations and afterward estimated, I prefer to use and report MAC
because it is simpler to understand, especially for practitioners and coaches.
By deriving the equation (1), we may obtain the equation (3) for horizontal acceleration.
a(t) = MSS
TAU ×et
TAU (3)
By integrating equation (1), we get the equation for distance covered (4).
d(t) = MSS ×(t+TAU ×et
TAU )MSS ×TAU (4)
1.2 Model parameters estimation using timing gates split times
Table 1 contains sample split times measured during 40 m sprint performance using timing gates positioned
at 5, 10, 20, 30, and 40 m.
To estimate model parameters using split times, distance is a predictor and time is the outcome
variable; hence, equation (4) takes the form of the equation (5).
t(d) = TAU ×W(ed
MSS×T AU 1)+ d
MSS +T AU (5)
W
in equation
(5)
represents Lambert’s W function (Goerg, 2022). Equation
(4)
, in which the time is
the predictor and distance is the outcome variable, is commonly employed in research (Morin, 2017; Morin
& Samozino, 2019; Stenroth & Vartiainen, 2020), This method should be avoided since reversing the
predictor and outcome variables in a regression model may create biased estimated parameters (Motulsky,
2018, p. 341). This bias may not be practically significant for profiling short sprints, but it is a statistically
flawed practice and should be avoided. It is thus preferable to utilize statistically correct equation
(5)
to
estimate model MSS and TAU.
Estimating MSS and TAU parameters using equation
(5)
as model definition is performed using
non-linear least squares regression. To the best of my knowledge, scientist, researchers, and coaches have
been performing short sprints modeling using the built-in solver function of Microsoft Excel (Microsoft
Corporation, Redmond, Washington, United States) (Samozino et al., 2016; Clark et al., 2017; Morin,
2017; Morin & Samozino, 2019; Morin et al., 2019; Stenroth, Vartiainen & Karjalainen, 2020; Stenroth
Table 1. Sample split times measured during 40 m sprint performance using timing gates positioned at 5,
10, 20, 30, and 40 m.
Distance (m) Split time (s)
5 1.34
10 2.06
20 3.29
30 4.44
40 5.56
2/14
& Vartiainen, 2020). These, and additional functionalities, have been recently implemented in the open-
source
{
shorts
}
package (Jovanovi
´
c & Vescovi, 2020; Vescovi & Jovanovi
´
c, 2021; Jovanovi
´
c, 2022)
for R-language (R Core Team, 2022), which utilizes the
nlsLM()
function from the
{
minpack.lm
}
package (Elzhov et al., 2022). Compared to the built-in solver function of Microsoft Excel,
{
shorts
}
package represents a more powerful, flexible, transparent, and reproducible environment for modeling
short sprints, and it is used in this study to estimate model parameters.
Using the split times from Table 1, estimated MSS, TAU, and MAC parameters are equal to 9.02
ms1
, 1.14
s
, and 7.94
ms2
, respectively. Maximal relative power (PMAX; expressed in
W/kg
) is an
additional parameter often estimated and reported (Samozino et al., 2016; Morin et al., 2019). PMAX is
calculated using equation
(6)
. This method of PMAX estimation disregards the air resistance and thus
represents net or relative propulsive power. Calculated PMAX using estimated MSS and MAC parameters
is equal to 17.91 W/kg.
PMAX =M SS ×MAC
4(6)
1.3
Problems with parameters estimation using split times due to flying start and reaction
time
To ensure accurate short sprint parameter estimates, the initial force production must be synced with start
time, often reffed to as “first movement” triggering [Haugen, Tønnessen & Seiler (2012); Haugen,
Breitsch
¨
adel & Seiler (2020); Haugen & Buchheit (2016); haugenSprintMechanicalVariables2019;
Haugen, Breitsch
¨
adel & Samozino (2020); Samozino et al. (2016)]. This represents a challenge when
collecting sprint data using timing gates and can substantially impact estimated parameters.
