Age and sex effects in physical fitness components of 108,295 third graders including 515 primary schools and 9 cohorts

Children’s physical fitness development and related moderating effects of age and sex are well documented, especially boys’ and girls’ divergence during puberty. The situation might be different during prepuberty. As girls mature approximately two years earlier than boys, we tested a possible convergence of performance with five tests representing four components of physical fitness in a large sample of 108,295 eight-year old third-graders. Within this single prepubertal year of life and irrespective of the test, performance increased linearly with chronological age, and boys outperformed girls to a larger extent in tests requiring muscle mass for successful performance. Tests differed in the magnitude of age effects (gains), but there was no evidence for an interaction between age and sex. Moreover, “physical fitness” of schools correlated at r = 0.48 with their age effect which might imply that "fit schools” promote larger gains; expected secular trends from 2011 to 2019 were replicated.

www.nature.com/scientificreports/ reanalysis). However, the absence of evidence for cross-sectional age differences and their interactions with sex within a single year of life at a prepubertal development stage must not be taken as evidence for their absence 16 , because statistical power may not have been sufficiently large.
Our study included a very large representative sample of 108,295 eight-year old third graders. This large sample allowed us to zoom into a cross-sectional short-term ontogenetic window. In this single year of life, we expected to detect age differences that were not significant in the insets of Fig. 1. Indeed, we expected that the component differences in developmental gains (i.e., the differences in slopes) will be in general agreement with those in the main panel of Fig. 1 because gains should be larger (favouring older children) in tests that do not demand continuous acceleration of the body as with the ball push test vs the 6 min run test 12 . Similarly, as in Fig. 1 (both in the main panel and in the insets) the test-related sex effects (i.e., the differences between the lines) will be larger, the more muscle mass is required to perform a physical fitness test 1,13 (favouring boys) and less for tests involving motor coordination 17 . In other words, in general, we expected to anticipate the long-term ontogenetic longitudinal trends across four years with short-term ontogenetic cross-sectional trends within a single year for younger children in the third grade.
There is one exception to this expectation of agreement between short-term and long-term ontogenetic trends: The only significant interaction between age and sex was reported for powerUP (see right panel in Fig. 1) indicating divergence between boys and girls. If physical fitness components carry an early prepubertal signal in eight-year olds, then we should observe convergence of scores because girls will benefit earlier than boys from the rise of sex hormones 9,11 .
Cross-sectional analysis has been criticized for good reasons and in general preference is given to longitudinal analysis, mostly because only the latter delivers information about intraindividual growth 6 . However, we propose that, when the focus is on short-term ontogenesis, that is on changes within, not between years of life, a cross-sectional design is probably the only option to determine development-related gains because this design circumvents practice effects, a necessary consequence of repeated testing of physical fitness within a year. Another problem of cross-sectional designs are cohort effects (i.e., age-correlated cultural change). However, cohort effects are certainly negligible for children who attend the same grade but differ in age only within the same year of life. Thus, with the exception of loss of information about interindividual differences in intraindividual change, a cross-sectional design is more suitable to determine how within-year developmental profiles differ between health-related (e.g., cardiorespiratory endurance) 18 and skill-related components of physical fitness (e.g., power, speed, coordination) 18 .
Just as there are individual differences in physical fitness between children, there are also differences between the over 500 schools in how much they implement programs that facilitate gains in the development of physical fitness 19 . Explanatory hypotheses about these differences are beyond the scope of this article, but our use of a linear mixed model for statistical inference affords exploratory tests for their presence and adjusts test statistics for school-related sources of variance. Finally, as data collection for this cross-sectional study occurred annually from 2011 to 2019, secular trends are another source of variance in scores that must be taken into account. Here we expected that the results will be in line with recent original research and meta-analyses and show a decline in cardiorespiratory endurance and an increase in speed [20][21][22] .  www.nature.com/scientificreports/ In summary, the hypotheses about age-and sex-related differences in physical fitness tests were tested with a linear mixed model (LMM) that afforded the simultaneous consideration of children, schools, and cohorts as random factors and the estimation of variance components and correlation parameters for (a) test scores, (b) effects of contrasts between the tests, and (c) in the case of schools and cohorts also effects of age-related gains and sex differences. These model parameters serve primarily as measures of statistical control for the fixed effect estimates but may also yield substantive insights about the dynamics of development. For example, for correlations of scores, we expected the usual positive manifold between the tests, but for correlations of effects (i.e., the contrasts between tests) no directed hypotheses were formulated given the usually low reliability associated with difference scores. Although age and sex are between-child factors, they are also both within-school and within-cohort factors, providing us with the opportunity to detect reliable variance components and correlation parameters for these factors. No directed hypotheses were formulated for schools. However, secular trends are well documented, and we expected to replicate them.

