## Introduction

Coaches and athletes sometime change measurement/testing equipment. The problem is that the estimates from different equipment might differ (and they usually do). There are couple of soutions to this problem – (1) disregard the old scores (in which case coaches losses precious historical data) or (2) convert the old scores to the new and ideally maintain the measurement database with as little error as possible.

Since we are going for the second option, the practical question to be answered is how big is the error, or in other words signal vs. noise? I will present one way to evaluate the error of the conversion using Smallest Worthwihle Change metric.

Friend of mine, Sam Leahey is switching from Just Jump to Smart Jump contact mats and wants to convert the older Just Jump scores to the new Smart Jump scores.

We proposed the following protocol for data collection:

1. Get couple of athletes (or all of them)
2. Put mats close to each other and place left foot on one and right foot on another (see the picture)
3. Perform the vertical jump and take the scores for each
4. Switch the legs and perfom the jump again (this should remove any leg/differences)
5. Randomize the start, to avoid any fatigue related issues
6. Multiple athletes can repreat the procedure multiple times so we get as much data points as possible

Here is the picture of the procedure:

## The data

The testing data was saved to .csv file. Let’s load it and show first couple of rows

``````# Load needed libraries
suppressPackageStartupMessages(library(reshape2))
suppressPackageStartupMessages(library(caret))
suppressPackageStartupMessages(library(dplyr))
suppressPackageStartupMessages(library(rethinking))
suppressPackageStartupMessages(library(boot))

stringsAsFactors = FALSE)

``````##    Attempt    Date    Name JJ.LEFT JJ.RIGHT SJ.RIGHT SJ.LEFT
## 1        1 17.3.16     Joe    23.8     24.3    18.56   19.65
## 2        2 21.3.16     Sam    19.0     17.1    13.87   12.56
## 3        3 21.3.16    Zach    12.5     12.8    10.80    8.40
## 4        4 21.3.16    Erik    14.2     15.4    11.31   11.31
## 5        5 24.3.16     Sam    19.9     20.7    14.82   16.30
## 6        6 24.3.16  Ashton    14.5     13.4    12.51   10.57
## 7        7 24.3.16 Jillian    17.1     16.7    11.69   11.69
## 8        8 25.3.16     Joe    22.5     24.6    17.61   19.22
## 9        9 25.3.16    Zach    13.7     13.6     8.89   10.98
## 10      10 25.3.16    Erik    16.1     14.3    10.56   12.26
## 11      11 25.3.16  AidanH    14.8     13.6    10.35    9.95
## 12      12 26.3.16  Ashton    15.4     16.7    11.59   11.97
## 13      13 26.3.16    Kara    10.9     12.7     7.38    9.86
## 14      14 28.3.16     Sam    19.3     20.4    14.29   15.96
## 15      15 30.3.16     Sam    19.9     18.4    14.82   14.23``````

Let’s plot the JustJump left and right scores and SmartJump left and right scores (when left/right foot is placed on the mat).

``````# Just Jump
gg <- ggplot(jumpDataRaw, aes(x = JJ.LEFT, y = JJ.RIGHT)) +
geom_smooth(method = "lm", color = "grey", alpha = 0.3) +
geom_abline(slope = 1, color = "red", linetype = "dashed") +
geom_point(color = "blue", alpha = 0.7) +
theme_bw() +
ggtitle("Just Jump leg difference scatterplot")
print(gg)``````

``````gg <- ggplot(jumpDataRaw, aes(x = JJ.LEFT - JJ.RIGHT)) +
geom_density(fill = "blue", alpha = 0.3) +
geom_histogram(aes(y=..density..), bins=20, alpha = 0.5) +
geom_vline(xintercept = mean(jumpDataRaw\$JJ.LEFT - jumpDataRaw\$JJ.RIGHT),
color = "red", linetype = "dashed") +
theme_bw() +
ggtitle("Just Jump leg difference distribution")
print(gg)``````

``````# Smart Jump
gg <- ggplot(jumpDataRaw, aes(x = SJ.LEFT, y = SJ.RIGHT)) +
geom_smooth(method = "lm", color = "grey", alpha = 0.3) +
geom_abline(slope = 1, color = "red", linetype = "dashed") +
geom_point(color = "blue", alpha = 0.7) +
theme_bw() +
ggtitle("Smart Jump leg difference scatterplot")
print(gg)``````