To demonstrate this impact, let us imagine we have three twin brothers with the same short sprint
characteristics: MSS equal to 9
ms1
, TAU equal to 1.125
s
, MAC equal to 8
ms2
, and PMAX equal to
18
W/kg
(these represent true short sprint parameters). Let us call them Mike, Phil, and John. They all
perform a 40
m
sprint from a standing start using timing gates set at 5, 10, 20, 30, and 40
m
. For Mike
and Phil, the timing system is activated by the initial timing gate (i.e., when they cross the beam) at the
start of the sprint (i.e., d=0m). For John, the timing system is activated after the gunfire.
Mike represents the theoretical model, in which we assume that the initial force production and the
timing initiation are perfectly synchronized. We have already utilized Mike’s split times in Table 1.
On the other hand, Phil decides to move slightly behind the initial timing gate (i.e., for 0.5
m
) and use
body rocking to initiate the sprint start. In other words, Phil uses a flying start, a common scenario when
testing field sports athletes. From a measurement perspective, flying start distance is often recommended
to avoid premature triggering of the timing system by lifted knees or swinging arms (Altmann et al., 2015,
2017, 2018; Haugen & Buchheit, 2016; Haugen, Breitsch
¨
adel & Samozino, 2020). Flying start can also
result from body rocking during the standing start. Clearly, any flying start with a difference between the
initial force production and the start time can lead to skewed parameters and predictions. Since it is hard
to get faster at a sprint, inconsistent starts can hide the effects of the training intervention.
Since the gunfire triggers John’s start, his split times have an additional reaction time of 0.2
s
. This
is similar to a scenario where the athlete prematurely triggers a timing system when standing too close
to the initial timing gate. We can thus use John’s data to demonstrate the effects of this scenario on the
estimated parameters as well.
Timing gates utilized in this theoretical example provide precision to two decimals (i.e., closest 10
ms)
, representing a measurement error source. A graphical representation of the sprint splits can be found
in Figure 1.
Estimated sprint parameters can be found in Table 2. As seen from the results (Table 2), estimated
short sprint parameters for all three brothers differ from the true parameters used to generate the data (i.e.,
their true short sprint characteristics). All three brothers have a bias in estimated parameters due to timing
gates’ precision to 2 decimals (i.e., 10
ms
). Bias in estimated parameters in Phil’s case is due to the flying
start involved, while in John’s case, it is due to the reaction time involved in the split times.
1.4 How to overcome missing the initial force production when using timing gates?
The literature suggests using a correction factor of +0.5
s
as a viable solution (i.e., simply adding +0.5
s
to
split times) to convert to “first movement” triggering when utilizing recommended 0.5
m
flying distance
3/14
John (gunfire)
Mike (theoretical)
Phil (flying start)
1
2
3
4
5
5 10 20 30 40
Distance (m)
Time (s)
Figure 1. Phil, Mike, and John split times over a 40
m
distance. All three brothers have identical sprint
performances but utilize different sprint starts, resulting in different split times.
Table 2. Estimated sprint parameters for Mike, Phil, and John using data from Figure 1. All three
brothers have identical sprint performance but utilize different sprint starts, which results in different split
times, and thus different sprint parameter estimates. Due to the timing gates’ precision to 2 decimals (i.e.,
10 ms), estimated Mike’s parameters also differ from the true values.
Athlete MSS TAU MAC PMAX
True 9.00 1.12 8.00 18.0
Mike (theoretical) 9.02 1.14 7.94 17.9
Phil (flying start) 8.60 0.61 14.00 30.1
John (gunfire) 9.59 1.62 5.93 14.2
4/14
behind the initial timing gate (Haugen, Tønnessen & Seiler, 2012; Haugen & Buchheit, 2016; Haugen,
Breitsch
¨
adel & Seiler, 2019, 2020). Intriguingly, the average difference between the standing start with
a photocell trigger and a block start to gunfire for a 40-meter sprint was 0.27
s
(Haugen, Tønnessen &
Seiler, 2012). Consequently, although a timing correction factor is required to prevent further inaccuracies
in estimates of kinetic variables (e.g., overestimate power), a correction factor that is too big would have
the opposite effect (e.g., underestimate power).
1.4.1 Estimated time correction model
Instead of using apriori time correction from the literature, this parameter may be estimated using the
supplied data, together with MSS and TAU. Stenroth, Vartiainen & Karjalainen (2020) proposes the same
approach, titled the time shift method, and the estimated parameter, named the time shift parameter. In
accordance with the current literature, we have termed this parameter time correction (TC) (Vescovi &
Jovanovi´
c, 2021).