Results
Overview. Table 1 displays statistics for fixed effects of age (linear) and sex as well as their interactions with the four test contrasts for LMM m2. Test-specific z-transformations eliminated main effects of contrasts H1 to H4 (all z ≤ 0.99). Neither the age x sex interaction nor any of the interactions of this term with the four test contrasts were significant (all |z|≤ 1.74). Adding quadratic trends of age and their associated interactions to the model, did not significantly contribute to goodness of fit; χ 2 (10) = 9.17, p = 0.52. None of the age x sex interaction was significant when tested separately for the five tests (all |z|≤ 1.04; see Supplement A for details about both control LMMs).
Age-related gains. Figure 2 displays both the gains in physical fitness with age and boys' higher scores than girls' in each of the five physical fitness tests. The parallel lines in each panel also visualize the lack of significant evidence for age x sex as well as age x sex x test interactions.
Despite the small age range, the differences were large and visible for all five physical fitness tests; the overall linear trend for age was significant (z = 31.60). The LMM tested the interactions of age with the four test contrasts, that is whether slopes in neighbouring panels (averaged across sex) were parallel. The left two panels (H1) and Table 1. Fixed-effect estimates of linear mixed model. H1 to H4 = hypothesis 1-4, endurance = cardiorespiratory endurance (i.e., 6 min run test), coordination = star run test, speed = 20-m linear sprint test, powerLOW = power of lower limbs (i.e., standing long jump test), powerUP = power of upper limbs (i.e., ball push test), * = z-value > 3.0, linear mixed model random factors: cohorts (9), schools (515), children (108,295), observations = 525,126 (missing = 3%). For estimates of variance components and correlation parameters see Table 3. www.nature.com/scientificreports/ the right two panels (H4) show that this is clearly not the case-even without a statistical test. Indeed, three of four expected interactions were significant (see second block of Table 1): the age slope was larger for coordination than cardiorespiratory endurance (H1; z = 18.44), larger for coordination than speed (H2; z = − 6.22), and larger for powerUP than powerLOW (H4; z = 25.87). The difference between age slopes for powerLOW and speed (H3) was not significant (z = − 0.69).

Source of variance
Counter to this profile of differences, Fig. 2 which is based on observed scores suggests that speed gain is larger than powerLOW gain. Indeed, this contrast was significant for LMM m1, that is as long as VCs for cohortrelated differences between tests were not in the LMM. The large differences in secular trends are shown in Fig. 4 (below); adjusting for them in the LMM yielded the reported partial effects.
Sex-related effects. The difference between lines in Fig. 2 displays the expected differences between boys and girls for the performance in the five physical fitness tests; the overall sex effect was significant with z = 86.52. The third block of Table 1 lists statistics for the interactions between sex and the tests contrasts. All interaction terms were significant and in agreement with a priori expectations. Boys performed better than girls on cardiorespiratory endurance than coordination (H1; z = − 39.03), better on speed than coordination (H2; z = 12.35), better on powerLOW than speed (H3; z = 11.24), and better on powerUP than powerLOW (H4; z = 43.84). The magnitude of these interactions is shown in Fig. 2 in the differences between the parallel lines for boys and girls for neighbouring panels (averaged over age).

