Use of X means and C4.5 algorithms on lateral cephalometric measurements to identify craniofacial patterns

Authors: Merve Gonca, Mehmet Birol Özel

Abstract

Background

Craniofacial phenotyping is essential for individualized orthodontic diagnosis and treatment planning. Traditional skeletal classifications, such as the ANB angle, may oversimplify complex relationships among malocclusion types. Machine learning-based unsupervised methods may allow for more nuanced sub-phenotypic classification.

Methods

A total of 330 pre-treatment LCRs (110 each from Class 1, Class 2, and Class 3 based on ANB (°)) were assessed in this study. The X-means method was used to create clusters. The relationship between the clusters and cephalometric variables was evaluated using the C4.5 decision tree. X-means clustering was employed to identify natural groupings within the dataset, followed by C4.5 decision tree analysis to determine key discriminative variables. After post-pruning, 288 LCRs were included in the final analysis. One-way ANOVA and Kruskal-Wallis tests were used to assess differences among clusters.

Results

A total of four clusters were obtained using the X-means algorithm. Decision trees were used to identify the most discriminative variables among clusters. These clusters exhibited distinctive sagittal and vertical skeletal and dental features, particularly differences in individualized ANB, interincisal angle, and mandibular plane inclination. The root node in the second decision tree was the Individualized ANB (°). The interincisal angle was the main parameter determining the distinction between Clusters 0 and 1. The main parameter that determined the distinction between Cluster 2 and Cluster 3 was N-Go-Gn (°). Significant differences were found in all measurements except N-Go-Ar (°), FH/PP (°), and S-Ar-Go (°) angles (p < 0.05).

Conclusion

The combination of X-means clustering and C4.5 decision tree analysis enabled the identification of four distinct craniofacial sub-phenotypes across all skeletal malocclusion classes. Four sub-phenotypic categorizations of all skeletal malocclusions were obtained. Mandibular plane inclination and interincisal angle were the most critical variables distinguishing these phenotypes. Assessing various forms of skeletal malocclusions may improve clinical outcomes and diagnostics by showing how different skeletal classes interact.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12903-025-06651-6.

Background

Lateral cephalometric radiographs (LCRs) play a significant role in the diagnostic process in orthodontics [1]. Various intracranial reference planes were used in the measurements that constituted the foundation of cephalometric analyses. Various measurements assess different categories exhibiting interactions due to their shared intracranial reference planes [2]. The impact of growth patterns in various directions extends beyond the measurements within a specific group, influencing the measurements observed in other groups. For example, in an individual with a more vertical growth pattern, measurements of the maxillomandibular relationship may show that the existing relationship is more prone to a Class 2 relationship [3].

Data mining enables identifying and analyzing potentially valuable information within a dataset using various algorithms [4, 5]. In orthodontics, various methodologies, including the clustering method, decision tree analysis, principal component analysis (PCA), and network analysis, have been employed for data interpretation [6–12]. Auconi et al. [8, 9] used fuzzy cluster and network analysis to estimate the risk of imbalance in growth patterns and unsuccessful treatment outcomes in individuals with Class 3 malocclusions. Another study by Auconi et al. [7] also predicted favorable/unfavorable growth prediction in individuals with Class 3 malocclusions using discriminant analysis and classification trees. Uribe et al. [11, 12] used PCA and cluster analysis to determine the phenotypic characteristics of individuals with Class 2 and Class 3 malocclusions.

The primary objective of the clustering algorithm is to optimize intra-class similarity while minimizing inter-class similarity. This is achieved by grouping similar objects in clusters or subgroups without any predetermined classification [13, 14]. X-means clustering technology is considered an advancement of K-means clustering technology because it incorporates the automatic calculation of cluster numbers [15, 16]. One of the most common techniques used in data mining is decision trees, which have nodes, branches, and leaves for variables, conditions, and outcomes. The decision tree root node contains the most predictive variable. The algorithms in this study gradually purify clusters at each node by reducing disorder or impurity in the initial data set. Entropy and information gain can quantify disorder and impurity. Decision trees use classification or if-then rules to extract and represent knowledge. Each system rule represents a pathway from the root node to each leaf node. Decision trees, also known as tree diagrams, are useful decision support tools in operations research, particularly in decision analysis. A decision tree determines the strategy with the highest probability of success. Trees can also describe conditional probabilities. A decision tree is a hierarchical structure like a flow chart with branches representing test outcomes and terminal nodes representing classes. The highest information gain test attribute is chosen for the current node. This trait reduces specimen categorization data [4, 17, 18].

