A total of 1253 CBCTs were evaluated against the eligibility criteria until finding 9 sub-groups of 40 cases each (360 cases). There were no missing data. There were 243 women and 117 men with a mean age of 22.28 ± 2.80 years. The mean (SD) age of patients of skeletal Classes I, II, and III were 22.27 ± 2.89, 22.28 ± 2.72, and 22.31 ± 2.81 years, respectively (min = 18, max = 28 for each of the 3 groups). There was not a significant difference between the ages of these 3 groups (t-test, P = 0.993). The mean (SD) age of patients in vertical growth patterns ‘short face, normal face, and long face’ were respectively 22.35 ± 3.09, 22.22 ± 2.57, and 22.28 ± 2.75 years (min = 18, max = 28 for each of the 3 groups). There was not a significant difference between the ages of these 3 groups either (t-test, P = 0.935).

In vertical growth patterns ‘short face, normal face, and long face’, there were 91, 80, and 72 women, respectively. In these groups, there were 29, 40, and 48 men, respectively. The chi-square test showed a slight but statistically significant difference between the distributions of sexes across these 3 groups (P = 0.032). In skeletal Classes I, II, and III there were respectively 85, 81, and 77 women, and 35, 39, and 43 men. The sexes were similarly distributed across these 3 Classes (P = 0.545, chi-square).

Mandibular plane angle

In vertical growth groups ‘short face, normal face, and long face’, there were respectively 79, 13, and 0 cases with low mandibular plane angles, 41, 60, 44 cases with normal mandibular plane angles, and 0, 47, and 76 cases with high mandibular plane angles. The distributions of mandibular plane angles differed significantly across the groups ‘short face, normal face, and long face’ (P = 0.000, chi-square). The mandibular plane angles in horizontal skeletal Classes were as follows: In Classes I, II, and III, there were respectively 33, 40, and 19 low angles, 52, 47, and 46 normal angles, and 35, 33, and 55 high angles. These mandibular angle distributions were significantly different among skeletal Classes I, II, and III (P = 0.004).

Cephalometrics

Descriptive statistics and 95% CIs for the cephalometric variables as well as the results of the one-way ANOVA comparisons across different skeletal Classes and also among different vertical growth patterns are presented as Tables 1 and 2. The Tamhane post hoc test showed that all the pairwise comparisons performed after the significant ANOVAs were significant (all P values < 0.000001). The ANOVA showed that only FMA and the lower facial height were not significantly different across Classes I, II, and III (Table 1). The Tamhane post hoc test showed that all the ensued pairwise comparisons were significant (all P values < 0.000001). Regarding the vertical growth patterns, only the ANOVA comparisons pertaining to the FMA and the lower facial height had become significantly different among the vertical growth patterns (Table 2). The Tamhane post hoc pairwise comparisons were all significant (all P values < 0.000001).

MF position

Descriptive statistics and 95% CIs for the MF parameters in all the sub-groups are presented in Table 3; Figs. 4, 5 and 6.

For the parameter S (width), the 3-way ANCOVA (adjusted R-squared = 0.974, Fig. 4) showed that the effects of age (P = 0.078) and sex (P = 0.170) were insignificant. The effects of horizontal skeletal Classes (P < 0.000001) and vertical growth patterns (P < 0.000001) were significant. Regarding skeletal Classes, all pairwise comparisons became significant (all P values < 0.000001, Bonferroni); S was the smallest in Class II and longest in Class III cases. Similarly, all pairwise comparisons between different vertical growth patterns became significant (all P values ≤ 0.00008); S was the shortest in long faces and the largest in short faces. The only significant interaction was that of horizontal skeletal Classes and vertical growth patterns (P < 0.000001).

Fig. 4

Means (and 95% CI) for the variable S (in mm, the perpendicular distance to the symphysis on the axial plane) in different Classes and vertical growth patterns. This distance shows the horizontal “width” dimension

For the parameter C (height), the 3-way ANCOVA (adjusted R-squared = 0.922, Fig. 5) showed that the effects of age (P = 0.198) and sex (P = 0.886) were non-significant. The effects of horizontal skeletal Classes (P = 0.00002) and vertical growth patterns (P < 0.000001) were significant. Regarding skeletal Classes, pairwise comparisons between Class III and each of Classes I or II became significant (both P values ≤ 0.003, Bonferroni), C being larger in Class III than both Classes I and II. However, there was not a significant difference between Classes I and II in terms of the parameter C (height) (P = 0.684). All pairwise comparisons between different vertical growth patterns became significant (all P values < 0.000001); C was the largest in long faces and smallest in short faces. The only significant interaction was between horizontal skeletal Classes and vertical growth patterns (P = 0.00002).