Variance components and correlation parameters. Test scores. Table 2 lists estimates of VCs and
CPs for the five test scores from a re-parameterized version of LMM `m2` with the same goodness of fit and the same estimates for fixed-effects. The test-related VCs were large for children (0.72-0.78), of medium-size for schools (0.24-0.37), and small for cohorts (0.03-0.06). VCs for the age-related gains (slopes 0.10) and the sex effect (0.05) were also small for schools. It is noteworthy that the differences between schools in the age-related gain of their children is larger than the differences between cohorts.
In the first block of Table 2, CPs between tests scores for children are shown in black and the corresponding zero-order correlations (ZOCs) are shown in light grey. The ZOCs are based on 96,529 children with complete test scores; they came from 512 different schools. In the second block, we list corresponding results for schools. For this analysis, we also added the criterion that a school had to report complete data from more than 30 boys and 30 girls to ensure stable estimation of age and sex effects within schools. This criterion left us with 421 schools, 93,661 children, and 468,305 scores.
There were three noteworthy patterns of results. First, as expected, all child-and school-related CPs and ZOCs between test scores were positive. Thus, the five tests represent a latent construct "physical fitness" both for differences between children and for differences between schools. This was also supported by a random-effects principal component analyses (rePCA) of the two orthogonal random-effect structures. The first principal component (PC1) loadings ranged from 0.49 to 0.35 for the child-related PC1 and from 0.45 to 0.29 for the school-related PC1, accounting for 65% and 38% of the respective variances (see Supplement A for details). www.nature.com/scientificreports/ Second, the tests did not correlate equally highly with each other. Most notably, the child-related CPs of cardiorespiratory endurance, coordination, speed, and powerLOW correlated very highly between 0.56 and 0.77, but their correlations with powerUP were distinctly smaller (CPs: 0.25-0.50). This observation holds also for the other three correlation matrices, but overall correlations were smaller (see Table 2 for all CPs).
Again, this interpretation was supported by rePCAs (see Supplement A for details). The smallest loading on child-and school-related PC1s was obtained for powerUP (0.35, 0.29). Moreover, for children the loadings for the second PC2s (15%) represented the difference between cardiorespiratory endurance (0.53) and powerUP (− 0.84). Similarly, for schools the third PC3 (12%) represented the difference between the average of cardiorespiratory endurance (0.31) and coordination (0.47) and powerUP (− 0.71). We take this PC2/PC3-based difference score as support for the hypothesis that powerUP favoured heavier children for who strength of arms may mask reduced cardiorespiratory endurance and coordination (see Discussion).
Third, child-related CPs ( Table 2, upright numbers) were larger than child-based ZOCs ( Table 2, numbers). This was a rather striking pattern because one might expect the opposite given that ZOCs were confounded with large effects of age and sex. Conversely, CPs were larger despite adjustment for sex and age differences in the fixed effects and for differences due to schools and cohorts in the random-effect structure of the LMM. The reason for the result is LMM-based shrinkage of conditional means of the units of the random factors in the direction of the GM. Thus, the entire data set was used to "correct" unreliable observations, also called "borrowing strength" to improve predictions. Due to this shrinkage correction for unreliability, CPs revealed the latent relations between measures much more clearly than ZOCs.
Effects of test contrasts. In the random-effect structure of LMM m2, estimates were returned for child-, school-, and cohort related VCs for GM and the four test contrasts; VCs of age and sex were also estimated for school. CPs for child and school reflect correlations between the contrasts (i.e., effect correlations). The results are shown in Table 3. As in Table 2, CPs are shown in upright numbers and corresponding ZOCs are italics.
VCs for test contrasts were larger (0.47-0.72) for children and somewhat smaller, but still highly reliable (0.30-0.40) for schools, especially when compared to VCs estimated for school-related age (0.10) and sex (0.05) effects, and especially when compared to cohort-related effects (0.04-0.09). CPs and ZOCs of effects are smaller than CPs and ZOCs based on test scores because, with the exception of those involving GM, they are all based on difference scores.