Methods

Aim and hypothesis

This study aims to show the relationship between the clusters and the dataset through C4.5 decision tree analysis of the clusters generated using the X-means algorithm.

The study’s objective of phenotype differentiation through unsupervised learning, the following hypotheses were

Null hypothesis (H₀): Cephalometric measurements do not significantly differ among the clusters derived from skeletal Class 1, 2, and 3 individuals.

Alternative hypothesis (H₁): Cephalometric measurements show statistically significant differences among the clusters derived from skeletal Class 1, 2, and 3 individuals.

This retrospective study was conducted in accordance with the ethics defined in the Declaration of Helsinki. The present study was approved by the research ethics committee of Karadeniz Technical University (number: 2016/106). The study sample was selected through retrospective screening of LCRs and panoramic radiographs (PRs) taken routinely from patients who were referred for orthodontic treatment in the Department of Orthodontics, Faculty of Dentistry, Technical University. Informed written consent was obtained for each patient routinely at the beginning of treatment, including consent to use patient records in scientific studies. This retrospective study was conducted between June 2016 and May 2017.

The inclusion criteria were as (1) LCRs and PRs of individuals whose growth and development have ended or are close to ending (cervical vertebral maturation stage 5–6); (2) good quality of LCRs and PRs; (3) no excessive facial asymmetries; (4) no diagnosed systemic disease; (5) no congenital deformities such as cleft lip and palate or syndrome affecting the bone; (6) no history of dental and maxillofacial trauma; (7) no mandibular surgery or previous orthodontic treatment; (8) no missing or impacted teeth other than third molars.

The study included radiographs of 330 individuals, with 110 individuals in each group. These groups consisted of individuals with Class 1, Class 2, and Class 3 malocclusions, classified according to ANB angles. In the initial sample (n = 330), the participants’ chronological ages ranged from 14.10 to 28.37 years (mean ± SD: 17.46 ± 2.77).

LCRs were digitized using the AudaxCeph Imaging (Advantage Cephalometric X-Ray Analysis Software Ver 4.2.0.3101). Seventy-four measurements were used in the study. The measurements are demonstrated in supplementary content. LCR measurements were performed by an orthodontist (X) with 4 years of experience.

Measurement error

One month after the first measurement, 33 randomly selected LCRs were reassessed by the same examiner (X) to evaluate measurement errors. The intraobserver agreement was analyzed using the intraclass correlation coefficient (ICC).

Statistical analysis

The X-means clustering algorithm, an extension of the conventional K-means method, was employed to determine the optimal number of clusters in the dataset. Unlike K-means, which requires a predefined number of clusters, X-means automatically estimates the optimal cluster count using the Bayesian Information Criterion (BIC) to evaluate model quality at each splitting step. This allowed the model to converge on a four-cluster solution based on internal data structure. Following clustering, the C4.5 decision tree algorithm was applied to identify the most informative cephalometric parameters distinguishing the clusters. The algorithm uses entropy-based information gain to determine node splits and builds a hierarchical classification tree. Post-pruning was used to reduce model complexity and prevent overfitting [15].

The RapidMiner (version 7.0.3) software (RapidMiner, Germany) was used for the X-means and C4.5 algorithms. The X-means algorithm was used to generate clusters from the dataset, and the association between the clusters and the dataset was demonstrated using C4.5 decision tree analysis. The data pool underwent post-pruning. This procedure reduced the data size from 330 to 288 people. Four distinct clusters were formed by iteratively applying the same methods to a sample size of 288 individuals. The association between these clusters and the dataset was subsequently elucidated by constructing a secondary decision tree. After post-pruning, the final dataset (n = 288) included individuals aged 14.10 to 28.37 years (mean ± SD: 17.48 ± 2.85). Each of the four clusters identified by the X-means algorithm was characterized by distinct craniofacial patterns based on the distribution of cephalometric measurements. Cluster 0 included individuals with relatively average sagittal and vertical proportions, reflecting mild Class 1–2 features. Cluster 1 exhibited pronounced Class 2 characteristics, such as increased ANB angles and proclined incisors. Cluster 2 corresponded to skeletal Class 3 patterns, characterized by a prognathic mandible, reduced vertical facial height, and negative ANB values. Cluster 3 also represented skeletal Class 3 features but with increased vertical dimension and dentoalveolar compensation. These definitions were derived from the structure of the decision tree and supported by statistical differences between clusters.

