The average age of individuals in our cohort was 40 years. Of the 400 CBCT images, 10 individuals (2.5%) presented with discrepancies in the measurements of ANS between the researchers. Eight of these individuals were from the white population group and 6 of these were males. Table 1 shows the correlations of population group and sex with the different ANS measurements according to age.

The relationship between the two population groups and the different ANS measurements according to age can be visualised in Fig. 3a–e.

a–e Scatterplots showing the relationship of the different ANS measurements for the two population groups according to age. In these scatterplots, light blue indicates BSA individuals and dark blue indicates WSA individuals

The relationship between sex and the different ANS measurements according to age can be visualised in Fig. 4a–e. These included all females and males irrespective of population group.

a–e Scatterplots showing the relationship of the different ANS measurements for sex and age. In these scatterplots, light blue represents FSA and dark blue represents MSA

The data was further divided into separate population and sex groups. Classification decision trees (pruned) were fitted to ascertain the relationship between population group, sex, age and the measurements (Sag1, Sag2, Sag3, Ax1, Ax2). We further performed a 100-fold cross-validation with training data of size 300 and testing data of size 100. The analysis was conducted in R v 4.1.3 (R Core Team (2022)) using the rpart package [26].

When female sex was considered with population group as the dependent variable, we found that Ax1 presented with a Pearson correlation that was less than 0.5 (0.4959155) and a p-value that was statistically significant (p-value = 8.22e-14). The findings show that when Ax1 had a measurement of less than 6 mm, this indicated that it was a black South African female (BSAF) (Fig. 5a). Whereas, when Ax1 was longer than 15 mm, all of the individuals were white South African females (WSAF). Furthermore, in the female population, the Ax2 angle was found to be statistically significant (p-value < 2.2e-16, Pearson correlation of 0.5493407). When females presented with an Ax2 angle of less than 100°, then the majority of individuals were WSAF and if the angle was greater than 130°, then it was more likely to be a BSAF (Fig. 5b).

a and b Scatterplots for FSA with population group and the axial measurements Ax1, Ax2. Light blue represents BSAF and dark blue represents WSAF

In males, the Sag1 measurement showed a Pearson’s correlation of − 0.4389073 and p-value of 7.974e-11 with the population group. The findings showed that white South African male (WSAM) individuals were more likely to present with a Sag1 measurement longer than 9 mm (Fig. 6a), while the majority of individuals that presented with a Sag1 length shorter than 3 mm were black male South African (BSAM) individuals. A statistically significant association between the length of the ANS (Ax1) and the angle of the ANS with the canine bulges (Ax2) was seen in males (Pearson’s correlation of − 0.5067143 and 0.5571469 respectively). More BSAM individuals presented with an Ax1 length of less than 10 mm, with all of individuals that presented with Ax1 less than 5 mm being BSAM. White South African male (WSAM) individuals presented with a longer Ax1 and only WSAM presented with an Ax1 longer than 15 mm (Fig. 6b). Furthermore, the majority of BSAM presented with an Ax2 angle (Fig. 6c) that was obtuse (larger than 130°) when compared to that of WSAM (100° and smaller). Significantly, individuals that presented with an Ax2 measurement of less than 90° were strictly WSAM.

a–c Scatterplots for MSA against the population group indicating Sag1, Ax1, Ax2 measurements. Light blue represents BSAM and dark blue represents WSAM

A regression decision tree was created for the two population groups with sex as the dependent variable. This was performed using all the ANS variables and separately including and excluding age. When considering the decision trees where age was included, the only ANS measurements that showed significant correlation were Sag2 and Ax1 (Root node error = 0.49749).

In BSA individuals (Fig. 7), the Sag2 measurement was the first measurement to segment the data for sex. When Sag2 was ≥ 51 mm, 43% of the population was BSAM (with a probability of 28%) and when the Sag2 measurement was smaller than 51 mm, 57% of black population were BSAF (with a probability of 67%). Of this 57% BSAF with a Sag2 smaller than 51 mm, 34% presented with an Ax1 that was smaller than 9.2 mm (probability of 75%) and a further 28% of this 34% BSAF had Sag2 < 49 mm (with a probability of 82%).