Using the original equation
(5)
to implement the TC parameter now provides the new equation
(7)
.
Equation
(7)
is utilized as the model definition in the Estimated TC model, as opposed to the model using
equation
(5)
, which I have termed No correction model. The model in which TC is fixed (i.e., by simply
adding TC to split times) is termed the Fixed TC model.
t(d) = TAU ×W(ed
MSS×T AU 1)+ d
MSS +T AU TC (7)
From a regression perspective, the TC parameter can be viewed as an intercept. It can be beneficial
when we assume a fixed time shift is involved (i.e., reaction time or premature triggering of the timing
equipment). If we compare the split times of Mike and John in Figure 1, we can notice that the lines
are parallel. In this scenario, the Estimated TC model can remove bias between Mike and John. The
Estimated TC model can also help remove bias in estimated parameters in Phil’s case. However, if we
look closely at Figure 1, we will notice that Phil’s and Mike’s lines are not parallel. This is because there
is already some velocity when the initial timing gate is triggered; thus, the time shift is not constant.
These models (i.e., Fixed TC of +0.3, +0.5
s
, and Estimate TC model) are applied to Mike, Phil, and
John’s split times. The estimated model parameters can be found in Table 3 and previously estimated
parameter values using No correction model. As can be noted from Table 3, adding +0.3
s
worked well
for Phil in terms of approaching true parameter values, while adding +0.5
s
was detrimental in un-biasing
estimated parameters.
The Estimated TC model worked well for all three athletes in terms of un-biasing the parameter
estimates. The estimated TC parameter for John was also very close to the true reaction time of 0.2 s.
1.4.2 Estimated flying distance model
Although the Estimated TC model performed well in Phil’s case (brother doing flying start), instead of
assuming time shift (which helps in un-biasing the estimates compared to the No correction model), we
can utilize model definition that assumes flying start distance (FD) involved in the data-generating-process
(DGP). This Estimated FD model utilizes equation (8) as the model definition.
t(d) = (TAU ×W(ed+FD
MSS×T AU 1)+ d+F D
MSS +T AU)
(TAU ×W(eFD
MSS×T AU 1)+ F D
MSS +T AU)
(8)
Table 3 contains all model estimates for three brothers, including the Estimated FD model. We can
notice that the Estimated FD model unbiased estimates for Phil, but failed to be estimated for John (brother
that starts at gunfire and has reaction time involved in his split times). This is because the Estimated FD
model is ill-defined under that scenario and cannot have a negative flying distance.
Overall, each model definition has assumed the mechanism of the data generation. No correction
model assumes perfect synchronization of the sprint initiation with the start of the timing.The Estimated
TC model introduces a simple intercept that can help estimate parameters when an assumed time shift is
involved (e.g., when reaction time is involved or premature triggering of the initial timing gate). Estimated
TC can also be used when flying start is utilized, but it assumes the constant time shift, which is not the
5/14
case in that scenario due to already gained velocity at the start. The Estimated FD model assumes there is
a flying sprint involved in the DGP and, as shown in Table 3, can be ill-defined when there is no flying
distance involved, but there is a time shift. All three models assume athlete accelerates according to the
mono-exponential equation (1).
This work aims to explore the behavior of these three models under simulated and known conditions.
This is needed to provide a theoretical understanding of the limits and expected errors of the short sprints
modeling, which can later inform more practical studies involving athletes.
2 METHODS
2.1 Simulation design
In this simulation, data is generated using true MSS (ranging from 7 to 11
ms1
, in increments of 0.05
ms1
, resulting in a total of 81 unique values), MAC (ranging from 7 to 11
ms2
, in increments of 0.05
ms2
, resulting in a total of 81 unique values), and flying distance (ranging from 0 to 0.5
m
, in increments
of 0.01
m
, resulting in a total of 51 unique values). Each flying sprint distance consists of 6,561 MSS and
MAC combinations.
Splits times are estimated using timing gates positioned at 5, 10, 20, 30, and 40
m
, with the rounding
to the closest 10 ms. In total, there are 334,611 sprints simulated.