The data conformity to normal distribution was analyzed using the Kolmogorov-Smirnov test. Descriptive statistics were calculated for all parameters in the study and presented with tables using mean and standard deviation for parametric data and median, minimum, and maximum for non-parametric data. One-way analysis of variance (ANOVA) was used to determine the difference among the groups in the parametric data obtained, and the post-hoc Tukey (Tukey HSD) test was used for paired comparisons between the groups. P-values of < 0.05 were accepted as statistically significant. The Kruskal-Wallis test was used to determine the difference among the groups in non-parametric data, and the Mann-Whitney U test was used for paired comparisons between groups. P-values of < 0.0083 were accepted as statistically significant.

Results

ICC values less than 0.5 indicate poor reliability, values between 0.5 and 0.75 indicate moderate reliability, values between 0.75 and 0.9 indicate good reliability, and values over 0.90 indicate excellent reliability [19]. The intra-rater agreement was estimated using the ICC and was found to be excellent for all LCR measurements (ICC ≥ 0.906) except 180-(DC/Xi-PM)°, SN/PP°, FH/PP°, SN/FH°, N-S-Ar°, S-Ar-Go°. The ICCs of these measurements were 0.877, 0.893, 0.895, 0.795, 0.847, and 0.859, respectively, and had good reliability.

Four clusters showing different skeletal craniofacial patterns were created in the first part of the statistical analysis using the X-means algorithm. The number of individuals in clusters was calculated as 95 individuals in Cluster 0 (female; 69, male; 26), 93 individuals in Cluster 1 (female; 61, male; 32), 62 individuals in Cluster 2 (female; 42, male; 20), and 80 individuals in Cluster 3 (female; 53, male; 27). To determine the relationship between the dataset and clusters, the first decision tree was created by targeting four clusters (Fig. 1).

Fig. 1First decision tree

In the second part of the analysis, post-pruning was applied to the decision tree, and 42 individuals were extracted. The same methods as in the first part of the analysis were repeated for 288 individuals. Four new clusters were 89 individuals in Cluster 0 (female; 63, male; 26), 83 individuals in Cluster 1 (female; 57, male; 26), 53 individuals in Cluster 2 (female; 39, male; 14), and 63 individuals in Cluster 3 (female; 43, male; 20). The second decision showed the relationship between clusters and the dataset (Fig. 2). Fig. 3 presents the number and percentage of individuals in each skeletal class to illustrate the distribution of individuals across clusters in both parts of the analysis.

Fig. 2Second decision tree

Fig. 3Malocclusion Class Percentages

The root node in the second decision tree was the calculation of individualized ANB. The internal node where the individualized ANB calculation was greater than − 2.3 pointed to Cluster 0 and Cluster 1, and the internal node where it was smaller pointed to Cluster 2 and Cluster 3. The main parameter determining the distinction between Cluster 0 and Cluster 1 was the interincisal angle°, and the main parameter determining the distinction between Cluster 2 and Cluster 3 was N-Go-Gn° (Fig. 2).

Differences between clusters showed statistically significant differences in all measurements except N-Go-Ar°, FH/PP°, and S-Ar-Go° (p < 0.05) (Tables 1 and 2, and Table 3). Although the S-Go measurement showed a statistically significant difference, no significant difference was observed among the groups in the post-hoc comparison (Table 2).

Table 1Intercluster comparison of lateral cephalometric measurementsCluster 0 (n = 89)Cluster 1 (n = 83)Cluster 2 (n = 53)Cluster 3 (n = 63)Post-hoc comparisonLCR measurementsOrt ± SSOrt ± SSOrt ± SSOrt ± SS p 0–10–20–31–21–32–3SNA°78.98 ± 3.2580.97 ± 3.7580.11 ± 3.7277.73 ± 3.45 < 0.001

0.002 0.2560.1430.508 < 0.001

0.002 FH/NA°90.7 ± 391.52 ± 3.6789.7 ± 3.888.52 ± 3.59 < 0.001 0.4170.351 0.001

0.017

< 0.001 0.265Co-A (mm)79.88 ± 4.2681.83 ± 5.0678.62 ± 4.3977.4 ± 4.48 < 0.001

0.028 0.384 0.006

< 0.001

< 0.001 0.482SNB(°)74.73 ± 3.2575.67 ± 3.6682.13 ± 3.4379.02 ± 3.64 < 0.001 0.295 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 FH/N-Pg(°)^+^87.87 ± 3.0987.05 ± 3.6893.06 ± 2.7490.47 ± 3.79 < 0.001 0.537 < 0.001