Decision tree for BSA individuals using sex group and ANS measurements Sag2 and Ax1. In this figure green represents BSAF and blue represents BSAM

When BSA individuals were considered excluding age and including all the ANS variables, and when considering only the sagittal variables, Sag2 ≥ 51 mm remained the first variable to separate the data by sex (Fig. 8). When excluding age and only considering the axial measurements, a similar distribution between BSAM and BSAF was observed with the Ax1 measurement serving as the first separator of the data.

Decision tree for BSA individuals excluding age, using sex group and the sagittal variables only. In this figure green represents BSAF and blue represents BSAM

When considering the ANS variables independently, Sag1, Sag3 and Ax2 showed an observable distinction between BSAF and BSAM (Fig. 9a–c).

a Decision tree for BSA excluding age and only considering Sag1. When Sag1 was 3.1 mm and longer, 74% of the BSA population were BSAM and BSAF (26%) presented with a Sag1 of less than 3.1 mm. In this figure green represents BSAF and blue represents BSAM. b Decision tree for BSA excluding age and only considering Sag3. When Sag3 was larger than 68 ̊, 67% presented as BSAF and when Sag3 was smaller than 68 ̊, 33% were BSAM. In this figure green represents BSAF and blue represents BSAM. c Decision tree for BSA excluding age and only considering Ax2. When Ax2 is larger than 104 degrees, 92% were found to be BSAF. In this figure green represents BSAF and blue represents BSAM

The decision tree for WSA individuals with sex as the dependent variable and using age, Sag2 and Ax1 (Root node error = 0.49751) showed the following. When the Sag2 measurement was less than 53 mm, 62% of the individuals were WSAF (with a probability of 70%) and 38% of the WSA had a Sag2 measurement equal or larger than 53 mm and were WSAM. Of the 62% WSAF that presented with Sag2 less than 53 mm, 41% of them were older than 37 years (with a probability of 82%). Of these WSAF that were older than 37 years, 27% had a Sag2 of smaller than 52 mm (with a probability of 89%). Of the WSAF older than 37 years, 14% had a Sag2 52 mm and greater. Of these WSAF, 9% were younger than 62 years (Fig. 10).

Decision tree for WSA individuals using age, Sag2 and Ax1. In this figure green represents WSAF and blue represents WSAM

The decision tree for WSA individuals when age was excluded still showed Sag2 to be relevant in addition to all the other ANS measurements. With age excluded, 62% were WSAF when Sag2 was less than 53 mm. A further 54% of these WSAF presented with an Ax2 less than 120° and a further 32% of these has a Sag3 equal or larger than 62°. WSAM (38%) on the other hand presented with Sag2 53 mm and larger, and 29% of these WSAM had an Ax2 of 92° and greater (Fig. 11).

Decision tree for WSA individuals excluding age, using sex and all the ANS measurements. In this figure green represents WSAF and blue represents WSAM

When only the sagittal ANS measurements were considered, Sag2 was still the most prominent measurement in discriminating sex. When only the axial ANS measurements were considered, Ax1 was found to be 7.9 mm or longer in 95% of WSAM a further 36% of these males presented with an Ax2 107° and larger. Only 5% of WSAF presented with Ax1 smaller than 7.9 mm (Fig. 12).

Decision tree for WSA individuals excluding age, using sex and the axial ANS measurements only. In this figure green represents WSAF and blue represents WSAM

When considering the ANS measurements for WSA independently excluding age, all the ANS measurements showed observables differences for sex (Fig. 13a–e).

a Decision tree for WSA excluding age and including only Sag1. When Sag1 was 4.2 mm or larger, 82% of the WSA population presented as WSAM. When Sag1 was less than 4.2 mm, 18% of the WSA population presented as WSAF. b Decision tree for WSA excluding age and including only Sag2. When Sag2 is 53 mm and longer, only 38% of the WSA population presented as WSAM. When Sag2 was less than 53 mm, 62% of the WSA population presented as WSAF. c Decision tree for WSA excluding age and including only Sag3. When Sag3 is less than 100 ̊, 97% of the WSA population presented as WSAM. When Sag3 was more than 100 ̊, 3% of the WSA population presented as WSAF. d Decision tree for WSA excluding age and including only Ax1. When Ax1 is 7.9 mm and larger, 95% of the WSA population presented as WSAM. e Decision tree for WSA excluding age and including only Ax2. When Ax2 was less than 121 ̊, 90% of the WSA population presented as WSAM

