## 3 Results  
### Remodeling Process of the ACL Graft  
The Bayesian multilevel skew-normal regression was used to model SNQs. Skew-normal distribution was used to control for any skewness observed in the data. Multilevel structure was used to control for differences between patient average SNQ values and patient variability. Model coefficients are presented in Table 3. Selected predicted posterior means of SNQ measurements in the sagittal plane are presented on Figure 6.  
The results of modelling SNQs from both sagittal and axial planes were similar. For both SNQs, the change over weeks value had a nonlinear nature where the clinical parameter increased until week 36 and then decreased. Our results of predicted means demonstrated that the SNQ at week 96 does not reach the level from week 2 (Figure 6). In addition, both SNQs were moderately and positively correlated to the corresponding rSD indices, indicating that increase in rSD was associated with increase in SNQ. SD represents a measure of the amount of variation or dispersion of a set of values. Therefore, we assumed greater SD is associated with less graft homogeneity. The change of SNQ over weeks was weakly dependent on weight bearing, for measurements in the sagittal plane. The initial increase in SNQ values was noticeably lower among patients who fully weighted their leg from the beginning. In addition, SNQ change was also related to age, where the increase in parameter values slightly higher among younger patients. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Remodeling Process of ACL Graft**} \\ \cline{3-8}
**Parameter** & \multicolumn{6}{c}{**SNQ at Sagittal Plane**} & \multicolumn{6}{c}{**SNQ at Axial Plane**} \\ \cline{3-8}  & **Me** & **LI** & **UI** & **Me** & **LI** & **UI** \\ \hline Intercept & 0.07 & \(-0.11\) & 0.25 & 0 & \(-0.18\) & 0.19 \\ \hline Week [L] & **0.68** & **0.46** & **0.88** & **0.96** & **0.73** & **1.18** \\ \hline Week [Q] & **-0.95** & **-1.15** & **-0.73** & **-0.67** & **-0.9** & **-0.43** \\ \hline rSD & **0.33** & **0.26** & **0.4** & **0.28** & **0.2** & **0.35** \\ \hline Sex & \(-0.14\) & \(-0.29\) & 0.01 & \(-0.13\) & \(-0.28\) & 0.03 \\ \hline Age & \(-0.07\) & \(-0.22\) & 0.09 & \(-0.05\) & \(-0.22\) & 0.1 \\ \hline \(\beta\) & Weight bearing & **0.2** & **0.02** & **0.37** & 0.05 & \(-0.13\) & 0.23 \\ \hline Week [L] : Sex & \(-0.03\) & \(-0.2\) & 0.15 & 0.1 & \(-0.08\) & 0.29 \\ \hline Week [Q] : Sex & 0.09 & \(-0.08\) & 0.26 & 0.11 & \(-0.07\) & 0.31 \\ \hline Week [L] : Age & \(-0.1\) & \(-0.27\) & 0.06 & **-0.19** & **-0.37** & **-0.01** \\ \hline Week [Q] : Age & **0.2** & **0.01** & **0.37** & **0.27** & **0.07** & **0.47** \\ \hline Week [L] : Weight bearing & **0.24** & **0.02** & **0.45** & **0.26** & **0.05** & **0.47** \\ \hline Week [Q] : Weight bearing & **-0.29** & **-0.52** & **-0.07** & 0.07 & \(-0.15\) & 0.3 \\ \hline \(\sigma\) & Intercept & \(-0.52\) & \(-0.65\) & \(-0.41\) & \(-0.52\) & \(-0.65\) & \(-0.4\) \\ \hline \(\tau\) & \(\beta\) Intercept & 0.45 & 0.34 & 0.59 & 0.46 & 0.35 & 0.61 \\ \hline \(\sigma\) Intercept & 0.25 & 0.11 & 0.41 & 0.31 & 0.17 & 0.47 \\ \hline \(\alpha\) & & 04.02 & 2.06 & 7.54 & 2.57 & 1.24 & 4.34 \\ \hline R\({}^{2}\) & Population level & 0.39 & 0.32 & 0.46 & 0.37 & 0.3 & 0.43 \\ \hline Total & 0.6 & 0.55 & 0.64 & 0.57 & 0.51 & 0.62 \\ \hline \hline \end{tabular} Note: Me is median of posterior, and LI and UI are lower and upper bounds of 95% credible interval. [L] and [Q]—linear and quadratic trends. The ’\(\cdot\)’ symbol indicates interaction term. \(\beta\)—regression weight; \(\tau\)—patient random effect; \(\sigma\) residual standard deviation; \(\alpha\)—skewness coefficient. The coefficients are on a standard normal scale. Statistically credible effects are bolded.  
\end{table} 
Table 3: Results of Bayesian multilevel skew-normal regressions of SNQs measured at sagittal and axial planes as dependent variables. Figure 6: **Remodeling process of ACL Graft**. Note SNQ has a non-linear structure.