5  Analyses of maturity-related effects on physical fitness

Code 
suppressWarnings(suppressMessages(library(tidyverse)))
suppressWarnings(suppressMessages(library(lme4)))
suppressWarnings(suppressMessages(library(ggplot2)))
suppressWarnings(suppressMessages(library(sjPlot)))
suppressWarnings(suppressMessages(library(performance)))
suppressWarnings(suppressMessages(library(flextable)))
suppressWarnings(suppressMessages(library(remef)))

load(file="data/info.rda")
load(file="data/ages.rda")
load(file="data/emo.rda")

data <- info |>
  left_join(ages) |>
  left_join(emo, multiple = "all") |>
  mutate(across(where(is.character), as.factor),
         gender=factor(gender, levels = c("girl","boy")),
         m1rg=(scale(age)+scale(m_mirwald))/2,
         m1gd=(scale(age)-scale(m_mirwald)),
         m2rg=(scale(age)+scale(m_moore))/2,
         m2gd=(scale(age)-scale(m_moore))) |> 
  select(Child, gender, Test, zScore,  m1rg, m1gd, m2rg, m2gd) |> 
  fill(m1rg:m2gd)

cbPalette <- c( "#0072B2", "#D55E00", "#009E73", "#CC79A7",
                "#F0E442", "#56B4E9", "#999999", "#E69F00")
                
tab_to_flex <- function(tab,x){
tab2 <- tab |> 
  sjtable2df::mtab2df(n_models = 1) |> 
  mutate(p=ifelse(p==Estimates,"",p))
b <- unlist(suppressWarnings(tab2 |> select(p) |> mutate(p = ifelse(p=="<0.001",1,ifelse(as.numeric(p)<0.05,1,0)))))
r <- tab2[1,which(tab2$Predictors=="Random Effects")]
l <- length(b)
flex <- tab2 |> 
  rename(Est.="Estimates",
         SE="std. Error",
         z="Statistic") |> 
  flextable() |> 
  autofit() |> 
  bold(j=5,i=which(b==1))|> 
  bold(j=1,i=r) |> 
  add_footer_row(values = paste0(x,"Est. = estimate; SE = standar error"), colwidths  = 5) |> 
  hline(i = r-1, part = "body") |> 
  width(j=1,width = 4) |> 
  width(j=2:5, width = .5) |>
  fontsize(size=10, part = "all") |> 
  height_all(0.5, part = "all", unit = "cm")
return(flex)
}

5.1 Introduction

Finding effects of body height and body mass on physical fitness raises the question how growth-related changes are associated with different components of physical fitness. To elucidate these effects, this chapter analyses the effects of age and maturity offset (Mirwald et al., 2002; Moore et al., 2015) on physical fitness in children with deficits in their physical fitness.

5.2 Statistical analysis and model fitting

Effects of maturity offset, age, and gender were analysed for four physical fitness tests, namely 6 min run, star run, 20m sprint, and standing long jump (Fühner et al., 2021). Maturity offset was estimated using formulas by Mirwald et al. (2002) and Moore et al. (2015). Details of data pre-processing of physical fitness data can be found in Section 8.1.4 and of maturity offset data in Section 8.1.2. To avoid collinearity for age and maturity offset (since maturity offset is calculated using age, see Section 8.1.2), the first and second unrotated principal components were calculated based on z-scores of these variables (Hotelling, 1933) (for an application, see (Kliegl, 1982)) to achieve uncorrelated constructs (Section 8.4.1). The first principal component is calculated as mean of z-scores for age and maturity offset. This proxies relative growth, as higher values represent older children who are also closer to their individual growth spurt, while lower values represent younger children further away from it. The second principal component is calculated as difference between age and maturity offset z-scores, which represents delay in growth relative to chronological age. Positive values therefore represent children with greater skeletal growth delays, while negative values represent children with advanced skeletal growth.

The analyses involved two separate LMMs, one for each estimate of maturity offset. In both LMMs, physical fitness tests, relative growth, growth delay, and gender were included as fixed effects. Using LRTs and a parsimonious model selection approach (Bates et al., 2018), the best fitting LMMs including Test, relative growth, growth delay, and gender were selected and reported. Child was included as a random factor. Successive difference contrasts were set for the variables gender and Test. Residual-based diagnostics (e.g., q-q plot, standardized residuals over fitted values) were carried out for both final LMMs to assess model performance and are reported in Section 5.3.1.3 and Section 5.4.1.3. Means and standard deviations are reported in the Appendix Section 8.5.4.