There were two results of theoretical relevance. First, there was a negative CP for r GM.H4 for children (− 0.31): If we invert the difference score to convert to a positive correlation, then large values of GM correspond to a large www.nature.com/scientificreports/ difference between powerLOW and powerUP. In other words, the larger powerLOW relative to powerUP, the larger is the expected value for GM. Thus, in line with the special status of powerUP reported above, powerUP is better thought of as an adjustment or sharpening of physical fitness as indicated by PowerLOW than as a genuine indicator of physical fitness by itself. The corresponding ZOC was only − 0.13. Second, we note three large negative CPs (r H1.H2, r H2.H3, and r H3.H4). However, these correlations are ambiguous because the contrasts had a test in common (i.e., coordination is part of H1 and H2; speed is part of H2 and H3; powerLOW is part of H3 and H4).
Third, the largest effect CP was observed between age and GM for schools (+ 0.48) suggesting that "fitter" schools promote more developmental change across the school year. Figure 3A displays a visualization of the CP using a scatterplot of the conditional means of age-related gain over conditional means of GMs of physical fitness for the 515 schools.
The scatterplot is not identical with the CP, in fact the correlation is 0.75, because conditional means are "predictions" of the school age effect and GM using the school data and all model parameters to correct for unreliability in the scores. Indeed, as shown in Fig. 3B, there is no evidence for this relation in the uncorrected scatterplot corresponding to the ZOC of 0.19. A significant CP corresponds to a simple interaction in the data and this interaction can be visualized by sorting schools into those of high and low physical fitness (split at z-score = 0) and those with a high and small gain for their children in the third grade (split at the age slope of + 0.25). In an ANOVA of schools' observed mean physical fitness values the interaction between these two post-hoc grouping factors was significant; F(1, 417) = 4.22, MSe = 0.012, p < 0.05. Of course, from a correlation we cannot infer the direction of causality. "Fitter schools" (e.g., schools offering extracurricular sport-related activities) may facilitate gains in children's fitness. Alternatively, if we assume that fitter children gain more in a year, then a school's high fitness as well as the associated large gain could be the result of being attended by fitter children (e.g., due to the school's location in a high-SES region).
Cohort-related variance components. The small, but significant VCs related to the random factor cohort indicate that there were reliable test x cohort interactions; they are shown in Fig. 4. Across the nine years from 2011 to 2019 there was a performance decline for cardiorespiratory endurance and an increase for speed. The other three components exhibit an initial increase followed by a decline of performance in recent years.  Table 3). These tests did not reveal any problems.

Discussion
The aim of this study was to examine short-term ontogenetic cross-sectional developmental differences in physical fitness for five tests tapping health-and skill-related components of physical fitness in a large sample of 108,295 German eight-year old children. Even in a single prepubertal year of life (1) performance increases linearly with chronological age in all physical fitness tests, (2) boys outperform girls in all physical fitness tests with sex differences being larger for tests requiring muscle mass and being smaller for tests requiring motor coordination, (3) the tests differ, mostly as expected, in the size of age and sex effects, (4) four of the five tests represent a common construct (i.e., correlate strongly positively with each other)-the exception is the ball push test that requires powerUP, (5) "physically fit schools" apparently promote more developmental gains within a year (but this is only www.nature.com/scientificreports/ correlational evidence) and (6) diverging secular trends for cardiorespiratory endurance (negative) and speed (positive) are in agreement with other research and meta-analyses. Furthermore, there was no evidence for an interaction of age and sex-with each other or with the test contrasts despite an abundance of statistical power. Counter to folklore, a very large data set (i.e., 108,295 subjects and 525,126 observations) did not "automatically" render everything significant! Given this statistical power, we are strongly inclined to interpret the absence of evidence for interactions as evidence of absence of interactions for these five tests of physical fitness 16 . Boys outperformed girls in all four physical fitness components (cardiorespiratory endurance, coordination, speed, power [LOW/UP]). This is in line with other studies reporting normative values [2][3][4][5]14,[23][24][25][26] . For instance, Tambalis et al. 5 reported that boys aged 6-18 years showed significantly better performances for cardiorespiratory endurance (i.e., 20 m shuttle run test), powerLOW (i.e., standing long jump test), and agility (i.e., 10 × 5 m agility shuttle run test) compared to girls. Furthermore, De Miguel-Etayo et al. 3 described that boys aged 6-10 years significantly outperformed girls in cardiorespiratory endurance (i.e., 20 m shuttle run test), speed (i.e., 40 m sprint test), and powerLOW (i.e., standing long jump test).
