## 2 Materials and Methods
The time measured included, for the RDI group, the patient's positioning in the dental chair, application of the rubber dam, bonding, and removal of the rubber dam; and for the TI group, the patient's positioning in the chair, placement of the mouth retractor, bonding, and removal of the retractor. Figure 1: Sample size calculation. For the individuals included in this study, before-bracketing scan models (BB) and after-bracketing intraoral scan models (AB) were obtained using the 3 Shape intraoral scanner device (Trios 3, Copenhagen, Denmark). The scan data in STL format were transferred to the Orthoanalyzer software (Orthoanalyzer 1.11.1.1, Copenhagen, Denmark). On the BB models, 0.022-inch MBT Prescription Gemini Twin brackets were placed according to MBT criteria by an orthodontic expert and checked by three experts (KCK, TSE, IM) [14]. The  
Figure 3: The application of rubber dam. Figure 2: The flowchart of the study. digital bracket placement led to the creation of virtual bonding models (VB) (Figure 4) and these models are considered the gold standard. In both groups (TI-RDI), superimposition was performed using the GOM Inspect software (GOM, V8 SR1, Braunschweig, Germany) to enable comparison of the AB models with the VB models (Figure 5). A local coordinate system was defined on each tooth in order to ensure the standardization of the measurements made. Planes were defined by selecting three points on the bracket surfaces to create local coordinate systems (Figure 6). For each local coordinate system, the X-axis was designated to represent the mesiodistal direction, the Y-axis the occlusoginyal direction, and the Z-axis the buccolingual direction. Figure 4: (**A,B**) The placement of brackets on BB models and the creation of VB models. Figure 5: The visual representation of superimposed models in the GOM program. For each tooth, the angle between the planes in the superimposed model was created. The measurements of the resulting angle were taken along the XY, XZ, and YZ axes of the tooth's local coordinate system, and the angular deviations for each bracket were calculated in degrees. The data were analyzed using IBM SPSS Statistics for Windows, version 29.0.1.0 (IBM Corp., Armonk, NY, USA). Descriptive statistics for continuous data included mean, standard deviation, median, minimum, and maximum values, while for discrete data, frequency and percentage values were provided. The normality of continuous data was assessed using the Shapiro-Wilk test. The analysis revealed that more than half of the variables did not follow a normal distribution. In the variables that were normally distributed, it was observed that the variances were not equal, leading to the decision to use non-parametric methods. Among the non-parametric test methods, the Mann-Whitney U test was chosen, considering the need to examine two independent variables, the equality of sample sizes in the test groups, and the continuous and quantitative nature of the data obtained in this study. After superimposing the models, the Mann-Whitney U test was used to compare the linear and angular values between the RDI and TI groups. A \(p\)-value of less than 0.05 indicates a significant difference within the limits of 0.5 mm for linear measurements and 2\({}^{\circ}\) for angular measurements, while a \(p\)-value greater than 0.05 indicates no significant difference. These limits were selected based on the professional standards of the Objective Grading System of the American Board of Orthodontics. The chair time measurements with each patient were compared using the parametric \(t\)-test for independent groups.