Content uploaded by Mladen Jovanović

Author content

All content in this area was uploaded by Mladen Jovanović on Jul 18, 2022

Content may be subject to copyright.

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 ×(1−e−t

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 ×e−t

TAU (3)

By integrating equation (1), we get the equation for distance covered (4).

d(t) = MSS ×(t+TAU ×e−t

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(−e−d

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

ms−1

, 1.14

s

, and 7.94

ms−2

, 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

ms−1

, TAU equal to 1.125

s

, MAC equal to 8

ms−2

, 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(−e−d

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(−e−d+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

ms−1

, in increments of 0.05

ms−1

, resulting in a total of 81 unique values), MAC (ranging from 7 to 11

ms−2

, in increments of 0.05

ms−2

, 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 =s∑N

i=1(100 ×yi−ˆyi

ˆyi)2

N−2(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%

9/14

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.

10/14

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 %

11/14

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.

REFERENCES

Allaire J, Xie Y, McPherson J, Luraschi J, Ushey K, Atkins A, Wickham H, Cheng J, Chang W, Iannone

R. 2022. Rmarkdown: Dynamic documents for r.

Altmann S, Hoffmann M, Kurz G, Neumann R, Woll A, Haertel S. 2015. Different Starting Distances

Affect 5-m Sprint Times. Journal of Strength and Conditioning Research 29:2361–2366. DOI:

10.1519/JSC.0000000000000865.

Altmann S, Spielmann M, Engel FA, Neumann R, Ringhof S, Oriwol D, Haertel S. 2017. Validity of

Single-Beam Timing Lights at Different Heights. Journal of Strength and Conditioning Research

31:1994–1999. DOI: 10.1519/JSC.0000000000001889.

Altmann S, Spielmann M, Engel FA, Ringhof S, Oriwol D, H

¨

artel S, Neumann R. 2018. Accuracy of

single beam timing lights for determining velocities in a ﬂying 20-m sprint: Does timing light height

matter? Journal of Human Sport and Exercise 13. DOI: 10.14198/jhse.2018.133.10.

Brown TD, Vescovi JD, Vanheest JL. 2004. Assessment of linear sprinting performance: A theoretical

paradigm. Journal of Sports Science & Medicine 3:203–210.

Buchheit M, Samozino P, Glynn JA, Michael BS, Al Haddad H, Mendez-Villanueva A, Morin JB. 2014.

Mechanical determinants of acceleration and maximal sprinting speed in highly trained young soccer

players. Journal of Sports Sciences 32:1906–1913. DOI: 10.1080/02640414.2014.965191.

Clark KP, Rieger RH, Bruno RF, Stearne DJ. 2017. The NFL Combine 40-Yard Dash: How

Important is Maximum Velocity? Journal of Strength and Conditioning Research:1. DOI:

10.1519/JSC.0000000000002081.

Edwards T, Piggott B, Banyard HG, Haff GG, Joyce C. 2020. Sprint acceleration characteris-

tics across the Australian football participation pathway. Sports Biomechanics:1–13. DOI:

10.1080/14763141.2020.1790641.

Elzhov TV, Mullen KM, Spiess A-N, Bolker B. 2022. Minpack.lm: R interface to the levenberg-marquardt

nonlinear least-squares algorithm found in MINPACK, plus support for bounds.

Furlan L, Sterr A. 2018. The Applicability of Standard Error of Measurement and Minimal Detectable

12/14

Change to Motor Learning Research—A Behavioral Study. Frontiers in Human Neuroscience 12:95.

DOI: 10.3389/fnhum.2018.00095.

Furusawa K, Hill AV, Parkinson JL. 1927. The dynamics of ”sprint” running. Proceedings of the

Royal Society of London. Series B, Containing Papers of a Biological Character 102:29–42. DOI:

10.1098/rspb.1927.0035.

Goerg GM. 2022. LambertW: Probabilistic models to analyze and gaussianize heavy-tailed, skewed data.

Haugen TA, Breitsch

¨

adel F, Samozino P. 2020. Power-Force-Velocity Proﬁling of Sprinting Athletes:

Methodological and Practical Considerations When Using Timing Gates. Journal of Strength and

Conditioning Research 34:1769–1773. DOI: 10.1519/JSC.0000000000002890.

Haugen TA, Breitsch

¨

adel F, Seiler S. 2019. Sprint mechanical variables in elite athletes: Are force-velocity

proﬁles sport speciﬁc or individual? PLOS ONE 14:e0215551. DOI: 10.1371/journal.pone.0215551.

Haugen TA, Breitsch

¨

adel F, Seiler S. 2020. Sprint mechanical properties in soccer players accord-

ing to playing standard, position, age and sex. Journal of Sports Sciences 38:1070–1076. DOI:

10.1080/02640414.2020.1741955.

Haugen T, Buchheit M. 2016. Sprint Running Performance Monitoring: Methodological and Practical

Considerations. Sports Medicine 46:641–656. DOI: 10.1007/s40279-015-0446-0.

Haugen TA, Tønnessen E, Seiler SK. 2012. The Difference Is in the Start: Impact of Timing and

Start Procedure on Sprint Running Performance: Journal of Strength and Conditioning Research

26:473–479. DOI: 10.1519/JSC.0b013e318226030b.

Jim

´

enez-Reyes P, Samozino P, Garc

´

ıa-Ramos A, Cuadrado-Pe

˜

naﬁel V, Brughelli M, Morin J-B. 2018.