2.2 Statistical analysis
MSS, MAC, TAU, and PMAX are estimated for each simulated sprint using No correction,Estimated
TC, and Estimated FD models. The agreement between true and estimated parameter values is evaluated
using the percent difference (%Di f f ) estimator (equation (9)).
%Di f f =100 ×estimated t rue
true (9)
The distribution of the simulated
%Di f f
is summarized using
median
and 95% highest-density
continuous interval (
HDCI
) (Kruschke, 2015, 2018; Kruschke & Liddell, 2018a,b; Makowski, Ben-
Shachar & L¨
udecke, 2019).
To provide magnitude interpretation of the
%Di f f
,region of practical equivalence (
ROPE
), as well as
the proportion of the simulations that lie within
ROPE
(
inside ROPE
; expressed as percentage) (Kruschke,
2015, 2018; Kruschke & Liddell, 2018a,b; Makowski, Ben-Shachar & L
¨
udecke, 2019; Jovanovi
´
c, 2020),
are calculated. For the purpose of this paper,
ROPE
is assumed to be equal to 95%
HDCI
of the
%Di f f
using the No correction model and no flying distance. Theoretically,
ROPE
represents the lowest error
(i.e., the best agreement) that can be achieved. It is limited purely by the timing gates measurement
precision (i.e., rounding to the closest 10 ms) and simulated parameters.
In addition to estimating agreement between true and estimated parameter values, practitioners are
often interested in whether they can use estimated measures to track changes in the true measures. A
minimal detectable change estimator with 95% confidence (
%MDC95
) (Furlan & Sterr, 2018; Jovanovi
´
c,
2020) is utilized to estimate this precision. The
%MDC95
value might be regarded as the minimum amount
of change that needs to be observed in the estimated parameter for it to be considered a true change.
In this study,
%MDC95
is calculated using percent residual standard error (
%RSE
; equation
(10)
) of
the linear regression between true (predictor) and estimated parameter values (outcome) (equation
(11)
).
Since simulated data with the known true values are utilized,
%RSE
represents the percent standard error
of the measurement (%SEM) in the estimated parameters.
%RSE =sN
i=1(100 ×yiˆyi
ˆyi)2
N2(10)
%MDC95 =%RSE ×2×1.96 (11)
In addition to providing
%MDC95
for the estimated parameters, the lowest
%MDC95
is estimated using
the No correction model and no flying distance (
%MDClowest
95
). Theoretically,
%MDClowest
95
represents the
6/14
Table 3. Estimated sprint parameters for Mike, Phil, and John using data from Figure 1 using No
correction,Fixed time corrections (TC), Estimated TC, and Estimated FD models.
Model Athlete MSS TAU MAC PMAX TC FD
True True 9.00 1.12 8.00 18.0
Mike (theoretical) 9.02 1.14 7.94 17.9
Phil (flying start) 8.60 0.61 14.00 30.1No correction
John (gunfire) 9.59 1.62 5.93 14.2
Mike (theoretical) 10.01 1.93 5.19 13.0
Phil (flying start) 9.05 1.13 8.02 18.2Fixed +0.3s TC
John (gunfire) 11.29 2.79 4.05 11.4
Mike (theoretical) 11.29 2.79 4.05 11.4
Phil (flying start) 9.62 1.61 5.98 14.4Fixed +0.5s TC
John (gunfire) 13.67 4.26 3.21 11.0
Mike (theoretical) 9.04 1.15 7.86 17.8 0.01
Phil (flying start) 9.00 1.08 8.35 18.8 0.28Estimated TC
John (gunfire) 9.04 1.15 7.86 17.8 -0.19
Mike (theoretical) 9.04 1.15 7.86 17.8 0.00
Phil (flying start) 9.03 1.16 7.82 17.7 0.54Estimated FD
John (gunfire)
lowest
%MDC95
that can be achieved, and it is limited purely by the timing gates measurement precision
(i.e., rounding to the closest 10
ms
) and simulated parameters.
%MDClowest
95
is used only as a reference to
evaluate estimated parameters’ %MDC95.
The analyses, as mentioned earlier, are performed on both pooled dataset (i.e., using all flying distance)
and across every flying distance. It is hypothesized that the Estimated FD model will have the highest
inside ROPE estimates and the lowest %MDC95 estimates.