The process of LMM selection for both models is documented in Section 8.5.2.1 and Section 8.5.3.1. The final LMMs included Test, relative growth, growth delay, and gender as fixed effects, with relative growth, growth delay, and gender specified as nested within the factor levels of Test. Child was included as a random factor with relative growth and growth delay further added as individual variance components.

5.3 Model 1: Effects of relative growth, growth delay, and gender on physical fitness (maturity offset estimate with Mirwald formula)

Code 
dat1 <- data |> 
  subset(Test %in% c("Run", "Star_r", "S20_r", "SLJ")) |>
  select(Child, gender, m1rg, m1gd, Test, zScore) |> 
  mutate(Test=factor(Test, levels = c("Run", "Star_r", "S20_r", "SLJ"))) |> 
  na.omit()

contrasts(dat1$Test)   <- MASS::contr.sdif(4)
contrasts(dat1$gender) <- MASS::contr.sdif(2)

m3 <- lmer(zScore ~ 0 + Test/(m1rg + m1gd + gender) + (1 + m1rg + m1gd | Child) , data = dat1, REML = FALSE, control = lmerControl(calc.derivs = FALSE))

5.3.1 Results

5.3.1.1 Fixed effects

The LMM revealed a significant positive effect of relative growth on performance for star run (β = .32; SE = .06; z = 5.34; p < .001), 20 m sprint (β = .13; SE = .06; z = 2.21; p = .027), and standing long jump performances (β = .24; SE = .06; z = 4.03; p < .011). No significant effects were found for 6 min run performance (β = -.11; SE = .06; z = -1.8; p = .071), however, z-values indicate a negative trend.

Positive growth-delay effects were found for 6 min run (β = .54; SE = .13; z = 4.24; p < .001), 20 m sprint (β = .38; SE = .13; z = 3; p = .003), and standing long jump (β = .33; SE = .13; z = 2.6; p = .009), with better performances in children with a larger growth delay (i.e., smaller maturity offset score relative to their chronological age).

Gender-related differences were found for 6 min run (β = -.75; SE = .17; z = -4.49; p < .001), and 20 m sprint (β = -.55; SE = .17; z = -3.29; p = .001), with better performances in girls compared to boys.

Results, also including those for other tests, are reported in Table 5.1 and shown in Figure 5.1.

Code 
p1_pc1<- dat1 |> 
  mutate(Test=case_when(Test=="Run"~"6 min run",
                        Test=="Star_r"~"star run",
                        Test=="S20_r"~"20 m sprint",
                        Test=="SLJ"~"standing long jump"),
         Test=factor(Test, levels= c("6 min run", "star run", "20 m sprint", "standing long jump")),
         gender=ifelse(gender=="girl", "girls", "boys"),
         gender=factor(gender, levels = c("girls", "boys"))) |> 
  ggplot(aes(y = zScore, x = m1rg, group = gender, colour = gender)) +
  geom_smooth(method = "lm") +
  facet_grid(.~Test, scales="free") +
  ylab("") +
  xlab("relative growth [mean ± SE]") +
  theme_bw() +
  theme(legend.position = "none") +
  scale_colour_manual(values = c(cbPalette[3:4]))

p1_pc2 <- dat1 |>
  mutate(Test=case_when(Test=="Run"~"6 min run",
                        Test=="Star_r"~"star run",
                        Test=="S20_r"~"20 m sprint",
                        Test=="SLJ"~"standing long jump"),
         Test=factor(Test, levels= c("6 min run", "star run", "20 m sprint", "standing long jump")),
         gender=ifelse(gender=="girl", "girls", "boys"),
         gender=factor(gender, levels = c("girls", "boys"))) |> 
  ggplot(aes(y = zScore, x = m1gd, group = gender, colour = gender)) +
  geom_smooth(method = "lm") +
  facet_grid(.~Test, scales="free") +
  ylab("") +
  xlab("growth delay [mean ± SE]") +
  theme_bw() +
  theme(legend.position = "bottom") +
  scale_colour_manual(values = c(cbPalette[3:4]))

gridExtra::grid.arrange(p1_pc1, p1_pc2, heights=c(3.5,4), left = "z-score [±SE]", top = "Maturity offset - Mirwald")
Figure 5.1: Plots of overall effects for relative growth, growth delay, and gender [Mirwald]
Code 
tab41 <- tab_model(m3, digits=2, digits.re=2, show.se=TRUE, show.stat=TRUE, show.ci=FALSE,CSS = css_theme("cells"),
                   pred.labels = c("Test [Run]", "Test [Star_r]", "Test [S20_r]", "Test [SLJ]", "Test [Run] : m1rg", "Test [Star_r] : m1rg", "Test [S20_r] : m1rg", "Test [SLJ] : m1rg", "Test [Run] : m1gd", "Test [Star_r] : m1gd", "Test [S20_r] : m1gd", "Test [SLJ] : m1gd", "Test [Run] : gender2-1", "Test [Star_r] : gender2-1", "Test [S20_r] : gender2-1", "Test [SLJ] : gender2-1"))

tab_to_flex(tab41, "Run = 6 min run, Star_r = star run, S20_r = 20 m sprint, SLJ = standing long jump, m1rg = relative growth (Mirwald) [z-score], m1gd = growth delay (Mirwald) [z-score], ")
Table 5.1: Effects of relative growth, growth delay, and gender on physical performances [Mirwald]