What are the reasons for the observed large test-specific sex differences? Sex-related differences in body composition appear to be likely candidates to account for the observed findings. The larger sex effect in pow-erUP compared to powerLOW can be explained by a better proportion of strength relative to body mass especially in the upper limbs in boys compared to girls 27,28 . The large difference in cardiorespiratory endurance can be explained by physiological factors such as boys' better mechanical efficiency and fractional utilisation of oxygen 2,29 . Furthermore, muscle mass 1 and muscle cross-sectional area 13 favour boys especially in physical fitness tests that recruit muscle mass. Beside these anthropometric factors and physiological demands, sociocultural aspects may also explain the sex difference. Haywood and Getchell 30 reported that girls usually participate in sports that require balance and flexibility (e.g., gymnastics, figure skating) compared to boys who rather participate in strength-related activities.
The sex effect was also significantly stronger for powerLOW than for speed. PowerLOW is determined much more by muscle mass where boys usually outperform girls 2,3,5 . In contrast, speed is less influenced by muscle mass than by motor coordination where sex differences are comparatively small 31 or were not found at all 17 . Therefore, the sex effect in powerLOW might be larger than in speed. Obviously, the demand of coordination relative to power and cardiorespiratory endurance is even larger in the star run test than in the 20-m linear sprint test [32][33][34] and this could be a reason why the sex effect is smaller for the star run test than for speed.
The observed decrease of sex differences in performances with increased demands for coordination seems to be plausible. Coordination draws on many different central (e.g., brain) and peripheral sites (e.g., motor units) within the nervous system 35 . The more a test engages the brain, the less relevant sex is a performance limiting factor. In summary, the decrease of size of sex effects across tests can most likely be explained by a decrease in the relevance of muscle mass (favouring boys over girls) 1 and an increase in the relevance in motor coordination (associated with small or no sex effects) 17,31 .
Within their ninth year of life, older children significantly outperform younger ones in all five fitness tests. This was expected 3,5,14 , but there were no significant gains for a sample of 240 children in the tenth year of life (see insets of Fig. 1). The absence of evidence for nonlinear relations between performance and chronological age as well as the similarity of gains for boys and girls are quite remarkable (see Fig. 2), especially in light of what long-term ontogenetic longitudinal research reveals when children are a few years older. Obviously, the current test battery does not detect the onset of puberty in the ninth year of life! Nevertheless, the different age-related grains for tests are compatible with what is known about the development of basic physiological parameters: Growth of body mass and height positively influence the performance in skill-related physical fitness components of coordination, speed and power. In contrast, increases in body mass have a negative impact on performance in tests such as cardiorespiratory endurance (e.g.., 6 min run test) where the own body mass has to be accelerated continuously. The smaller age effect for cardiorespiratory endurance compared to skill-related components to the Points are observed means with 95% CIs. Lines are third-order polynomial trends fitted to children's scores along with 95% error. Note the much smaller range of the y-axis (i.e., from − 0.10 to + 0.10) compared to age effects shown in Fig. 2 www.nature.com/scientificreports/ remaining physical fitness components is in line with a study conducted by de Miguel-Etayo et al. 3 . The authors could not find significant age differences in cardiorespiratory endurance (i.e., 20 m shuttle run test) but amongst other in powerLOW (i.e., standing long jump test) and strength (i.e., absolute handgrip strength test) in a sample of children aged 6-10 years. Similar, Viru et al. 36 stated that an accelerated improvement in cardiorespiratory endurance occurs at the ages of 11-15 years in boys and 11-13 years in girls. As far as the differential age effects between the components of physical fitness are concerned, tests of coordination, speed, and powerLOW share the highest correlations among the five tests. These three tests share the relevance of muscle mass yielding power, but they differ in the relevance of coordination. As mentioned above, the sex effect within these three tests (i.e., star run test < 20-m linear sprint test < standing long jump test; see differences between lines in Fig. 2) is in line with their ranking on coordination. For age-related gains within the school year, the partial effects yielded star run test > 20-m linear sprint test and standing long jump test, corresponding (roughly) to their ranking on motor coordination. The special status of powerUP (i.e., ball push test) is not only evident with respect to its lower correlations with other tests, but also with respect to the size of the age effect-by far the largest of the five tests (see Fig. 2).