Statistical analyses and graph construction were performed using the software R 4.2.1 (R Core Team,
2022) in RStudio (version 2022.02.3).
3 RESULTS
3.1 Model fitting
Table 4 contains failed model fitting for the Estimated FD model. These were disregarded from further
analysis.
The reason for these failed model fittings is probably in the combination of the very small flying
distance and the measurement precision of the timing gates, resulting in ill-defined model that cannot be
fitted.
Table 4. Failed model fittings for the Estimated FD model
Model Flying distance (m) Not fitted Total Not fitted (%)
Estimated FD 0.00 1765 6561 26.90
Estimated FD 0.01 12 6561 0.18
Estimated FD 0.02 16 6561 0.24
Estimated FD 0.03 10 6561 0.15
Estimated FD 0.04 4 6561 0.06
Estimated FD 0.05 1 6561 0.02
7/14
3.2 Percent difference
3.2.1 Region of practical equivalence
Estimated ROPEs are equal to -0.3 to 0.33% for MSS, -0.73 to 0.74% for MAC, -1.03 to 1% for TAU,
and -0.5 to 0.5% for PMAX (Table 5 and grey horizontal bars in Figures 2 and 3). An interesting finding
is that, given simulation parameters (particularly the precision of the timing gates to the closest 10
ms
),
MSS has the lowest
ROPE
compared to other short sprint parameters. Since
ROPE
represents the lowest
estimation error, MSS is the parameter that could be, given this theoretical simulation, estimated with the
most precision. In contrast, TAU and MAC can be estimated with the least precision.
3.2.2 Pooled analysis
The pooled analysis is performed using all flying distances pooled together. As such, the pooled analysis
represents the overall estimate of the agreement between true and estimated parameter values across
simulated conditions.
Figure 2 depicts the distribution of the pooled
%Di f f
. As expected, the Estimated FD model
performed with the highest
inside ROPE
parameter values (from 20 to 72%), with the most narrow 95%
HDCIs (from -5 to 5%), and no bias involved.
On the other hand, the No correction model performed poorly, with the lowest inside
ROPE
parameter
values (from 2 to 2%), with the widest 95%
HDCI
s (from -46 to 80%), and with the apparent bias
indicated with the
median
parameter values being outside of
ROPE
(from -35 to 49%). In addition,
visual inspection of Figure 2 indicates a non-normal distribution of estimated
%Di f f
parameter values,
demanding further analysis across flying distance values.
The Estimated TC model performed similarly to the Estimated FD model with slightly lower
inside ROPE
parameter values (from 9 to 67%), wider 95%
HDCI
s (from -9 to 8%), and with obvious
bias, although much smaller than the No correction model bias (from -3 to 3%).
Table 5 contains the pooled analysis results summary for every model and short sprint parameter.
3.2.3 Analysis across flying distances
Figure 3 depicts the esults of the analysis for every flying distance in the simulation.
inside ROPE
parameter estimates are calculated and depicted in Figure 4 for easier comprehension.
As expected, the No correction model demonstrated increasing bias as the flying distance increases
(from -46 to 76%), the widest 95%
HDCI
s (from -47 to 84%), and the lowest
inside ROPE
estimated
parameter values.
Estimated TC showed a small bias trend across flying distances (from -6 to 6%), resulting in decreasing
inside ROPE
performance (from 0 to 75%; see Figure 4), although with much smaller 95%
HDCI
s (from
-10 to 11%) compared to No correction model.
Estimated FD, as hypothesized, showed no bias and thus a stable
inside ROPE
performance across
flying distances (see Figure 4), with minimal 95% HDCIs (from -5 to 6%).
3.3 Minimal detectable change
3.3.1 Lowest Minimum Detectable Change
Estimated
%MDCslowest
95
are equal to 0.45% for MSS, 1.06% for MAC, 1.47% for TAU, and 0.7% for
PMAX (column lowest in Table 6 and dashed grey horizontal lines in Figure 5). An interesting finding is
that, given simulation parameters (particularly the precision of the timing gates to the closest 10
ms
), MSS
has the lowest
%MDCslowest
95
compared to other short sprint parameters. Since
%MDCslowest
95
represents
the lowest minimal detectable change, MSS is the parameter whose change could be, given this theoretical
simulation, estimated with the most precision. In contrast, TAU and MAC changes can be estimated with
the least precision.