Predictors

Est.

SE

z

p

Test [Run]

-0.81

0.08

-10.64

<0.001

Test [Star_r]

-0.54

0.08

-7.07

<0.001

Test [S20_r]

-0.76

0.08

-10.03

<0.001

Test [SLJ]

-0.63

0.08

-8.33

<0.001

Test [Run] : m1rg

-0.11

0.06

-1.80

0.071

Test [Star_r] : m1rg

0.32

0.06

5.34

<0.001

Test [S20_r] : m1rg

0.13

0.06

2.21

0.027

Test [SLJ] : m1rg

0.24

0.06

4.03

<0.001

Test [Run] : m1gd

0.54

0.13

4.24

<0.001

Test [Star_r] : m1gd

0.14

0.13

1.15

0.251

Test [S20_r] : m1gd

0.38

0.13

3.00

0.003

Test [SLJ] : m1gd

0.33

0.13

2.60

0.009

Test [Run] : gender2-1

-0.75

0.17

-4.49

<0.001

Test [Star_r] : gender2-1

-0.11

0.17

-0.69

0.493

Test [S20_r] : gender2-1

-0.55

0.17

-3.29

0.001

Test [SLJ] : gender2-1

-0.29

0.17

-1.76

0.079

Random Effects

σ2

0.39

τ00Child

0.19

τ11Child.m1rg

0.08

τ11Child.m1gd

0.31

ρ01

-0.06

0.51

ICC

0.33

N Child

76

Observations

1413

Marginal R2 / Conditional R2

0.150 / 0.430

Run = 6 min run, Star_r = star run, S20_r = 20 m sprint, SLJ = standing long jump, m1rg = relative growth (Mirwald) [z-score], m1gd = growth delay (Mirwald) [z-score], Est. = estimate; SE = standar error

5.3.1.2 Variance components and correlation parameters

Reliable individual variance components were found for relative-growth and growth-delay effects. A positive correlation was found between grand mean and growth delay effect (r = .51), showing children with better physical fitness also have a larger effect associated with growth delay. A negative correlation was found between relative-growth and growth-delay effects (r = -.74), revealing children with larger relative-growth effects tend to have smaller effects associated with growth delay. Results are reported in Table 5.2 and depicted in Figure 5.2.

Code 
p1_cor1 <- ranef(m3)$Child |> 
  data.frame() |> 
  ggplot(aes(x=X.Intercept., y=m1gd)) +
  geom_point()+
  geom_smooth(method = "lm", colour = "black", se=F) +
  ylab("grand mean") +
  xlab("growth delay effect") +
  theme_bw() 

p1_cor2 <- ranef(m3)$Child |> 
  data.frame() |> 
  ggplot(aes(x=m1rg, y=m1gd)) +
  geom_point()+
  geom_smooth(method = "lm", colour = "black", se=F) +
  ylab("relative growth effect") +
  xlab("growth delay effect") +
  theme_bw() 

gridExtra::grid.arrange(p1_cor1, p1_cor2, nrow = 1)
Figure 5.2: Correlations of grand mean and growth delay effect, and relative growth effect and growth delay effect in individual variance components [Mirwald]
Code 
vc1 <- VarCorr(m3)

as.data.frame(vc1, order="lower.tri") |> 
  filter(is.na(var2)==T) |> 
  select( -var2) |> 
  left_join(as.data.frame(vc1, order="lower.tri") |> 
              select(-vcov) |> 
              pivot_wider(names_from = var2, values_from = sdcor) |> 
              select(-c(grp,var1,`NA`)) |> 
              filter_all(any_vars(!is.na(.))) |> 
              t() |> 
              as.data.frame() |> 
              rownames_to_column(var="var1")) |> 
  mutate(across(where(is.numeric), \(x) round(x, 3)),
         var1=ifelse(var1=="(Intercept)","grand mean",var1)) |> 
  rename("Groups"=grp, "Name"=var1, "Variance"=vcov, "SD."=sdcor, "Corr"=V1) |> 
  flextable() |> 
  autofit() |> 
  add_footer_row(values="m1rg = relative growth (Mirwald), m1gd = growth delay (Mirwald)", colwidths = 6) |> 
  set_header_labels(V2 = "")
Table 5.2: Variance components and correlation parameters [Mirwald]

Groups

Name

Variance

SD.

