## 3 Results  
The explosive strength expressed by the SBJ revealed a mean value (SD) of 128.29 cm (22.76) and 121.17 cm (21.49), while the MO was, on average, 3.36 y (0.72) and 6.35 y (0.44) for boys and girls, respectively. A significant difference between sex was found (\(p<0.001\)). Figure 2 shows the results between sexes (with the covariates factored in), suggesting that females outperformed males in the SBJ; females had greater performances than males by up to 20 cm (on average). Furthermore, the parameter Log a that results from interaction between height, mass, MO, sex, and age was 0.973, 0.985, 0.982, and 0.995 for males and 1.199, 1.211, 1.208, and 1.211 for females, respectively, for 8, 9, 10, and 11-year-olds.  
The estimated parameters from the multiplicative model relating the SBJ distance to the body size components in Equation (1), incorporating the MO, are given in Table 2.  
Figure 1: (**a**) The association between SBJ and height in 8–11-year-old male children, and (**b**) the association between SBJ and height in 8–11-year-old female children.  
The model (Equation (1)) relating the SBJ distance to the body size components was  
\[\text{SBJ distance (cm)}=\text{a}\cdot\text{M}^{-0.435}\cdot\text{H}^{1.152} \tag{3}\]  
With a positive height (H) and negative mass (M), the model suggested that the optimal body shape associated with the SBJ is to be taller, but lighter (less body mass).  
Fittingly, the model (Equation (1)) revealed significant differences in the constant Log a parameter (Table 2) due to sex (\(p<0.001\)) and age (\(p<0.001\)), together with the interactions of age and sex (\(p<0.001\), Figure 3).  
In particular, the parameter Log a appears in the footnote of Table 2. Note that the maturity offset (MO) made significant positive contributions to predicting the log-transformed SBJ distance (\(p<0.001\)).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **B** & **Std. Error** & **t** & **Sig.** & **95\% Confidence Interval** \\ \hline  & & & & & **Lower Bound** & **Upper Bound** \\ \hline Intercept Log(a) & 0.995 & 0.327 & 3.043 & 0.002 & 0.354 & 1636 \\ Log(M) (k1) & 1.152 & 0.064 & 18.092 & 0.000 & 1027 & 1276 \\ Log(H) (k2) & \(-\)0.435 & 0.015 & \(-\)28.935 & 0.000 & \(-\)0.465 & \(-\)0.406 \\ MO (c) & 0.084 & 0.012 & 7.155 & 0.000 & 0.061 & 0.107 \\ Sex (Female) & 0.226 & 0.043 & 5.311 & 0.000 & 0.143 & 0.310 \\ Age (8.00 y) & \(-\)0.022 & 0.019 & \(-\)1.205 & 0.228 & \(-\)0.059 & 0.014 \\ Age (9.00 y) & \(-\)0.010 & 0.014 & \(-\)0.704 & 0.482 & \(-\)0.037 & 0.017 \\ Age (10.00 y) & \(-\)0.013 & 0.009 & \(-\)1.376 & 0.169 & \(-\)0.032 & 0.006 \\ Age (8.00) *Sex (Female) & \(-\)0.083 & 0.017 & \(-\)4746 & 0.000 & \(-\)0.117 & \(-\)0.049 \\ Age (9.00) *Sex (Female) & \(-\)0.065 & 0.014 & \(-\)4631 & 0.000 & \(-\)0.092 & \(-\)0.037 \\ Age (10.00) *Sex (Female) & \(-\)0.009 & 0.012 & \(-\)0.785 & 0.432 & \(-\)0.032 & 0.014 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of Equation (1) about SBJ performance. The parameter Log a for males is 0.973, 0.985, 0.982, and 0.995, and 1.199, 1.211, 1.208, and 1.211 for females, respectively, for 8, 9, 10, and 11-year-olds. Note that the baseline group is the 11-year-old boys from which all other sex and age groups are compared.  
Figure 2: Estimated marginal means of SBJ by sex. Covariates appearing in the model are evaluated as follows: Ln Height = 4.9147, LN Mass = 3.4928, and MO = \(-\)4.8206.