Obviously, the performance in powerUP was influenced by factors other than physical fitness. We propose that body mass contributes to performance in tests that assess powerUP. Heavier children usually have a higher muscle mass compared to normal weighted children 37 . While a higher muscle mass positively influences performances in non-weight-bearing tests (e.g., ball push test), it negatively influences performances in weight-bearing tests (e.g., standing long jump test) because the body mass has to be accelerated in contrast to non-weight-bearing tests 12 . Therefore, tests for the assessment of powerUP favour heavier children for whom strength of arms and trunk may mask reduced cardiorespiratory endurance and coordination. Thus, a high score for powerUP may be more indicative of a lack of overall physical fitness because it does not measure physical fitness to the same degree than the other four tests (i.e., small correlations with the other four tests and GM). Our results suggest that the best indicator of physical fitness is the average of the first four tests (cardiorespiratory endurance, coordination, speed, and powerLOW). Analyses including body mass (not measured in the present study) could support the interpretation of the special status powerUP in the assessment of physical fitness. The linear mixed model included school as a second random factor, supported the estimation of correlation parameters for test scores and age effects, and revealed a strong correlation (r = 0.48) between the age effect and overall physical fitness. The direction of causality is not clear, but in line with Hattie 19 and García-Hermoso et al. 38 . The results are in agreement with the hypothesis that schools differ in how much they qualitatively promote the development of children's physical fitness 38 . The development of physical fitness depends on the quality of the physical education lessons and especially on the quality of the physical education teacher teaching physical education 38 . These school differences in the age effect of physical fitness within the third grade were strong enough to yield reliable differences in the overall physical fitness of their children.
Our study is not without limitations. First, the five tests do not cover all components of physical fitness 18 and there are alternative tests for each component. Although our results do not necessarily generalize to other components like muscle strength, muscle endurance, or balance 18 , they represent those for whom an early detection of puberty was most likely. Second, from a cross-sectional study we cannot know whether the linear gains for tests hold at the individual level or are the result of averaging over individual differences in non-linear growth curves. Obviously, high-density monitoring within a year would be desirable, but such longitudinal data are not without their own problems. For example, learning effects could be reduced due to the completion of at least three familiarization sessions 39 to separate learning effects from growth. However, motivational factors may also play a role. The longitudinal cardiorespiratory endurance data in Fig. 1 suggest no further growth or even a decline in performance for 12 year old children 6 . This is obviously not in agreement with what we know about the objective development of cardiorespiratory endurance 2,4,5 . Third, divergence between cross-sectional and longitudinal profiles is usually cultural change (i.e., cohort effects). Data were accumulated from 2011 to 2019, which is long enough for cohort effects to materialize. The variance of between cohort differences in physical fitness was by far the smallest source examined in this study. Fourth, several important covariates (e.g., physical activity, sedentary behaviour, socioeconomic indicators) which affect physical fitness were not assessed but could provide an additional insight into the development of physical fitness. Fifth, physical fitness is highly related to biological maturity and more mature youth outperform less mature youth in physical fitness 40 . Our study indicates the strongest test of an early detection of puberty-related divergence with physical-fitness tests that we are aware of. We conclude that puberty-related development has not started or is not strong enough yet in eight-year old children to be picked up with physical fitness or some of its components. A joint analysis with anthropometric measures (i.e., body mass, body height) and status of biological maturity (e.g., peak height velocity, secondary sex characteristics) will allow a stronger test of the hypothesis that the onset of puberty can already be detected in the ninth year of life.