Corr

Child

grand mean

0.191

0.437

Child

m1rg

0.077

0.278

-0.062

Child

m1gd

0.314

0.560

0.506

-0.752

Residual

0.390

0.624

m1rg = relative growth (Mirwald), m1gd = growth delay (Mirwald)

5.3.1.3 Model diagnostics

Diagnostic checks of LMM residuals are plotted in Figure 5.2 and did not reveal any violations of LMM requirements.

Code 
dat1$resid  <- resid(m3)
dat1$sresid <- resid(m3)/sd(resid(m3))
dat1$fitted <- fitted(m3)

p1_qq <- dat1 %>% 
   ggplot(aes(sample = sresid)) + 
   stat_qq(size=.1, color="red") +
   stat_qq_line(color="blue") +
   geom_abline(intercept=0,slope=1) + 
   xlab("Standard normal quantile") + 
   ylab("Standardized LMM residual") + 
   coord_fixed(ratio=1, xlim = c(-3,+3), ylim=c(-3,3)) + theme_bw()

p1_f <- dat1 %>%  
  ggplot(aes(x=fitted, y=resid)) +   
  facet_grid(.~Test) +
  geom_point(shape=I(".")) +  
  xlab("Fitted values") + ylab("LMM residual") + theme_bw() +
  geom_hline(yintercept=0)  +  geom_density2d(linewidth=1) 

p1_box <- dat1 |> 
  ggplot(aes(resid, Test)) + 
  geom_boxplot() +
  ylab("LMM residual") +
  coord_flip() + theme_bw()

p1_at <- dat1 |>  
  pivot_longer(m1rg:m1gd) |> 
  ggplot(aes(value, resid)) + 
  geom_point(colour="black", size=.1) + 
  geom_hline(yintercept=0) + 
  geom_smooth(method="lm", color="black") + 
  facet_grid(name~Test) +
  ylab("LMM residual") + xlab("z-value") +theme_bw() +
  coord_cartesian(ylim=c(-3,+3))

gridExtra::grid.arrange(p1_qq, p1_f, p1_box, p1_at, nrow = 2)
Figure 5.3: Model performance parameters

5.4 Model 2: Effects of relative growth, growth delay, and gender on physical fitness (maturity offset estimate with Moore formula)

Code 
dat2 <- data |> 
  subset(Test %in% c("Run", "Star_r", "S20_r", "SLJ")) |>
  select(Child, gender, m2rg, m2gd, Test, zScore) |> 
  mutate(Test=factor(Test, levels = c("Run", "Star_r", "S20_r", "SLJ")),
         gender=factor(gender, levels=c("girl","boy"))) |> 
  na.omit()

contrasts(dat2$Test)   <- MASS::contr.sdif(4)
contrasts(dat2$gender) <- MASS::contr.sdif(2)

m6 <- lmer(zScore ~ 0 + Test/(m2rg + m2gd + gender) + (1 + m2rg + m2gd | Child) , data = dat2, REML = FALSE, control = lmerControl(calc.derivs = FALSE))

5.4.1 Results

5.4.1.1 Fixed effects

The LMM revealed significant positive effects of relative growth for star run (β = .34; SE = .05; z = 6.48; p < .001), 20 m sprint (β = .2; SE = .05; z = 3.84; p < .001), and standing long jump (β = .3; SE = .05; z = 5.74; p < .001) performances.

For growth delay a positive significant effect was found on 6 min run (β = .44; SE = .18; z = 2.44; p = .015), with children having a larger growth delay eliciting better performances.

Further, a significant effect for gender was found in 6 min run (β = -.63; SE = .22; z = -2.88; p = .004), with girls outperforming boys.

Results are reported in Table 5.3 and shown in Figure 5.4.

