# Geometric Morphometric Shape Analysis of Mandibular Post-Canine Dentition
## 2 Materials and Methods  
### Sample Size Calculation  
In a study conducted by [PERSON] et al. (2018), [8] it was found that there is a difference in the shape of human premolars between males and females. Specifically, the centroid sizes of the upper premolars were significantly larger in males (first premolar: 28.33 \(\pm\) 1.96; second premolar: 27.54 \(\pm\) 1.83) compared to those in females (first premolar: 27.82 \(\pm\) 1.93; second premolar: 26.93 \(\pm\) 2.05) [8]. The sample size was determined using Equation (1) to compare means of the two groups. The equation used was  
\[\mathrm{N}=\frac{2\Big{(}Z_{1-\frac{\alpha}{2}}+Z_{1-\beta}\Big{)}^{2}\sigma^{ 2}}{d^{2}} \tag{1}\]  
where N is the sample size, \(d\) is the clinically significant difference, and \(\sigma\) is the mean standard deviation. After considering the centroid values of males and females in the upper first and second premolars, with a 5% alpha error (z = 1.959964) and 80% power (z = 0.841621), taking a clinically significant difference of 1 unit, the required sample size for each sex was calculated to be 60 individuals. Prospective pretreatment dental casts (n = 160) were obtained from the archives of the Department of Orthodontics at the Manipal College of Dental Sciences, Mangalore, as per the sample size calculation mentioned above. The sample size determined was 120 individuals; but, in order to account for anatomic variation in the mandibular second premolar, an additional 40 casts were taken, for examination of the 60 male and 60 female casts of all post-canine teeth. This study was undertaken following clearance from the institutional ethics committee. Study casts of patients in the age range of 13-20 years were included in this study. After the age of 20, when the individual's occlusion has developed, the contacts and cusp tips tend to show attrition, leading to the development of proximal contact areas and occlusal wear facets. This can result in incorrect landmark acquisition, particularly in terms of proximal contacts and cusp tips. To prevent such errors, our sample was drawn from the age range of 13-20 years, where the landmarks are stable, and no physiological changes are observed so early in life. Comprehensive demographic history was recorded, including the birthplace and domicic of each individual, as well as those of their parents and grandparents. Individuals were included in this study if they, along with their parents and grandparents, were born and raised in the Dakshina Kannada region. This three-generation residency criterion was established to ensure a strong regional connection among participants. Verification of this information was conducted through patient interviews at the dental clinic, where questions about the birthplaces, residences, and occupations of their parents and grandparents were asked. In order to avoid confounding factors and errors in evaluation, the inclusion and exclusion criteria listed in Table 1 were closely adhered to. After obtaining the impressions (taken using Dentsply Aquasil Soft Putty and Kit, Dentsply, Noida, India) and pouring the casts using Type 4 Extra Hard Dental Die Stone (Zhermack Badia Polesine (RO), Italy), we promptly digitized them using the advanced inEOS X5-Lab scanner (Dentsply Sirona, Noida, India). To accurately define landmarks, we identified key landmarks based on both anatomic and geometric evidence. Our approach was substantiated by the seminal work of [PERSON] (1969), who proposed a method for describing the basal area of the posterior teeth [14]. We also drew on the landmark definitions suggested by [PERSON] et al. (2002) and [PERSON] et al. (2014), modifying them as needed to suit our specific needs [5; 15].