To sum up, the short-term ontogenetic results of this cross-sectional study revealed test-specific age and sex effects, but no interaction between age and sex despite an abundance of statistical power. According to Ortega et al. 14 physical fitness data of an individual should be compared with reference values of a sex and age-matched similar general population. Such norms are usually only available on annual or semi-annual basis. Our results suggest that physical education teachers, coaches, or researchers can use a proportional adjustment to adequately evaluate physical fitness of prepubertal school-aged children. Furthermore, especially muscle strength / powerUP should be promoted in sport activities for girls in order to reduce the large sex difference between boys and girls. Lastly, physical education teachers, coaches, or researchers should be careful in the interpretation of the ball push test and should consider that it does not measure physical fitness to the same degree than the other tests. We started with data from 144,045 children. Of those, we included only healthy children who had been enrolled within the legal key date of the Federal State of Brandenburg, that is in a given year of school enrolment they were at least 6.00 and at most 6.99 years old on September 30 th and, therefore, varied between 8.00 and 8.99 years in the third grade (n = 110,669). In addition to early-entry (n = 2,664), late-entry (n = 30,457) and children without information about birthdate (n = 255), we did not include children with signs of emotional (e.g., autism) and/or physical disorders (e.g., disabilities like infantile cerebral palsy) that were evaluated by the responsible and experienced physical education teacher based on a medical clearance (n = 28). After the first iteration, the LMM-based conditional means of the random effects identified one school as an extreme outlier on several tests across the years and the 171 children of that school were excluded as well. Finally, we applied a ± 3 SD criterion to individual test scores which led to the exclusion of another 2,175 children (2%). This left us with 108,295 children (i.e., 75% of those tested) from 515 different schools.
Physical fitness tests. Physical  Qualified physical education teachers of each school administered the tests according to standardized test protocols during the regular physical education classes in the participating schools (www. uni-potsd am. de/ en/ emoti kon/ proje kt/ metho dik for further information on the test protocols). Teachers were instructed in a standardized assessment through an advanced training. Tests were always conducted in the morning between 8 and 12 o' clock. Encouragement to achieve the best performance was permitted. Prior to testing, all third-graders performed a standardized warm-up program consisting of different running exercises (e.g., side-steps) and small games (e.g., playing tag).
Cardiorespiratory endurance. Cardiorespiratory endurance was assessed with the 6 min run test. Children had to run as far as they could within six minutes around an official volleyball field (9 × 18 m, every 9 m a pylon/ marker was set beside the running court [i.e., six pylons around the field]) at a self-paced velocity. Split time was given every minute. The maximal distance achieved during the six minutes in meters to the nearest ninemeters marker was used as dependent variable in the analysis. The 6 min run test was reliable (test-retest) in children aged 7-11 years with an intraclass correlation coefficient (ICC) of 0.92 42 . The 6 min run test correlated at r = 0.69 (p < 0.01) with VȮ 2max assessed via a gas analysis during a progressive treadmill test in children aged 9-11 years 43 .

Coordination.
Coordination under time pressure was tested with the star run test (see Fig. 5). Children had to complete a parkour with different movement directions and movement forms (i.e., running forward, running backward, side-steps to the left side, side-steps to the right side). The parkour had to be performed in a given order over a 9 × 9 m star-shaped area where each of the four spikes is marked by a pylon. After starting in the centre of the star, children had to complete the parkour as fast as they could by running in every movement form two times within the given sequence. They had to touch each pylon with the hand. The whole covered distance is 50.912 m. The faster of two test trials was used in the analysis. The shortest time for completing the parkour in seconds to the nearest 1/10 s was measured using a stopwatch and was used as dependent variable in the analysis. The star run test was reliable (test-retest) in 8-10 year old children with an ICC of 0.68 44 .
Speed. Speed was assessed with the 20-m linear sprint test. After an acoustic signal, the children had to sprint out of a frontal erect posture as fast as they could over a distance of 20 m for two times; the faster test trial was used in the analysis. The shortest time for sprinting the 20 m in seconds to the nearest 1/10 s was measured using a stopwatch and was used as dependent variable in the analysis. The 20-m linear sprint test was reliable (test-retest) in children aged 7 to 11 years with an ICC of 0.90 42 .
