# COVID-19 and Changes in the Model of Physical Fitness and Body Composition of Young Tennis Players
## 2 Materials and Methods
### Statistical Analysis  
The data were analysed using linear mixed-effects models (LMEs), separately for males and females, with the random intercept and random slope for age by subject ID included in the model. For each of the six outcomes (BEEPVO2M, BFAT, BMUSC, CMJ, OMBT, SJ), the fixed effects part of the model contained age (modelled as a non-linear association using cubic splines), period (2015-2019, 2020, 2021) as a three-level factor, and year difference (set to 0 for 2019-2021 and to \(-\)1, \(-\)2, \(-\)3, and \(-\)4 for 2018, 2017, 2016, and 2015, respectively), which was modelled as a non-linear association using natural splines to allow extrapolation. A square-root transformation was applied to the outcome variable to stabilize the variance. This model specification assures that only the data from 2015 to 2019 were used to estimate the general (age-adjusted) trend. This trend was then used to make predictions for the 2020-2021 period. The number and the position of the knots when constructing the splines were set in such a way that the fit of the model was adequate. Correct specification of the two design matrices (i.e., the goodness-of-fit (GoF) analysis) was verified as proposed by [PERSON] [25]: the considered subset F processes included the age (including its respective nonlinear terms) and year difference (including its respective nonlinear terms); note that a significant \(p\)-value indicates an inadequate model fit. It was specified a priori that only  
Figure 1: Overview of COVID-19 lockdowns and training/competition schedule for young tennis players. two contrasts will be considered: the difference between the age-adjusted effect in 2020 (and 2021) and the age-adjusted predicted effect in 2020 (and 2021). Positive values of the contrast suggest that the age-adjusted outcome in a particular year (2020 or 2021) was larger than it ought to have been given the age-adjusted, pre-COVID trend; negative values of the contrast suggest that the value was smaller. The family-wise-error rate (FWER) was controlled using the procedure proposed by [PERSON] [26]. Appropriate effect size measures (e.g., Cohen's _d_) were computed (Supplementary Material). A sensitivity analysis was also performed, in which the number of knots in the natural spline for the year difference was varied. Given that these sensitivity analyses did not affect the ultimate inferences, these results are not reported. An exploratory power analysis was conducted by performing a Monte Carlo simulation study. We assumed that the trend is given by using the following equation: \(y=5+0.1x-0.01x^{2},\ x\in\{0,1,\ldots,6\}\), with a normally distributed random error around this trend with a mean of 0 and a standard deviation of 0.5. A study with 300 independent observations pre-COVID (uniformly distributed over the interval \(x\in\{0,1,\ldots,4\}\)) and 30 and 50 observations one and two years after COVID (\(x\in\{5\}\) and \(x\in\{6\}\)), respectively, would have a 95% power to detect a 10% decline from the estimated trend (using only pre-COVID data) one year after COVID (\(x\in\{5\}\)). A study with 225 independent observations pre-COVID (uniformly distributed over the interval \(x\in\{0,1,\ldots,4\}\)) and 20 and 40 observations one and two years after COVID (\(x\in\{5\}\) and \(x\in\{6\}\)), respectively, would have a 85% power to detect a 10% change. Note that the first and second sample sizes are approximately equal to the sample sizes obtained in our study for males and females, respectively. Also, note that the power calculation is conservative since independent observations are assumed (in practice, the power is expected to be larger as we can reasonably assume that any two observations from the same individual are positively correlated). The analysis was performed using R (R version 3.6.3.) [27]. LMEs were fitted using the R package [28]. Basis matrices for splines were generated using the package **splines**. The GoF analysis was performed using the package **gofLMM**. The contrasts were estimated using the package **mulcomp**.