3.3.2 Pooled analysis
Pooled
%MDCs95
represents an estimate of the sensitivity to detect true change with 95% confidence
when the flying start distance is not standardized (but within simulation parameter limits (ranging from 0
to 0.5
m
). As expected, the No correction model demonstrates the highest
%MDCs95
(from 3 to 44%),
while Estimated TC and Estimated FD demonstrated much smaller
%MDCs95
(from 1 to 8% and from 1
to 7%, respectively) (Table 6).
An interesting finding is that the MSS parameter showed very low
%MDCs95
across models (from
1 to 3%), even for the No correction model. This indicates that even the non-standardized short sprint
8/14
TAU
PMAX
MSS
MAC
No correction Estimated TC Estimated FD No correction Estimated TC Estimated FD
No correction Estimated TC Estimated FD No correction Estimated TC Estimated FD
0
25
50
75
0
25
50
75
−10.0
−7.5
−5.0
−2.5
0.0
−40
−20
0
% Diff
Figure 2. Pooled distribution of the %Di f f . Error bars represent the distribution median and 95%
HDCI
. A grey area represents parameter
ROPE
(assumed to be equal to 95%
HDCI
of the
%Di f f
using
No correction model and no flying distance).
Table 5. ROPEs, a summary of %Di f f distribution, and inside ROPE for pooled analysis.
Parameter ROPE (%) Model % Diff Inside ROPE (%)
No correction median -3%, 95% HDCI [-7 to 0%] 2%
Estimated TC median 0%, 95% HDCI [-1 to 0%] 67%
MSS -0.3 to 0.33%
Estimated FD median 0%, 95% HDCI [-1 to 1%] 72%
No correction median 49%, 95% HDCI [11 to 80%] 2%
Estimated TC median 3%, 95% HDCI [-2 to 8%] 12%
MAC -0.73 to 0.74%
Estimated FD median 0%, 95% HDCI [-4 to 4%] 25%
No correction median -35%, 95% HDCI [-46 to -11%] 2%
Estimated TC median -3%, 95% HDCI [-9 to 2%] 16%
TAU -1.03 to 1%
Estimated FD median 0%, 95% HDCI [-5 to 5%] 31%
No correction median 44%, 95% HDCI [6 to 73%] 2%
Estimated TC median 3%, 95% HDCI [-2 to 8%] 9%
PMAX -0.5 to 0.5%
Estimated FD median 0%, 95% HDCI [-4 to 4%] 20%
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MSS
MAC
TAU
PMAX
No correction
Estimated TC
Estimated FD
0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4
0
25
50
75
−5
0
5
10
15
−10
0
10
20
−50
−40
−30
−20
−10
0
−15
−10
−5
0
5
−20
−10
0
10
0
25
50
75
−5
0
5
10
15
−10
0
10
20
−10.0
−7.5
−5.0
−2.5
0.0
−2
−1
0
1
−5.0
−2.5
0.0
Flying distance (m)
% Diff
Figure 3. Distribution of the
%Di f f
across every flying distance in the simulation. Error bars represent
the distribution
median
and 95%
HDCI
. A grey area represents parameter
ROPE
(assumed to be equal to
95% HDCI of the %Di f f using No correction model and no flying distance). For the less crowded
visualization, flying distance in increments of 0.05 mis plotted.
No correction
Estimated TC
Estimated FD
No correction
Estimated TC
Estimated FD
No correction
Estimated TC
Estimated FD
No correction
Estimated TC
Estimated FD
TAU
PMAX
MSS
MAC
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6
0
25
50
75
0
25
50
75
Flying distance (m)
Inside ROPE (%)
Figure 4. insid e ROPE estimated across every flying distance in the simulation.
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monitoring (i.e., without standardized flying distance) using the No correction model, given simulation
parameters, can be used to track changes in MSS. TAU, MAC, and PMAX parameters, on the other hand,
demand much larger %MDCs95 (from 7 to 44%, from 6 to 37%, and from 6 to 36%, respectively).