Code 
p2_pc1<- dat2 |> 
  mutate(Test=case_when(Test=="Run"~"6 min run",
                        Test=="Star_r"~"star run",
                        Test=="S20_r"~"20 m sprint",
                        Test=="SLJ"~"standing long jump"),
         Test=factor(Test, levels= c("6 min run", "star run", "20 m sprint", "standing long jump")),
         gender=ifelse(gender=="girl", "girls", "boys"),
         gender=factor(gender, levels = c("girls", "boys"))) |> 
  ggplot(aes(y = zScore, x = m2rg, group = gender, colour = gender)) +
  geom_smooth(method = "lm") +
  facet_grid(.~Test, scales="free") +
  ylab("") +
  xlab("relative growth [mean ± SE]") +
  theme_bw() +
  theme(legend.position = "none") +
  scale_colour_manual(values = c(cbPalette[3:4]))

p2_pc2 <- dat2 |>
  mutate(Test=case_when(Test=="Run"~"6 min run",
                        Test=="Star_r"~"star run",
                        Test=="S20_r"~"20 m sprint",
                        Test=="SLJ"~"standing long jump"),
         Test=factor(Test, levels= c("6 min run", "star run", "20 m sprint", "standing long jump")),
         gender=ifelse(gender=="girl", "girls", "boys"),
         gender=factor(gender, levels = c("girls", "boys"))) |> 
  ggplot(aes(y = zScore, x = m2gd, group = gender, colour = gender)) +
  geom_smooth(method = "lm") +
  facet_grid(.~Test, scales="free") +
  ylab("") +
  xlab("growth delay [mean ± SE]") +
  theme_bw() +
  theme(legend.position = "bottom") +
  scale_colour_manual(values = c(cbPalette[3:4]))

gridExtra::grid.arrange(p2_pc1, p2_pc2, heights=c(3.5,4), left = "z-score [±SE]", top = "Maturity offset - Moore")
Figure 5.4: Plots of overall effects of relative growth, growth delay, and gender [Moore]
Code 
tab42 <- tab_model(m6, digits=2, digits.re=2, show.se=TRUE, show.stat=TRUE, show.ci=FALSE,CSS = css_theme("cells"),
                   pred.labels = c("Test [Run]", "Test [Star_r]", "Test [S20_r]", "Test [SLJ]", "Test [Run] : m2rg", "Test [Star_r] : m2rg", "Test [S20_r] : m2rg", "Test [SLJ] : m2rg", "Test [Run] : m2gd", "Test [Star_r] : m2gd", "Test [S20_r] : m2gd", "Test [SLJ] : m2gd", "Test [Run] : gender2-1", "Test [Star_r] : gender2-1", "Test [S20_r] : gender2-1", "Test [SLJ] : gender2-1")) 

tab_to_flex(tab42, "Run = 6 min run, Star_r = star run, S20_r = 20 m sprint, SLJ = standing long jump, m2rg = relative growth (Moore) [z-score], m2gd = growth delay (Moore) [z-score], ")
Table 5.3: Effects of relative growth, growth delay, and gender on physical performance [Moore]

Predictors

Est.

SE

z

p

Test [Run]

-0.82

0.09

-9.30

<0.001

Test [Star_r]

-0.56

0.09

-6.43

<0.001

Test [S20_r]

-0.79

0.09

-9.06

<0.001

Test [SLJ]

-0.67

0.09

-7.72

<0.001

Test [Run] : m2rg

-0.02

0.05

-0.36

0.722

Test [Star_r] : m2rg

0.34

0.05

6.48

<0.001

Test [S20_r] : m2rg

0.20

0.05

3.84

<0.001

Test [SLJ] : m2rg

0.30

0.05

5.74

<0.001

Test [Run] : m2gd

0.44

0.18

2.44

0.015

Test [Star_r] : m2gd

0.01

0.18

0.04

0.965

Test [S20_r] : m2gd

0.20

0.18

1.09

0.276

Test [SLJ] : m2gd

0.11

0.18

0.60

0.548

Test [Run] : gender2-1

-0.63

0.22

-2.88

0.004

Test [Star_r] : gender2-1

-0.00

0.22

-0.00

0.998

Test [S20_r] : gender2-1

-0.34

0.22

-1.58

0.115

Test [SLJ] : gender2-1

-0.05

0.22

-0.23

0.817

Random Effects

σ2

0.40

τ00Child

0.28

τ11Child.m2rg

0.04

τ11Child.m2gd

0.40

ρ01

0.00

0.71

ICC

0.42

N Child

76

Observations

1413

Marginal R2 / Conditional R2

0.105 / 0.478

Run = 6 min run, Star_r = star run, S20_r = 20 m sprint, SLJ = standing long jump, m2rg = relative growth (Moore) [z-score], m2gd = growth delay (Moore) [z-score], Est. = estimate; SE = standar error

5.4.1.2 Variance components and correlation parameters

Similar to the LMM reported above a positive correlation of variance components were found for the grand mean and the second principal component (r = .71), with children who have larger growth-delay effects also having better overall physical fitness, and a negative correlation was found between the first and second principal component (r = -.5), with children who have larger relative growth-related improvements exhibiting smaller effects of growth delay on physical fitness, and vice versa. Results are reported in Table 5.4 and depicted in Figure 5.5.