``````gg <- ggplot(jumpDataRaw, aes(x = SJ.LEFT - SJ.RIGHT)) +
geom_density(fill = "blue", alpha = 0.3) +
geom_histogram(aes(y=..density..), bins=20, alpha = 0.5) +
geom_vline(xintercept = mean(jumpDataRaw\$SJ.LEFT - jumpDataRaw\$SJ.RIGHT),
color = "red", linetype = "dashed") +
theme_bw() +
ggtitle("Smart Jump leg difference distribution")
print(gg)``````

As can be seen there are differences in the left and right at each mat. This can be due numerous reasons: (1) measurement error, (2) typical error/variation, (3) change in the subjects states (i.e. fatigue, effort), (4) assymetrical landing/jumping.

Depending on the way the tests were done, change in subject states can be minimizied by athletes starting on different left/right position (randomization) or giving enough of rest between trial. Let’s assume that this was done.

In the Smart Jump there was clearly one outlier with big difference between jumps. Let’s find him and remove him

``````jumpDataRaw <- mutate(jumpDataRaw, JJ.Difference = JJ.LEFT - JJ.RIGHT)
jumpDataRaw <- mutate(jumpDataRaw, SJ.Difference = SJ.LEFT - SJ.RIGHT)

outlier <- which.min(jumpDataRaw\$SJ.Difference)
print(jumpDataRaw[outlier,])``````
``````##    Attempt   Date   Name JJ.LEFT JJ.RIGHT SJ.RIGHT SJ.LEFT JJ.Difference
## 24      25 5.4.16 AidanP    13.5     13.4    16.63    9.52           0.1
##    SJ.Difference
## 24         -7.11``````

We will remove this lad from the sample

``jumpDataRaw <- jumpDataRaw[-outlier,]``

What is important to check is if there are correlations between SJ.Difference and JJ.Difference. If there is a correlation, that would imply that there was a change in athlete state between trials (either fatigue or change in effort) or there was assymmetry in landing (favoring one leg over the other).

Before plotting the relationship between two differences, we will normalize them them so they end up being on the same scale

``````jumpDataRaw\$JJ.Difference <- as.numeric(scale(jumpDataRaw\$JJ.Difference))
jumpDataRaw\$SJ.Difference <- as.numeric(scale(jumpDataRaw\$SJ.Difference))``````

Here is the plot

``````gg <- ggplot(jumpDataRaw, aes(x = SJ.Difference, y = JJ.Difference)) +
geom_smooth(method = "lm", color = "grey", alpha = 0.3) +
geom_abline(slope = -1, color = "red", linetype = "dashed") +
geom_point(color = "blue", alpha = 0.7) +
theme_bw() +
ggtitle("Jump differences")
print(gg)``````

``summary(lm(SJ.Difference ~ JJ.Difference, jumpDataRaw))``
``````##
## Call:
## lm(formula = SJ.Difference ~ JJ.Difference, data = jumpDataRaw)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -2.60580 -0.48194 -0.00346  0.57192  1.89890
##
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)
## (Intercept)    3.220e-17  1.227e-01   0.000  1.00000
## JJ.Difference -3.965e-01  1.238e-01  -3.203  0.00226 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9263 on 55 degrees of freedom
## Multiple R-squared:  0.1572, Adjusted R-squared:  0.1419
## F-statistic: 10.26 on 1 and 55 DF,  p-value: 0.002262``````
``cor(jumpDataRaw\$SJ.Difference, jumpDataRaw\$JJ.Difference)``
``## [1] -0.3965054``

There appears to be negligable correlation between the two, so we will assume there were no change in athlete state between the trials.

Let’s inspect athletes that show highest “assymmetry”