3.3.3 Analysis across flying distances
When estimated across flying distances,
%MDCs95
shows interesting and surprising patterns (Figure
5). For every short sprint parameter, Estimated TC showed stable and lower
%MDCs95
compared to
Estimated FD (from 1 to 6% and from 1 to 8%, respectively). This is surprising because even if it
has biased estimates of short sprint parameters (see Percent difference results section, mainly Figure 4)
compared to the Estimated FD,Estimated TC might be more sensitive to detect changes, given simulation
parameters.
Another surprising finding is that the No correction model, even if shown to be highly biased in
estimating short sprint parameter values (see Percent difference results section, mainly Figure 3), showed
the lowest
%MDCs95
for the MAC and TAU parameters (from 1 to 5% and from 1 to 3% respectively).
This indicates that, when short sprint measurement is standardized (i.e., athlete perform with the same
flying distance), given the simulation parameters, the No correction model can be the most sensitive model
to detect changes in MAC and TAU parameters. This is the case for the MSS and PMAX parameters
(from 0 to 3% and from 1 to 9%, respectively).
When it comes to estimating changes in short sprint parameters, change in MSS is the most sensitive
to be detected (from 0 to 3%) compared to MAC (from 1 to 7%), TAU (from 1 to 8%), and PMAX (from
1 to 9%).
4 CONCLUSION
The simulation study employed in this paper demonstrated some expected and unexpected theoretical
findings. Among the expected findings are (1) the bias and low
inside ROPE
performance in estimating
short sprint parameters using the No correction model, (2) more negligible bias and higher
inside ROPE
for the Estimated TC model, and (3) no bias and highest inside ROPE for the Estimated FD model.
The unexpected finding of this study is the performance of the No correction model in sensitivity of
estimating the change of the MAC and TAU parameters, which outperformed the other two models.
When estimating short sprint parameters across models, given simulation parameters, MSS and change
in MSS can be estimated more precisely compared to TAU, MAC, and PMAX parameters and their
changes.
In addition to model performances, this simulation study provided the
ROPE
s and
%MDCslowest
95
.
These could be useful for further validity and reliability studies evaluating short sprint model performance
involving real athletes and timing gates positioned at the same distances with the same rounding.
The take-away message for the practitioners is that besides standardizing the sprint starting technique
for the short sprint performance monitoring, it would be wise to utilize and track the results of all three
models. The Estimated FD model will provide unbiased estimates of the current performance, but the No
correction model might be more sensitive in detecting changes in TAU and MAC parameters.
This practical conclusion should be taken with caution, since it based on the results of this theoretical
simulation. Additional studies involving real athletes in evaluating the performance of these three
models are needed. These should involve estimating the short sprint parameters agreement between
gold-standard (i.e., criterion) measure (e.g., radar gun, laser gun, or video analysis) and practical timing
gates measure. One such study is currently in preparation. In addition to theoretical findings, such a study
will provide model performance estimates when biological variability is involved in short sprints, which
is not considered in the current study.
Table 6. Pooled %MDCs95 estimated using pooled simulation dataset.
Parameter lowest No correction Estimated TC Estimated FD
MSS 0.45 % 3 % 1 % 1 %
MAC 1.06 % 37 % 7 % 6 %
TAU 1.47 % 44 % 8 % 7 %
PMAX 0.7 % 36 % 7 % 6 %
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Estimated TC
Estimated FD
No correction
No correction
Estimated TC
Estimated FD
No correction
Estimated TC
Estimated FD
Estimated TC
Estimated FD
No correction
TAU
PMAX
MSS
MAC
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6
2
4
6
2.5
5.0
7.5
1
2
2
4
6
8
Flying distance (m)
Minimal Detectable Change (%)
Figure 5. Estimated %MDCs95 across every flying distance in the simulation. The dashed line
represents %MDCslowest
95
5 SUPPLEMENTAL MATERIAL
The R Markdown (Xie, Allaire & Grolemund, 2018; Xie, Dervieux & Riederer, 2020; Allaire et al.,
2022; Xie, 2022) source code for this paper and analysis can be found on the GitHub repository:
https://github.com/mladenjovanovic/shorts-simulation-paper.
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