Code 
p2_cor1 <- ranef(m6)$Child |> 
  data.frame() |> 
  ggplot(aes(x=X.Intercept., y=m2gd)) +
  geom_point()+
  geom_smooth(method = "lm", colour = "black", se=F) +
  ylab("grand mean") +
  xlab("growth delay effect") +
  theme_bw() 

p2_cor2 <- ranef(m6)$Child |> 
  data.frame() |> 
  ggplot(aes(x=m2rg, y=m2gd)) +
  geom_point()+
  geom_smooth(method = "lm", colour = "black", se=F) +
  ylab("relative growth effect") +
  xlab("growth delay effect") +
  theme_bw() 

gridExtra::grid.arrange(p2_cor1, p2_cor2, nrow = 1)
Figure 5.5: Correlations of grand mean and growth delay effect, and relative growth effect and growth delay effect in individual variance components [Moore]
Code 
vc2 <- VarCorr(m6)

as.data.frame(vc2, order="lower.tri") |> 
  filter(is.na(var2)==T) |> 
  select(-var2) |> 
  left_join(as.data.frame(vc2, order="lower.tri") |> 
              select(-vcov) |> 
              pivot_wider(names_from = var2, values_from = sdcor) |> 
              select(-c(grp,var1,`NA`)) |> 
              filter_all(any_vars(!is.na(.))) |> 
              t() |> 
              as.data.frame() |> 
              rownames_to_column(var="var1")) |> 
  mutate(across(where(is.numeric), \(x) round(x, 3)),
         var1=ifelse(var1=="(Intercept)","grand mean",var1)) |> 
  rename("Groups"=grp, "Name"=var1, "Variance"=vcov, "SD"=sdcor, "Corr"=V1) |> 
  flextable() |> 
  autofit() |> 
  add_footer_row(values="m2rg = relative growth (Moore), m2gd = growth delay (Moore)", colwidths = 6) |> 
  set_header_labels(V2 = "")
Table 5.4: Variance components and correlation parameters [Moore]

Groups

Name

Variance

SD

Corr

Child

grand mean

0.283

0.532

Child

m2rg

0.042

0.204

0.001

Child

m2gd

0.402

0.634

0.708

-0.501

Residual

0.396

0.629

m2rg = relative growth (Moore), m2gd = growth delay (Moore)

5.4.1.3 Model diagnostics

Diagnostic checks of residuals are plotted in Figure 5.5 and did not reveal any violations of LMM requirements.

Code 
dat2$resid  <- resid(m6)
dat2$sresid <- resid(m6)/sd(resid(m6))
dat2$fitted <- fitted(m6)

p2_qq <- dat1 %>% 
   ggplot(aes(sample = sresid)) + 
   stat_qq(size=.1, color="red") +
   stat_qq_line(color="blue") +
   geom_abline(intercept=0,slope=1) + 
   xlab("Standard normal quantile") + 
   ylab("Standardized LMM residual") + 
   coord_fixed(ratio=1, xlim = c(-3,+3), ylim=c(-3,3)) + theme_bw()

p2_f <- dat2 %>%  
  ggplot(aes(x=fitted, y=resid)) +   
  facet_grid(.~Test) +
  geom_point(shape=I(".")) +  
  xlab("Fitted values") + ylab("LMM residual") + theme_bw() +
  geom_hline(yintercept=0)  +  geom_density2d(linewidth=1) 

p2_box <- dat2 |> 
  ggplot(aes(resid, Test)) + 
  geom_boxplot() +
  ylab("LMM residual") +
  coord_flip() + theme_bw()

p1_at <- dat2 |>  
  pivot_longer(m2rg:m2gd) |> 
  ggplot(aes(value, resid)) + 
  geom_point(colour="black", size=.1) + 
  geom_hline(yintercept=0) + 
  geom_smooth(method="lm", color="black") + 
  facet_grid(name~Test) +
  ylab("LMM residual") + xlab("z-value") +theme_bw() +
  coord_cartesian(ylim=c(-3,+3))

gridExtra::grid.arrange(p1_qq, p1_f, p1_box, p1_at)
Figure 5.6: Model performance parameters [Moore]

5.5 Summary of results

The main findings were (1) positive effects of relative growth for performances in star run, 20 m sprint, and standing long jump for both maturity offset formulas, with children of higher relative growth performing better, (2) positive effects of growth delay for performances in 6 min run for both maturity offset formulas, with children having larger growth delays showing better performances, (2.1) a positive effect of growth delay for performances in 20 m sprint and star run in the first LMM (i.e., Mirwald estimate) only, (3) gendered differences in 6 min run for both maturity offset formulas, with girls performing better than boys, and (3.1) gendered differences in 20 m sprint performances in the first LMM (i.e., Mirwald estimate) only, also with girls performing better than boys.