``````jumpDataRaw <- mutate(jumpDataRaw, Assymmetry = sqrt(SJ.Difference^2 + JJ.Difference^2))
``````##    Attempt    Date      Name JJ.LEFT JJ.RIGHT SJ.RIGHT SJ.LEFT
## 1       26  8.4.16      Alex    18.0     12.9    13.31   11.78
## 2       13 26.3.16      Kara    10.9     12.7     7.38    9.86
## 3        3 21.3.16      Zach    12.5     12.8    10.80    8.40
## 4       32  9.4.16 Christina    15.8     11.5     8.76    8.19
## 5       46 15.4.16      Erik    10.9     14.5    10.39   10.57
## 6       48 15.4.16        Ty    15.9     12.1    10.39    9.39
## 7       22  2.4.16    Ashton    16.5     14.9    12.61   10.66
## 8        8 25.3.16       Joe    22.5     24.6    17.61   19.22
## 9        6 24.3.16    Ashton    14.5     13.4    12.51   10.57
## 10       9 25.3.16      Zach    13.7     13.6     8.89   10.98
##    JJ.Difference SJ.Difference Assymmetry
## 1      2.8666089   -1.59281546   3.279407
## 2     -1.3263223    2.31812911   2.670740
## 3     -0.4148155   -2.44132464   2.476315
## 4      2.3804719   -0.65652948   2.469348
## 5     -2.4201305    0.07494394   2.421291
## 6      2.0766363   -1.07590758   2.338802
## 7      0.7397597   -2.00244058   2.134716
## 8     -1.5086237    1.46961994   2.106117
## 9      0.4359241   -1.99268760   2.039812
## 10    -0.1717471    1.93776293   1.945359``````

I will remove everyone who is above 2 for the further analysis

``````inconsistent <- jumpDataRaw\$Assymmetry > 2
jumpDataRaw <-  jumpDataRaw[!inconsistent, ]``````

And here is the plot one more time with those lads removed.

``````jumpDataRaw <- mutate(jumpDataRaw, JJ.Difference = JJ.LEFT - JJ.RIGHT)
jumpDataRaw <- mutate(jumpDataRaw, SJ.Difference = SJ.LEFT - SJ.RIGHT)
jumpDataRaw\$JJ.Difference <- as.numeric(scale(jumpDataRaw\$JJ.Difference))
jumpDataRaw\$SJ.Difference <- as.numeric(scale(jumpDataRaw\$SJ.Difference))

gg <- ggplot(jumpDataRaw, aes(x = SJ.Difference, y = JJ.Difference)) +
geom_smooth(method = "lm", color = "grey", alpha = 0.3) +
geom_abline(slope = -1, color = "red", linetype = "dashed") +
geom_point(color = "blue", alpha = 0.7) +
theme_bw() +
ggtitle("Jump differences")
print(gg)``````

``summary(lm(SJ.Difference ~ JJ.Difference, jumpDataRaw))``
``````##
## Call:
## lm(formula = SJ.Difference ~ JJ.Difference, data = jumpDataRaw)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -1.80747 -0.69757  0.09858  0.58990  2.24886
##
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)
## (Intercept)   -4.344e-17  1.417e-01   0.000    1.000
## JJ.Difference -2.368e-01  1.432e-01  -1.653    0.105
##
## Residual standard error: 0.9821 on 46 degrees of freedom
## Multiple R-squared:  0.0561, Adjusted R-squared:  0.03558
## F-statistic: 2.734 on 1 and 46 DF,  p-value: 0.1051``````
``cor(jumpDataRaw\$SJ.Difference, jumpDataRaw\$JJ.Difference)``
``## [1] -0.2368485``

## SWC and TE

Let’s calculate Smallest Wortwhile Change (SWC) and Typical Error (TE) for each platform. Since couple of athletes repeated the test couple of times, the best we can do is to average them.

``````jumpDataAveraged <- jumpDataRaw %>%
select(3:7) %>%
group_by(Name) %>%
summarise_each(funs(mean)) %>%
ungroup()``````

To calculate SWC in a given test, we need to average them since we have two performed (left and right).