Power of lower limbs (PowerLOW). PowerLOW was tested using the standing long jump test. Out of a standing frontal posture the children had to jump as far as they could. The participants had to land with both feet together. They were allowed to swing their arms prior to and during the jump, but after landing the hands were not allowed to touch the floor. The distance in meters to the nearest one centimeter between toes at take-off and For each test, we determined the ± 3 SD boundary separately for boys and girls. Measurement outside these boundaries were usually implausible (i.e., recording errors) or extreme outliers. They were treated as missing values (3%). Finally, we converted scores within tests (aggregated over boys and girls) to z-scores to facilitate comparison of test, age and sex effects.
Statistical inference was based on a linear mixed model (LMM) estimated with the MixedModels package 49 in the Julia programming language 50 . The LMM included child (N = 108,295), school (N = 515), and cohort (N = 9) as three random factors; the total number of observations (i.e., max = 5 per child) was 525,126. Fitted model objects were processed with random-effects principal component analysis to obtain loadings of the variance-covariance matrix of the random effects and facilitate its interpretation.
As fixed effects, we specified four sequential-difference contrasts for the five tests: (H1) coordination vs. cardiorespiratory endurance, (H2) speed vs. coordination, (H3) powerLOW vs. speed, and (H4) powerUP vs. powerLOW. Also included were the effect of age (centered at 8.5 years) as a second-order polynomial trend, the effect of sex (boys-girls), and all interactions between contrasts, age, and sex. Given the large number of observations, children, and schools, we adopted a two-sided z-value > 3.0 as significance criterion for the interpretation of fixed effects.
Child, school, and cohort were included as random factors. With three random factors there was a need for selecting a random-effect structure that included theoretically relevant and reliable variance components (VCs) and correlation parameters (CPs), but was also still supported by the data (i.e., was not overparameterized). Tests varied within children, schools, and cohorts; age and sex varied between children, but within schools and within cohorts. Therefore, in principle, VCs and CPs of linear effects of age and sex could be estimated for schools and cohorts, but not for children.
Parsimonious model selection occurred in two major steps without knowledge or consideration of fixed-effect estimates 51 ; details are provided in Supplement A. We started with a model including Grand Mean (varying intercepts) for all three random factors and, given the large numbers of 108,926 children and 515 schools and the small number of nine cohorts, included also test-related VCs and CPs for child and school and age-related and sex-related VCs and CPs for school, but not for cohort. This LMM m1 was well supported by the data. In the second major step, we increased the complexity of the random-effect structure for cohort by adding test-related VCs (LMM m2), then test-related CPs (LMM m3), and finally age-and sex-related VCs and CPs (LMM m4). www.nature.com/scientificreports/ LMM m4 was not supported by the data (i.e., the fit was singular) and did not significantly improve the goodness of fit over LMM m3; delta χ 2 (13) = 14.11, p = 0.37. LMM m3 improved the goodness of fit over LMM m2 according to the likelihood ratio test, χ 2 (10) = 48.45, p < 0.001, but not when the increase in model complexity is penalized according to LMM m2 = 0,00012345 and LMM m3 = 0,00012345). As we had no directed hypotheses relating to test-related CPs for the factor cohort, we stayed with LMM m2 which represented a very large improvement in goodness of fit relative to LMM m1; χ 2 (4) = 1489.57, p < 0.001. We also estimated LMM m2 with two alternative parameterizations that did not change the goodness of fit, but yielded information about CPs between test scores instead of test effects (i.e., contrasts). Finally, we fitted two control LMMs to test the significance of quadratic age trends for fixed effects and the absence of evidence for sex x age interactions separately for each fitness component (i.e., nested within the five levels of the factor test).

Data availability
The datasets generated and analysed during the current study as well as Julia and R scripts are available in the Open Science Framework (OSF) repository : https:// osf. io/ 2d8rj/.