## 3 Results
### Landmark-Based Hard Tissue Comparison  
The mean difference in centroid size between the preoperative and postoperative point configurations was \(-1.09\), indicating a reduction in size due to surgery. The maximum difference was 12.5 and the minimum was \(-12.8\). A Wilcox test showed no statistically significant differences between pairs of size measures (\(p\)-value = 0.191), nor did the Mann-Whitney test (\(p\)-value = 0.728). In contrast, sex-related differences were found to be significant (Mann-Whitney, \(p\)-value = 0.0001), as was an age-related correlation, indicating a negative relationship between size and patient age (Spearman's rho = \(-0.329\), \(p\)-value = 0.01). When the pooled data set was registered and analysed using one-way PERMANOVA, no statistically significant differences were found between the preoperative and postoperative groups (overall sum of squares: 2.941 \(\times\) 104, within-group sum of squares: 2.904 \(\times\) 104, F = 0.7419, \(p\)-value = 0.652). Similarly, testing for gender differences also revealed statistically significant differences (one-way PERMANOVA, overall sum of squares: 2.941 \(\times\) 104, within-group sum of squares: 2.448 \(\times\) 104, F = 11.69, \(p\)-value = 0.0001). Pairwise comparisons showed that Procrustes distances ranged from 0.022 to 0.168, with an average of 0.093 (Table 4). Although the median values suggested that females had greater differences between preoperative and postoperative configurations, the Mann-Whitney test did not confirm this finding. Similarly, no age-related correlation was observed (Spearman's rho = 0.12, \(p\)-value = 0.523). The most significant differences between preoperative and postoperative skulls were mainly distributed among the midfacial points, with the greatest variance observed at the upper incisor. Figure 4 further details the relocation of the points after surgery. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**PD** & **Total** & **Females** & **Males** \\ \hline Min & 0.022 & 0.022 & 0.049 \\ Max & 0.168 & 0.154 & 0.168 \\ Mean & 0.093 & 0.094 & 0.092 \\ SD & 0.034 & 0.037 & 0.033 \\ Median & 0.090 & 0.101 & 0.084 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Descriptive statistics of Procrustes distances computed between preoperative and postoperative point configurations. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Signed Post-to-Preoperative Differences**} & \multicolumn{3}{c}{**Absolute Post-to-Pre Operative Differences**} \\ \cline{2-9}  & **MEAN** & **STD** & **Z** & \(p\)**-Value** & **MEAN** & **STD** & **Z** & \(p\)**-Value** \\ \hline zy dx-sin & 0.256 & 2.042 & 0.871 & 0.384 & 1.364 & 1.528 & \(-0.794\) & 0.427 \\ mx dx-sin & 0.164 & 3.148 & \(-0.800\) & 0.424 & 2.148 & 2.284 & 0.062 & 0.951 \\ go dx-go sin & 1.983 & 3.394 & \(-0.310\) & 0.756 & 3.079 & 2.417 & \(-0.286\) & 0.775 \\ zy dx-subsp & 0.892 & 4.752 & 0.000 & 1.000 & 3.545 & 3.244 & 0.077 & 0.939 \\ zy sin-subsp & 2.275 & 6.797 & 0.179 & 0.858 & 5.313 & 4.753 & 1.717 & 0.086 * \\ mx dx-subsp & \(-1.492\) & 9.475 & \(-0.935\) & 0.350 & 7.020 & 6.449 & \(-0.148\) & 0.883 \\ mx sin-subsp & \(-1.581\) & 9.241 & 0.972 & 0.331 & 7.557 & 5.430 & 0.406 & 0.685 \\ go dx-pog & 0.530 & 6.275 & 1.006 & 0.314 & 4.424 & 4.429 & 1.180 & 0.238 \\ go sin-pog & 1.370 & 5.919 & 0.708 & 0.479 & 4.083 & 4.459 & 1.925 & 0.054 * \\ inc dx-go dx & \(-0.590\) & 9.572 & 1.611 & 0.107 & 6.660 & 6.825 & \(-1.540\) & 0.124 \\ inc sin-go sin & \(-0.792\) & 9.299 & 0.991 & 0.322 & 6.731 & 6.384 & \(-0.465\) & 0.642 \\ subsp-pog & \(-3.006\) & 12.271 & 0.370 & 0.771 & 5.523 & 11.326 & 0.025 & 0.980 \\ sto-pog & \(-2.986\) & 11.218 & 1.702 & 0.089 * & 5.515 & 10.175 & \(-1.850\) & 0.064 * \\ B-pog & \(-1.053\) & 9.678 & 0.139 & 0.890 & 7.649 & 5.821 & 2.746 & 0.006 ** \\ \hline \hline \end{tabular}  
*—significant at 10% level of significance, ***—significant at 5% level of significance.  
\end{table} 
Table 3: Signed post-to-preoperative differences between the linear measurements.