# Geometric Morphometric Shape Analysis of Mandibular Post-Canine Dentition
## 2 Materials and Methods
### Sample Size Calculation
Landmarks were identified based on \"Anatomic evidence\" (corresponding to cusp tips, fissure junctions, and the endpoints of common clinical measurements like mesiodistal width, buccolingual width, line angles, and point angles) and \"Geometric evidence\" (based on the crests of curvatures, line and point angles, and surface landmarks corresponding to occlusal surface landmarks). These landmarks were marked on the 3D models using the open-source 3D Slicer software, version 4.10.2 ([[http://www.slicer.org](http://www.slicer.org)]([http://www.slicer.org](http://www.slicer.org))) accessed on 25 December 2023 [16]. The number of landmarks varied according to the complexity of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Inclusion Criterion** & **Exclusion Criterion** \\ \hline \(\bullet\) & Individuals from Dakshina Kannada, between 13 and 20 years of age, were included in this study. & Any tooth in the posterior segment that has undergone restoration, or has fractures, caries, or crowns, was excluded from this study. \\ \(\bullet\) & Individuals having a full complement of posterior teeth from the first premolar to the tooth, as well as individuals included in the study & Patients exhibiting any developmental anomalies that affect the shape or size of the tooth, as well as individuals displaying symptoms of a syndrome, were excluded from this study. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Inclusion and exclusion criteria. tooth involved. The first premolar, the two-cusp-type second premolar, the three-cusp-type second premolar, the first molar, and the second molar had totals of 20, 19, 21, 32, and 27 landmarks, respectively. Of these landmarks, 11, 11, 13, 19, and 15 were based on geometric evidence. In total, 120 dental casts with the presence of all 8 of the relevant teeth, i.e., the first premolar to the second molar, bilaterally, were selected for shape analysis. These 120 casts consisted of those from 60 males and 60 females, and were specifically analyzed in relation to the mandibular first premolar, mandibular first molar, and mandibular second molar. In the sample of 120 patients, it was found that 37 had the two-cusp type of mandibular second premolar, while 83 had the three-cusp type. In order to increase the accuracy of our analysis, we expanded our sample size to include 40 additional casts (making the total sample number n = 160). This allowed us to examine the shape variations in both the two-cusp (n = 49) and three-cusp (n = 111) types of mandibular second premolars (Table 2). Using a random subsample of 20 dental models, the assessment of intra- and interobserver error in landmark location was carried out. Over the course of five days, landmarks were marked independently by two observers. A single investigator obtained the landmarks to study for intra-observer variation. Reliability was measured using the Interclass Correlation Coefficient (ICC) [17] (IBM, SPSS version 20.0, Chicago, IL, USA), which ranged from 0.982 to 1.000, showing excellent reliability. Intra-operator reproducibility at the digitization stage was assessed using a one-way nonparametric ANOVA, with a randomized permutation procedure (10,000 iterations), using MorphoJ software. The average variance related to digitization error for centroid size and shape among the selected teeth