Fig. 5

Means (and 95% CI) for the variable C (in mm, the perpendicular distance to the inferior cortex of the mandible on the cross-sectional plane) in different Classes and vertical growth patterns. This distance represents the inferior-superior “height” dimension

For the parameter R (length), the 3-way ANCOVA (adjusted R-squared = 0.960, Fig. 6) showed that the effects of age (P = 0.065) and sex (P = 0.979) were insignificant. The effects of horizontal skeletal Classes (P < 0.000001) and vertical growth patterns (P < 0.000001) were significant. Regarding skeletal Classes, all pairwise comparisons became significant (all P values < 0.000001, Bonferroni); R was the largest in Class III and smallest in Class II. In terms of vertical growth patterns, pairwise comparisons between R values in short-face people versus normal- or long-face individuals were significant (both P values ≤ 0.00003, Bonferroni) with short-face patients having the largest R values. However, there was not a significant difference between R values measured in long faces versus normal faces (P = 0.448). The only significant interaction was that of horizontal skeletal Classes and vertical growth patterns (P < 0.000001).

Fig. 6

Means (and 95% CI) for the variable R (in mm, the perpendicular distance to the anterior border of the ramus on the parasagittal plane) in different Classes and vertical growth patterns. This distance indicates the anterior-posterior “length” dimension

Table 1 Descriptive statistics and 95% CIs for cephalometric variables in different skeletal Classes. The P values are computed using the one-way ANOVA.Table 2 Descriptive statistics and 95% CIs for cephalometric variables in different vertical growth patterns. The P values are computed using the one-way ANOVA.Table 3 The MF parameters in all subgroups (in mm)Table 4 The position of mental foramen (in mm) in different skeletal Classes and vertical growth patterns. The P values are calculated using the one-way ANOVA.Correlations between age with the 3 MF parameters

The Pearson coefficient showed no significant correlations between age with each of the 3 MF parameters (each n = 360, coefficients ranged between − 0.015 and − 0.029, all 3 P values ≥ 0.579).

Differences in the MF position across horizontal classes I, II, and III

According to the one-way ANOVA, there were significant differences among skeletal Classes I to III in the case of the MF parameters S and R (Table 4). All the Tamhane post hoc comparisons were significant (all P values < 0.000001).

In each of the sexes, both parameters S and R were significantly different across Classes I, II, and III (Table 5). All Tamhane pairwise comparisons were significant (all P values < 0.000001). However, in either of the sexes assessed separately, the parameter C (height) was not different among the Classes (Table 5).

The only insignificant ANOVA comparison (P = 0.092) was for the parameter C (height) compared among the Classes within the ‘short-face’ group (Table 6). Except for this ANOVA comparison, all other ANOVA comparisons performed among Classes I, II, and III separately in each of the groups ‘short, long, and normal face’ were significant (Table 6). The Tamhane pairwise comparisons of the parameters S and R became all significant (all P values < 0.000001). In the case of the parameter C (height) in normal-face patients, the Tamhane test following the significant ANOVAs showed that the pairwise comparison of the parameter C (height) between Classes I and II were not significant (P = 0.992), but the other two were significant (both P values < 0.001). In the case of the parameter C (height) in long-face patients, the only significant pairwise comparison was seen between Classes I and II (P = 0.002), and the other two were insignificant (P ≥ 0.06).

Table 5 The position of mental foramen (in mm) in different skeletal Classes and vertical growth patterns, separately in males and females. The P values are calculated using the one-way ANOVA.Table 6 The position of mental foramen (in mm) in skeletal Classes in different vertical patterns. The P values are calculated using the one-way ANOVA.Table 7 The position of mental foramen (in mm) in vertical patterns within different skeletal Classes. The P values are calculated using the one-way ANOVA.Differences in the MF position across vertical growth patterns

There was a significant difference among the vertical growth patterns only in the case of the MF parameter C (height) (Table 4). All the post hoc comparisons were significant (all P values < 0.000001).

In each of the sexes, only the variable C was different across the ‘long, short, and normal’ vertical growth patterns (Table 5). All Tamhane pairwise comparisons were significant (all P values < 0.000001). However, in both sexes, the parameters S and R were not different among vertical growth patterns (Table 5).

The only ANOVA comparison that became insignificant was that of the parameter S (width) compared among vertical growth patterns within Class III patients (Table 7). All other ANOVA comparisons became significant (Table 7). Most of the Tamhane post hoc comparisons became significant (all significant P values ≤ 0.014). A few pairwise comparisons became insignificant: The parameter S (width) compared between long and normal face patterns within the Class II group (P = 0.933); the parameter R (length) compared between long and normal faces within the Class II group (P = 0.560); and the parameter R (length) compared between long and normal faces within the Class III group (P = 0.724).

Differences between MF parameters in men versus women

The independent-samples t-test showed that only the parameter C (height) was significantly greater in males than in females (Table 8). The other two parameters did not show sex dimorphism (Table 8).

Table 8 Sex dimorphism in mental foramen parameters (in mm). The P values are calculated using the independent-samples t-test