Both LMMs revealed relative growth and growth delay to be relevant individual variance components and further revealed (4) a positive correlation of individual variance components between overall physical fitness and growth delay, with children with greater growth-delay effects tending to have higher levels of physical fitness, and (5) a negative correlation of individual variance components between relative growth and growth delay, with children with larger relative growth-related effects tending to have smaller effects of growth delay on physical fitness and vice versa.

5.6 Discussion

To the author’s knowledge, no comparable approach utilising unrotated principal components of age and maturity offset is available to this date. Further, comparable assessments investigating the development of physical fitness in children with physical fitness deficits is scarce, thus, generalisation of these results is limited. However, considering age-related improvements in physical fitness in regular fit children aged 8 to 14 years in these tests (Albrecht, 2015; Andersen et al., 1976; Castro-Piñero et al., 2009; Castro-Piñero et al., 2011; Catley & Tomkinson, 2013; Fühner et al., 2021, 2022; Golle et al., 2014; Lundgren et al., 2011; Miguel-Etayo et al., 2014; Niessner et al., 2020; Oliveira et al., 2014; Santos et al., 2014; Tambalis et al., 2016; Teich et al., 2023), the positive effect of relative growth for test performance in star run, 20 m sprint, and standing long jump to are not surprising and show that relative growth yields comparable results. Especially as age and maturity offset similarly progress over the consecutive assessments in the longitudinal design of the SMaRTER study, as well as the assessed components of physical fitness (as shown in Chapter 3). Interestingly, the model with maturity offset estimates by Mirwald et al. (2002) indicated a non-significant but potentially relevant trend for 6 min run performances, indicating a decline in performances in children of higher relative growth. The results would contradict the general assumption that age- and growth-related physical progress in terms of height, stature, and body composition leads to an improvement in physical fitness (Deprez et al., 2013; Lovell et al., 2015). While the plateau of cardiorespiratory fitness has been demonstrated in boys aged 15-16 years and in girls aged 14-15 years in physically fit children (Niessner et al., 2020; Ortega et al., 2011; Tambalis et al., 2016), the trajectory/onset might be different in children with deficits in physical fitness. Considering that maturity offset is a proxy for skeletal growth (Mirwald et al., 2002; Moore et al., 2015), a possible asynchronous progression of bone and muscle growth (e.g. bone growth at constant muscle capacity) (Beunen et al., 2006) would lead to a decrease in achieved force output (Hawkins & Metheny, 2001). As children in this study were within the range of growth spurt and peak elevation velocity (i.e., peak elevation velocity of -2 and 0 years) (Malina, 2004), growth-related changes might be more visible in continuous whole body workloads such as the 6 min run, where they would accumulate and thus explain negative non-significant effects of relative growth on 6 min run performances found in one model.

Effects for growth delay differed between models. The model using maturity offset according to Mirwald et al. (2002) found a positive growth delay effect on 6 min run, 20 m sprint, and star run performances, with larger growth delays leading to better performance. The model using the formula by Moore et al. (2015) found these effects on 6 min run only.

Side note: Considering the novelty of this approach and variable, it might be worth elucidating growth delay using an example for the results for 6 min run. Older children (e.g., z-age = 1) that are further away from their age at peak height velocity (e.g., z-maturity-offset = -1) would yield higher growth-delay values (i.e., z-age - z-maturity offset = 1 - (-1) = 2), which would be indicative of a more delayed growth trajectory. In comparison younger children (e.g., z-age = -1) that are closer to their age at peak height velocity (e.g., z-maturity-offset = 1) would score lower growth-delay values (i.e., z-age - z-maturity offset = -1 - 1 = -2), which would be indicative of no delays or even an accelerated growth trajectory. Now regarding the positive effect found for growth delay on 6 min run performances, better performances would be found in children with larger growth delays, meaning for example same age children with a delayed growth trajectory would tend to show better 6 min run performances compared to children with an accelerated growth trajectory.