``````jumpDataAveraged <- mutate(jumpDataAveraged, JJ = (JJ.LEFT + JJ.RIGHT) / 2)
jumpDataAveraged <- mutate(jumpDataAveraged, SJ = (SJ.LEFT + SJ.RIGHT) / 2)``````

And to calculate TE we need to calculate the differences again:

``````jumpDataAveraged <- mutate(jumpDataAveraged, JJ.Difference = (JJ.LEFT - JJ.RIGHT))
jumpDataAveraged <- mutate(jumpDataAveraged, SJ.Difference = (SJ.LEFT - SJ.RIGHT))``````

And here is our averaged data that we will use for SWC and TE calculation (showing only first 5 rows):

``head(as.data.frame(jumpDataAveraged))``
``````##     Name  JJ.LEFT JJ.RIGHT SJ.RIGHT  SJ.LEFT       JJ       SJ
## 1 AidanH 14.35714 13.45714 10.06571 10.13714 13.90714 10.10143
## 2 AidanP 13.00000 14.10000  9.26000  9.62500 13.55000  9.44250
## 3 Amelia 16.00000 14.10000 11.45000 11.54000 15.05000 11.49500
## 4 Andrew 14.10000 14.50000 10.35000 10.53000 14.30000 10.44000
## 5 Ashley 17.20000 15.90000 11.69000 12.12000 16.55000 11.90500
## 6 Ashton 16.07500 15.92500 11.58000 11.42750 16.00000 11.50375
##   JJ.Difference SJ.Difference
## 1          0.90    0.07142857
## 2         -1.10    0.36500000
## 3          1.90    0.09000000
## 4         -0.40    0.18000000
## 5          1.30    0.43000000
## 6          0.15   -0.15250000``````

The SWC is calculate using between subjects standard deviation, multiplied by 0.2 (see Will Hopkins work). This is the smallest importance change in performance. Let’s calculate that for both JustJump and SmartJump

``````JJ.SWC <- 0.2 * sd(jumpDataAveraged\$JJ)
SJ.SWC <- 0.2 * sd(jumpDataAveraged\$SJ)``````

Here are the SWCs for our mats:

• JustJump SWC 0.63
• SmartJump SWC 0.52

Let’c calculate Typical Error/Variation for each. Here we are interested in the differences between two averaged trials (see Will Hopkins):

``````JJ.TE <- sd(jumpDataAveraged\$JJ.Difference) / sqrt(2)
SJ.TE <- sd(jumpDataAveraged\$SJ.Difference) / sqrt(2)``````

Here are the TEs for our mats:

• JustJump TE 0.61
• SmartJump TE 0.41

Let’s calculate signal to noise for each mat (this is SWC / TE).

• JustJump 1.03
• SmartJump 1.26

It seems that SmartJump has better signal to noise quality (i.e ability to discern smallest improvement in performance).

## Conversion between mats

Let’s create the conversion between the mats. I will use average between left and right jumps for each mat, but will not summarise for each athlete to get more data points (trials) as I have done for SWC and TE calculation.

``````jumpData <- jumpDataRaw[3:7]
jumpData <- mutate(jumpData, JJ = (JJ.LEFT + JJ.RIGHT) / 2)
jumpData <- mutate(jumpData, SJ = (SJ.LEFT + SJ.RIGHT) / 2)``````

Let’s plot the data:

``````gg <- ggplot(jumpData, aes(x = JJ, y = SJ)) +
geom_smooth(method = "lm", color = "grey", alpha = 0.3) +
geom_point(color = "blue", alpha = 0.7) +
theme_bw()
print(gg)``````

The simplest approach to conversion would be to use linear regression to get the formula:

``````linearModel <- lm(SJ ~ JJ, jumpData)
summary(linearModel)``````
``````##
## Call:
## lm(formula = SJ ~ JJ, data = jumpData)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -1.05121 -0.27433  0.09155  0.29789  1.20783
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.49285    0.34769  -4.294 8.98e-05 ***
## JJ           0.82383    0.02166  38.039  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4753 on 46 degrees of freedom
## Multiple R-squared:  0.9692, Adjusted R-squared:  0.9685
## F-statistic:  1447 on 1 and 46 DF,  p-value: < 2.2e-16``````

What we get is the following conversion formula:

SmartJump = -1.49 + 0.82 * JustJump

But we are also interested in the error of the model (in this case residual standard error or standard deviation of the residuals), which is equal to:

``sd(residuals(linearModel))``
``## [1] 0.4702623``

There is a slight difference between SD of residuals and residual standard error, becasue the latter is divided by degrees of freedom (N – 1 – k).

