## 2 Materials and Methods  
The subjects were 33 patients with open bite who visited the Tokyo Dental College Chiba Dental Center between 2012 and 2018. Criteria were defined to include cases that ended active treatment 1 year before, at which point the patients had overbite change within 1 mm and a positive overbite. The patients used a circumferential-type retainer after treatment. In addition, prior to treatment, ANB was more than 3.0\({}^{\circ}\), overbite was less than \(-2\) mm, and deviation of menton from the mid-facial line was less than 2 mm as measured by the frontal cephalogram. Exclusion criteria were patients with severe tongue thrust, TMJ symptoms, especially progressive condylar resorption, and congenital abnormality such as craniofacial syndrome, cleft lip palate, and so on. Surgical orthodontic treatment was applied to 15 patients (Group S), and orthodontic treatment was applied to 18 patients, of which 8 cases underwent extrusion of anterior teeth (Group E) and 10 cases underwent intrusion of molars (Group I; Table 1). All cases were diagnosed, treated, and selected by six orthodontic specialists with more than 10 years of experience. Measurements were made using a cephalometric radiograph and dental models of these subjects. Vertical and anterior-posterior measurements of skeletal and denture pattern in the cephalometric analysis were conducted, and overjet and overbite in the dental model analysis were measured. ANB and Wits appraisal were used for cephalometric anterior-posterior evaluation, while FMA, lower facial height, and palatal plane were used for vertical evaluation, which indicates the skeletal type (Figures 4 and 5). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Surgical Orthodontic** & **Intrusion of Molars** & **Extusion of Anterior** \\  & **Treatment (Group S)** & **(Group D)** & **Teeth** \\  & \(n\) = 15** & \(n\) = 10** & **(Group E)** \\ \hline sex & Male: 1 & Male: 3 & Male: 2 \\  & Female:14 & Female: 7 & Female: 6 \\ Age (average) & 29.3 \(\pm\) 13.8 & 20.6 \(\pm\) 7.8 & 17.4 \(\pm\) 10.4 \\ Treatment plan & & & \\ Extraction case & 4 & 7 & 5 \\ Non-extraction case & 11 & 3 & 3 \\ Type of surgery & & & \\ SSRO & 4 & & \\ SSRO + Le Fort 1 & 11 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Patients’ characteristics. All measurements were taken manually by the same single researcher. Any measurement errors were re-measured by the researcher after a one-week interval. These measurements were compared between the three groups by one-way ANOVA and Tukey's analysis. The measured items that showed significant difference (\(p<0.05\)) among the three groups were selected to perform receiver operating characteristic (ROC) analysis, and the area under the curve (AUC) of the ROC curve was used to determine appropriate cephalometric parameters for the decision of treatment--specifically, deciding between extruding anterior teeth or intruding molars, or between intruding molars or adopting surgical orthodontic treatment. The treatment option criteria were also examined based on the cutoff point, which is the best combination of specificity and sensitivity. The sample size was calculated using the mean and standard deviation of the pre-treatment mandibular plane angle for each group from our preliminary study, which consisted of 20 subjects, with an alpha error of 80% and a beta error of 20%, requiring 7 patients in each group. Figure 4: Cephalometric measurements (skeletal pattern). 1: nasion; 2: orbitale; 3: anterior nasal spine (ANS); 4: posterior nasal spine (PNS); 5: point A; 6: point B; 7: PM; 8: mentor; 9: xi; 10: portion; 11: ANB angle; 12: Wits appraisal; 13: lower facial height; 14: mandibular plane angle (FMA). Palatal plane angle: the angle between the palatal plane connecting ANS and PNS and the Frankfort plane. Figure 5: Cephalometric measurements (denture pattern).