This difference could be attributed to a difference in the complexity of formulas (see Section 8.1.2 for details). Moore et al. (2015) calculated the maturity offset based on age and standing or sitting height, while Mirwald et al. (2002) set up a more complex formula including multiple variables (i.e., standing height, sitting height, leg length, body mass, & age) while also giving age a less central role. Positive effects of growth delay on test performance could be explained by negative effects of skeletal growth discussed above. This means that children who enter the growth spurt later relative to their age experience fewer adverse effects of skeletal growth. One possible explanation could be that age, compared to skeletal growth, might be a better proxy for maturity- and other growth-related processes that improve fitness performance, such as muscle growth (Beunen et al., 2006). However, age-related improvements in physical fitness might also be influenced by social aspects related to age. Age, for example, regulates access to spaces that provide physical fitness services, such as school enrollment (Fühner et al., 2022) or access to organised sports facilities (Deprez et al., 2013; Lovell et al., 2015). Another example would be that all physical fitness recommendations in sport science include age-specific recommendations (Benjamin & Glow, 2003; Bull et al., 2020), which in turn are used to develop organised sport programs and curricula for physical education. This would mean that the environment might at least partially determine the magnitude of age-related effects on physical fitness (as argued in the previous chapter in the context of the positive correlation of individual levels of overall physical fitness and age-related improvements; see Section 4.5). Thus, age-related physical fitness improvements in children should be considered in context of growth processes, such as tissue growth, as well as the social environment available to children depending on their age.

LMMs also revealed gender differences, with girls performing better in 20 m sprint (in the model using Mirwald’s estimate) and 6 min run (both models). Differences in outcome between both LMMs could be attributed to both maturity offset formulas comprising different gendered estimates, which presumably behave differently in the LMM approach. The direction of these findings contradicts current evidence for children, according to which boys tend to perform better than girls, as was found in children from the EMOTIKON sample (Fühner et al., 2021, 2022; Teich et al., 2023). A common argument for gendered physical fitness differences with better performance in girls compared to boys is that girls enter growth spurt and achieve age at peak height velocity two years earlier than boys (Malina, 2004) (see e.g. Fühner et al. (2022) analysis of older than key-age children in terms of 20 min sprint performance). This argument would fit the distribution of maturity offsets between girls and boys, as maturity offset is on average ~1 year closer to age at peak velocity for girls than for boys (i.e., Mirwald: girls = -1.6 ± 1 & boys = -2.7 ±.8 years; Moore: girls = -1.8 ± .9 & boy = -2.7 ± .7 years). However, this argument would contradict results for growth delay, where greater growth delay leads to better physical fitness performances, identifying the growth spurt and associated growth-related changes as detrimental to 6 min run performances. This means that girls’ performances tend to stagnate or decline as they progress through growth spurt towards age at peak height velocity. Accordingly, another possible explanation could be that gendered differences in physical performance vary for children with deficits in physical performance. While there is ample evidence of physiological and social benefits for regularly fit boys compared to girls, such as better oxygen utilisation (Armstrong & Welsman, 2007; Tomkinson et al., 2018), higher muscle mass (Kanehisa et al., 1995; Malina, 2004), easier access to sport (Haywood & Getchell, 2014), and gender-specific self-perceptions and perceptions of sport through parents (Fredricks & Eccles, 2005), these factors may not affect children with deficits in physical fitness in a similar way. Accordingly, further research is needed to assess gender differences and their origins across the whole spectrum of physical fitness.

Finally, the inclusion of relative growth and growth delay as individual variance components allowed for an assessment of individual differences and their correlations. Considering age-related effects to be driven by anthropometrical growth-related changes (see Section 4.5), a positive correlation would have been expected for relative growth as well. Contradictory to this expectation, a negative correlation was found for overall physical fitness and growth-delay effects but no correlation for overall physical fitness and relative-growth effects. Additionally, a negative correlation between individual relative-growth and growth-delay effects was found, meaning that children with greater relative growth-related effects tend to show smaller growth-delay-related effects and vice versa. This could further underline potential adverse effects associated with skeletal growth discussed above and might reemphasize social components to drive positive effects associated with age on physical fitness discussed in Section 4.5 and Section 7.3. Further, considering only children with physical fitness deficits in their growth spurt before age at peak height velocity were included, these correlations might be specific to this period as well as the physical fitness level of the included children. Accordingly, further research is needed to explore the impact of different growth as well as maturity-related processes on development of physical fitness in children and their physical fitness levels.

5.7 Subsequent analyses

The following Chapter 6 is concerned with a different aspect of physical fitness. Utilising the multiple assessment for each child at each assessment in the Simon task (i.e., 60 data points each), the effects of physical fitness on executive function assessed using the Simon task were analysed. Implementing a LMM approach, the large number of measurements allowed to assess the effects of physical fitness on variance in the Simon task as well as accounting for individual differences in the effects of different physical fitness tests on Simon task reaction times.