## Prediction

Using residuals standard deviation (or RMSE – root mean square error, but this one is divided by N and not N – 1 as is in residuals standard error) we want to estimate how our model will generalize to the unseen data (because we will predict/convert for new athletes/trials).

Utilizing caret package I will evaluate our model performance (estiamted by RMSE) using repeated cross validation, or CV using 5 folds and 200 repeats.

``````ctr <- trainControl(method = "repeatedcv",
number = 5,
repeats = 200)

target = jumpData\$SJ
predictors = select(jumpData, JJ)

model <- train(y = target,
x = predictors,
method = "lm",
trControl = ctr)
print(model)``````
``````## Linear Regression
##
## 48 samples
##  1 predictor
##
## No pre-processing
## Resampling: Cross-Validated (5 fold, repeated 200 times)
## Summary of sample sizes: 36, 37, 39, 40, 40, 37, ...
## Resampling results
##
##   RMSE      Rsquared   RMSE SD     Rsquared SD
##   0.478878  0.9669502  0.09492898  0.02340058
##
## ``````

Using cross-validation of the simple linear regression, you can expect the following error (Root Mean Square Error; or residual standard deviation): 0.478878.

So here are our final model performance estimates (errors):

• Residual Standard Error = 0.4753464
• Residuals Standard Deviation = 0.4702623
• Training RMSE = 0.465338
• Cross-Validated RMSE = 0.478878

## How big is the prediction error?

One way to compare prediction error (CV-RMSE) is to compare it to SWC of the target variable (SmartJump), which is 0.52. The ratio between the two can give you signal vs. noise estimate for comparing the SmartJump predictions (from JustJump scores) in detecting the SWC in the SmartJump performance. In this case the ration between CV-RMSE and SWC is 0.93, which is good. This mean that we can use the conversion from JustJump to SmartJump without loosing the ability to detect the SWC.

## Bayesian computation

I recently started learning more about Bayesina Data Analysis, and will try to use it here. Bayesian data analysis is really flexible approach that can give us more “probabilistical” parameters (posteriors) and can “chew” a lot of things being thrown at it. In this case we can throw in SWC and TE, but this is still beyond my knowledge (might come back to this problem again in the next couple of months). For this reason I will use Bayesian approach to quantify to models ability to detect SWC in the target variable.

Please be free to let me know if I did any major/minor fuck ups here.

I will use rethinking package from the book I am reading (Statistical Rethinking – great book BTW)

Here is the definition of Bayesian model:

``````bayesianModel <- map(
alist(
SJ ~ dnorm(mu, sd), # Likelihood
mu <- intercept + slope * JJ,
slope ~ dnorm(1, 1), # Prior for slope
intercept ~ dnorm(0, 1), # Prior for intercept
sd ~ dunif(0, 1) # Prior for sigma (sd of residuals)
) , data = jumpData )``````

This is our data with 95% confidence interval and 95% prediction interval:

``````# Plot the model
JJ.seq <- seq(from = min(jumpData\$JJ),
to = max(jumpData\$JJ),
length.out = 100)

pred_dat <- data.frame(JJ = JJ.seq)

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``````mu.mean <- apply(mu, 2, mean)
mu.PI <- apply(mu, 2, HPDI, prob = 0.95)
sim.SJ <- sim(bayesianModel, data = pred_dat, 1e4)``````
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``````SJ.PI <- apply(sim.SJ, 2, HPDI, prob = 0.95)

# Plot the model
df <- data.frame(JJ.seq = JJ.seq,
SJ.mu = mu.mean,
SJ.CI.lower = mu.PI[1,],
SJ.CI.upper = mu.PI[2,],
SJ.PI.lower = SJ.PI[1,],
SJ.PI.upper = SJ.PI[2,])

gg <- ggplot(df, aes(x = JJ.seq, y = SJ.mu)) +
geom_line() +
geom_ribbon(aes(ymin = SJ.CI.lower, ymax = SJ.CI.upper),
fill = "grey", color = NA, alpha = 0.7) +
geom_ribbon(aes(ymin = SJ.PI.lower, ymax = SJ.PI.upper),
fill = "grey", color = NA, alpha = 0.4) +
geom_point(data = jumpData, aes(x = JJ, y = SJ), color = "blue", alpha = 0.5) +
theme_bw() +
xlab("JustJump") + ylab("SmartJump")
gg``````

What we need to do next is to simulate the data points using the model parameters and calculate the residuals. We will draw 10,000 samples.