Relationship between vertical and horizontal force-velocity-power proﬁles in various sports and levels

of practice. PeerJ 6:e5937. DOI: 10.7717/peerj.5937.

Jovanovi

´

c M. 2020. Bmbstats: Bootstrap Magnitude-based Statistics for Sports Scientists. Mladen

Jovanovi´

c.

Jovanovi´

c M. 2022. Shorts: Short sprints.

Jovanovi

´

c M, Vescovi JD. 2020. Shorts: An R Package for Modeling Short Sprints. Accepted to

International Journal of Strength and Conditioning (IJSC).

Kruschke JK. 2015. Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Boston: Academic

Press.

Kruschke JK. 2018. Rejecting or Accepting Parameter Values in Bayesian Estimation. Advances in

Methods and Practices in Psychological Science 1:270–280. DOI: 10.1177/2515245918771304.

Kruschke JK, Liddell TM. 2018a. Bayesian data analysis for newcomers. Psychonomic Bulletin & Review

25:155–177. DOI: 10.3758/s13423-017-1272-1.

Kruschke JK, Liddell TM. 2018b. The Bayesian New Statistics: Hypothesis testing, estimation, meta-

analysis, and power analysis from a Bayesian perspective. Psychonomic Bulletin & Review 25:178–

206. DOI: 10.3758/s13423-016-1221-4.

Makowski D, Ben-Shachar M, L

¨

udecke D. 2019. bayestestR: Describing Effects and their Uncertainty,

Existence and Signiﬁcance within the Bayesian Framework. Journal of Open Source Software 4:1541.

DOI: 10.21105/joss.01541.

Mangine GT, Hoffman JR, Gonzalez AM, Wells AJ, Townsend JR, Jajtner AR, McCormack WP, Robinson

EH, Fragala MS, Fukuda DH, Stout JR. 2014. Speed, Force, and Power Values Produced From Non-

motorized Treadmill Test Are Related to Sprinting Performance: Journal of Strength and Conditioning

Research 28:1812–1819. DOI: 10.1519/JSC.0000000000000316.

Marcote-Peque

˜

no R, Garc

´

ıa-Ramos A, Cuadrado-Pe

˜

naﬁel V, Gonz

´

alez-Hern

´

andez JM, G

´

omez M

´

A,

Jim´

enez-Reyes P. 2019. Association Between the Force–Velocity Proﬁle and Performance Variables

Obtained in Jumping and Sprinting in Elite Female Soccer Players. International Journal of Sports

Physiology and Performance 14:209–215. DOI: 10.1123/ijspp.2018-0233.

Morin JB. 2017.A spreadsheet for Sprint acceleration Force-Velocity-Power proﬁling. Available

at https://jbmorin.net/2017/12/13/a-spreadsheet-for-sprint-acceleration-force-velocity-power-

proﬁling/ (accessed October 27, 2020).

Morin J-B, Samozino P. 2019. Spreadsheet for Sprint acceleration force-velocity-power proﬁling.

Morin J-B, Samozino P, Murata M, Cross MR, Nagahara R. 2019. A simple method for computing sprint

acceleration kinetics from running velocity data: Replication study with improved design. Journal of

Biomechanics 94:82–87. DOI: 10.1016/j.jbiomech.2019.07.020.

Motulsky H. 2018. Intuitive biostatistics: A nonmathematical guide to statistical thinking. New York:

13/14

Oxford University Press.

R Core Team. 2022. R: A language and environment for statistical computing. Vienna, Austria: R

Foundation for Statistical Computing.

Samozino P, Rabita G, Dorel S, Slawinski J, Peyrot N, Saez de Villarreal E, Morin J-B. 2016. A simple

method for measuring power, force, velocity properties, and mechanical effectiveness in sprint running:

Simple method to compute sprint mechanics. Scandinavian Journal of Medicine & Science in Sports

26:648–658. DOI: 10.1111/sms.12490.

Stenroth L, Vartiainen P. 2020. Spreadsheet for sprint acceleration force-velocity-power proﬁling with

optimization to correct start time. DOI: 10.13140/RG.2.2.12841.83045.

Stenroth L, Vartiainen P, Karjalainen PA. 2020. Force-velocity proﬁling in ice hockey skating:

Reliability and validity of a simple, low-cost ﬁeld method. Sports Biomechanics:1–16. DOI:

10.1080/14763141.2020.1770321.

Vescovi JD, Jovanovi

´

c M. 2021. Sprint Mechanical Characteristics of Female Soccer Players: A Retro-

spective Pilot Study to Examine a Novel Approach for Correction of Timing Gate Starts. Frontiers in

Sports and Active Living 3:629694. DOI: 10.3389/fspor.2021.629694.

Ward-Smith AJ. 2001. Energy conversion strategies during 100 m sprinting. Journal of Sports Sciences

19:701–710. DOI: 10.1080/02640410152475838.

Xie Y. 2022. Bookdown: Authoring books and technical documents with r markdown.

Xie Y, Allaire JJ, Grolemund G. 2018. R markdown: The deﬁnitive guide. Boca Raton, Florida: Chapman;

Hall/CRC.

Xie Y, Dervieux C, Riederer E. 2020. R markdown cookbook. Boca Raton, Florida: Chapman; Hall/CRC.

14/14