Regression decision trees were created for the two sexes with population as the dependent variable. All correlations were found to be significant for FSA (Root node error = 0.5). When the Ax2 was larger than 106°, 64% of FSA were BSAF (probability of 71%) whereas 36% of were WSAF when the Ax2 was smaller than 106° (Fig. 14). Of the 64% BSAF with an Ax2 of greater than 106°, 45% of them were younger than 49 years. Of these BSAF younger than 49 years, 26% presented with a Sag3 greater than 78° (with a probability of 96%) and 22% had a Sag3 less than 78°. Of the 22% BSAF that had a Sag3 smaller than 78°, 20% (with a 71% probability) were older than 11 years of age. Of the WSAF that were 49 years and older, 17% presented with Ax1 that was ≥ 6.4 mm.

Decision tree for FSA including age and all the ANS measurements. In this figure green represents WSAF and blue represents BSAF

When looking at FSA excluding age for both population groups and all the ANS measurements, the findings were similar to when age was included. When Ax2 was larger than 106 ̊, 64% of the FSA population was found to be BSAF (Fig. 15).

Pruned decision tree for FSA excluding age and Ax2

When only the sagittal ANS measurements were considered, Sag1 was the first measurement to separate the data (Fig. 16). Sag2 and Sag3 did not show any large differences between the population groups.

Pruned decision tree of FSA, excluding age and only considering Sag1. WSAF (75%) presented with Sag1 3.5 mm and larger

When considering MSA with population group as the dependent variable and age was included, all correlations were found to be significant (Root node error = 0.495) when performed individually. For MSA, 58% were found to be BSAM when the Ax2 angle was 110° or larger (probability of 71%) and 42% were found to be WSAM when Ax2 was less than 110° (Fig. 17). Of the BSAM males that presented with Ax2 110° or larger, 52% were younger than 59 years (probability of 78%). In BSAM younger than 59 years, 31% had a Sag1 length of 3.8 mm or longer. Furthermore, 18% of these BSAM presented with an Ax2 larger than 116° (probability of 78%) and 16% of these additionally had a Sag3 angle greater than 55°. The WSAM on the other hand made up the other 42% of the MSA and presented with Ax2 smaller than 110°. Thirty percent of these WSAM had an Ax2 larger than 96° (with a probability of 28%). Of these WSAM with an Ax2 larger than 96°, 18% further presented with a Sag2 53 mm or longer. Of these WSAM, 16% had a Sag1 of 4.3 mm and greater and 10% of these were older than 29 years.

Decision tree for MSA including age and all the ANS measurements. In this image green indicates WSAM and blue indicated BSAM

However, when the classification tree was fitted for MSA, Sag2 was not found to be significant. When Sag2 was then removed, the following was found for MSA (Fig. 18). Fifty eight percent of the MSA that presented with Ax2 larger than 110° (71% probability) were BSAM. Of these BSAM, 52% were younger than 59 years of age and 31% of these BSAM younger than 59 years had a Sag1 3.8 mm and longer. Eighteen percent of these BSAM further presented with an Ax2 larger than 116° (65% probability).

Decision tree for MSA including age with population group with Sag2 removed. In this image green indicates WSAM and blue indicated BSAM

When age was excluded for MSA and all the ANS measurements included, 42% of the population were WSAM when Ax2 was less than 110 ̊. The remainder of the male population (58%) presented with an Ax2 larger than 110 ̊ and were BSAM. Of the 58% BSAM, 35% presented with a Sag1 4.3 mm and larger (Fig. 19).

Decision tree for MSA with age excluded for population group and all the ANS measurements. In this image green indicates WSAM and blue indicated BSAM

When only considering the sagittal variables, Sag1 was the first measurement to separate the data with 64% of MSA as WSAM when Sag1 ≥ 4.4 (Fig. 20a). When only the axial variables were considered, 58% of MSA presented as BSAM when Ax2 was larger than 110 ̊ (Fig. 20b). When considering the axial and sagittal variables independently, both Sag1 and Sag3 were relevant in separating WSAM and BSAM (Fig. 21). Sag2, Ax1 and Ax2 were fairly equally distributed in the black and white male populations

a and b Decision trees for MSA excluding age for population group and the sagittal and axial ANS measurements

a and b Decision trees for MSA excluding age for population group and Sag1 where 64% WSAM presented with Sag1 of 4.4 mm and longer and Sag3 larger than 77 ̊