``mu <- link(bayesianModel, data = jumpData, 1e4)``
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``````mu.mean <- apply(mu, 2, mean)
sim.SJ <- sim(bayesianModel, data = jumpData, 1e4)``````
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``modelResiduals <- t(sim.SJ) - mu.mean``

Let’s plot the distribution of the residuals over the SWC of the Smart Jump

``````gg <- ggplot(data.frame(Residuals = as.numeric(modelResiduals)),
aes(x = Residuals)) +
geom_vline(xintercept = -SJ.SWC, linetype = "longdash") +
geom_vline(xintercept =  SJ.SWC, linetype = "longdash") +
geom_density(fill = "blue", alpha = 0.5) +
theme_bw()
print(gg)``````

And we can easily calculate the probability of “capturing” SWC with converted jump height:

``````captured <- ifelse(modelResiduals < SJ.SWC & modelResiduals > -SJ.SWC, 1, 0)

SWCProbability <- sum(captured) / length(captured)
print(SWCProbability)``````
``## [1] 0.7235458``

So the probability of capturing the SWC with converted model is 0.72. With the signal to noise ration of around 1 this is the percentage we an expect (based on normal distribution).

In the future I might add Typical Errors in the Bayesian model. That is currently beyond my knowledge. Besides we can plug in SWC in the Bayesian models itself and estimate this “capture” probabilistically (using the Bayesian model), rathen than with the method I’ve used here. There is a recent article that argues about this approach.

## Bootstrap

We can also bootstrap our signal to noise ratio to get the idea of the distribution

``````signalToNoise <- function(data, i) {
data2 <- data[i,]
SWC <- sd(data2\$SJ) * 0.2

linearModel <- lm(SJ~JJ, data2)

residualStandardError <- summary(linearModel)\$sigma
return(SWC / residualStandardError)
}

stnBoot <- boot(jumpData, signalToNoise, R = 2000)
print(stnBoot)``````
``````##
## ORDINARY NONPARAMETRIC BOOTSTRAP
##
##
## Call:
## boot(data = jumpData, statistic = signalToNoise, R = 2000)
##
##
## Bootstrap Statistics :
##     original     bias    std. error
## t1* 1.127203 0.02099104   0.1994454``````
``confint(stnBoot)``
``````## Bootstrap quantiles, type =  bca
##
##       2.5 %   97.5 %
## 1 0.7844383 1.591124``````

Above are our boostrapped signal-to-noise estimates and here is the distribution

``````df <- data.frame(SignalToNoise = stnBoot\$t)
gg <- ggplot(df, aes(x = SignalToNoise)) +
geom_density(fill = "blue", alpha = 0.3) +
geom_histogram(aes(y=..density..), bins=20, alpha = 0.5) +
geom_vline(xintercept = mean(df\$SignalToNoise),
color = "red", linetype = "dashed") +
theme_bw() +
ggtitle("Distribution of the bootstrapped signal to noise ratio")
print(gg)``````

## Conclusion

Please note that this was a playbook – let me know if I did any errors in the reasoning. In the near future I will use Bayesian approach to estimate signal to noise (and “capture” percentage) ratios using both SWC and TE in the model.

I am a physical preparation coach from Belgrade, Serbia, grew up in Pula, Croatia (which I consider my home town). I was involved in physical preparation of professional, amateur and recreational athletes of various ages in sports such as basketball, soccer, volleyball, martial arts and tennis. Read More »

Welcome to Complementary Training Community! Forums [Playbook] Converting Just Jump to Smart Jump

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