# Geometric Morphometric Shape Analysis of Mandibular Post-Canine Dentition
## 3 Results  
Our study comprised 160 patient casts from 76 males and 84 females, with a mean age of 18.62 \(\pm\) 2.49 years (Table 2). Principal component analysis showed that the number of components, explaining 80% of the variation in the shapes of the mandibular first premolar, the two-cusp-type second premolar, the three-cusp-type second premolar, the first molar, and the second molar, were 15/53 (28.3%), 14/50 (28%), 15/56 (26.79%), 21/89 (23.6%), and 18/84 (21.43%) respectively. Procrustes superimposition showed a uniform scatter of landmarks round the median landmark in both the xy- and xz-axes. In the mandibular first premolar, the scatter plot of the principal components (PCs) 1 and 2 (PC1 and PC2) showed homogenous distribution (Figure 1) with the graphs, showing maximum variations in the mesiobuccal and distopalatal ends of the tooth outline, as well as in the mesiolingual groove ending (Figure 2). Analysis of shape and centroid size using Procrustes ANOVA showed that centroid sizes were not significantly different between the sexes (\(p>0.05\)) in all teeth (Table 3). The shape of the tooth, however, was significantly different between the sexes in the two-cusp-type mandibular second premolar (\(p=0.0035\)), mandibular first molar (\(p<0.0001\)), and mandibular second molar (\(p<0.0001\)). Regression analysis showed that size-dependent shape variation was noted in only 1.38% of cases, indicating no significant allometry (\(p=0.56\)) (Figure 3, Table 4). Discriminant function analysis also showed that there was 80.833% accuracy in distinguishing sexes, but this was not significant, with a \(p\) value of 0.3181 (Table 5). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \begin{tabular}{c} **Treatment** \\ **Mandibular I** \\ **Premolar** \\ \end{tabular} & \begin{tabular}{c} **Mandibular** \\ **Second Premolar** \\ \end{tabular} & \begin{tabular}{c} **Mandibular** \\ **Second Premolar** \\ \end{tabular} & \
\begin{tabular}{c} **Mandibular First** \\ **Molar** \\ \end{tabular} &
\begin{tabular}{c} **Mandibular** \\ **Second Mom** \\ \end{tabular} \\ \hline Total SS: & 2.308619 & 0.78446 & 1.5874 & 0.891322 & 1.162052 \\ \hline Predicted SS: & 0.031928 & 0.02264 & 0.05311 & 0.020819 & 0.034882 \\ \hline Residual SS: & 2.276692 & 0.76182 & 1.5343 & 0.870504 & 1.127169 \\ \hline \% predicted: & 1.38\% & 2.89\% & **3.35\%** & **2.34\%** & **3.00\%** \\ \hline \(p\) value & 0.056 & 0.1405 & **\textless{0.001}** & **\textless{0.001}** & **\textless{0.001}** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Regression of the shape parameters with the centroid size of the tooth. Figure 1: Principle component analysis for landmarks of the five types of mandibular post-canine dentition (red dots represent females and blue dots represent males). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \begin{tabular}{c} **Treatment** \\ **Mandibular I** \\ **Premolar** \\ \end{tabular} & \begin{tabular}{c} **Mandibular** \\ **Second Premolar**, \\ **Two-Cusp Type** \\ \end{tabular} & \begin{tabular}{c} **Mandibular** \\ **Second Premolar**, \\ **Three-Cusp Type** \\ \end{tabular} & \
\begin{tabular}{c} **Mandibular** \\ **First Mom** \\ \end{tabular} &
\begin{tabular}{c} **Mandibular** \\ **Second Mom** \\ \end{tabular} \\ \hline Procrustes distance & 0.02361 & 0.0472 & 0.0224 & 0.0205 & 0.0251 \\ \hline Mahalanobis distance & 1.8882 & 6.8036 & 2.7051 & 3.7158 & 3.0236 \\ \hline T-square & 106.96 & 547.91 & 198.30 & 414.21 & 274.27 \\ \hline \(p\) value & 0.3181 & 0.94 & **0.019** & 0.3073 & 0.1063 \\ \hline Prediction Accuracy, Female & 83.333 & 95 & 90.625 & 95.00 & 95.00 \\ \hline Prediction Accuracy, Male & 78.333 & 100 & 91.49 & 95.00 & 88.33 \\ \hline Overall Accuracy & 80.833 & 97.95 & 90.99 & 95.00 & 91.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Discriminant function analysis. Figure 2: Line graphs depicting landmark changes between sexes in the mandibular first and second premolars.