< 0.001

< 0.001

< 0.001

0.001 Xi-PM (mm)62.85 ± 3.3963.51 ± 3.9769.11 ± 3.9269.73 ± 4.28 < 0.001 0.676 < 0.001

< 0.001

< 0.001

< 0.001 0.826Co-Pg (mm)103.84 ± 5.56105.63 ± 6.13110.55 ± 5.88113.62 ± 6.06 < 0.001 0.195 < 0.001

0.001

< 0.001

< 0.001

0.028 Go-Me (mm)65.25 ± 4.2765.62 ± 4.7271.59 ± 4.6171.02 ± 4.54 < 0.001 0.952 < 0.001

< 0.001

< 0.001

< 0.001 0.907SNPg(°)76.14 ± 3.4876.5 ± 3.8583.47 ± 3.2479.68 ± 3.78 < 0.001 0.913 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 Go-Me/N-S (%)98.78 ± 6.6196.98 ± 6.07108.31 ± 6.36109.99 ± 7.81 < 0.001 0.295 < 0.001

< 0.001

< 0.001

< 0.001 0.536180-(DC/Xi-PM) (°)33.85 ± 5.5132.27 ± 6.0535.87 ± 5.5528.75 ± 5.96 < 0.001 0.2790.186 < 0.001

0.003

0.002

< 0.001 Ar-Go-Gn (°)124.25 ± 5.28124.92 ± 5.16123.14 ± 4.71132.49 ± 5.31 < 0.001 0.8310.602 < 0.001 0.206 < 0.001

< 0.001 N-Go-Ar (°)52.02 ± 3.9151.57 ± 3.6352.34 ± 3.6152.1 ± 4.250.6870.8650.9660.9990.6670.8390.988N-Go-Gn (°)^+^72.23 ± 4.373.35 ± 4.7770.81 ± 3.6980.38 ± 3.28 < 0.001 0.4940.212 < 0.001

0.004

< 0.001

< 0.001 Go-Ar (mm)41.43 ± 4.0842.6 ± 4.4444.13 ± 4.5942.48 ± 4.98 0.014 0.316 0.007 0.4860.2170.9980.201SN/AB (°)67.99 ± 4.6867.45 ± 5.0185.23 ± 5.1980.81 ± 5.98 < 0.001 0.906 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 A-B/Go-Me (°)76.41 ± 5.5876.71 ± 4.9263.44 ± 4.7757.02 ± 5.79 < 0.001 0.982 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 µ (°)13.59 ± 5.5813.29 ± 4.9226.56 ± 4.7732.98 ± 5.79 < 0.001 0.982 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 PP/AB (°)78.29 ± 5.0776.16 ± 4.8593.97 ± 5.5491.13 ± 5.89 < 0.001

0.043

< 0.001

< 0.001

< 0.001

< 0.001

0.022 FH/AB (°)^+^79.71 ± 4.5478 ± 4.7494.82 ± 4.1191.6 ± 6.29 < 0.001 0.098 < 0.001

< 0.001

< 0.001

< 0.001

0.008 (Co-Pg)-(Co-A) (mm)23.96 ± 3.8623.8 ± 3.931.93 ± 3.9836.23 ± 4.68 < 0.001 0.994 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 + 1/NA (°)^+^17.3 ± 7.5528.52 ± 6.5330.04 ± 6.6626.92 ± 5.09 < 0.001

< 0.001

< 0.001

< 0.001 0.7250.461 0.037 + 1/FH (°)108.01 ± 7.69119.15 ± 6.94119.75 ± 7.11115.44 ± 5.33 < 0.001

< 0.001

< 0.001

< 0.001 0.995 < 0.001

0.005 + 1/SN (°)96.28 ± 7.19109.49 ± 6.87110.15 ± 7.09104.65 ± 5.72 < 0.001

< 0.001

< 0.001

< 0.001 0.946 < 0.001

< 0.001 + 1/PP (°)106.59 ± 7.44118.2 ± 6.97118.89 ± 6.25114.97 ± 5.69 < 0.001

< 0.001

< 0.001

< 0.001 0.937 0.023

0.011 + 1/APg (mm)4 ± 2.58.75 ± 2.083.64 ± 2.353.61 ± 2.45 < 0.001

< 0.001 0.8170.743 < 0.001

< 0.001 1-1/APg (°)21.76 ± 4.7528.35 ± 4.8427.48 ± 5.7624.92 ± 4.26 < 0.001

< 0.001

< 0.001

< 0.001 0.738 < 0.001

0.027 -1i/APg (mm)-0.28 ± 2.262.56 ± 2.453.17 ± 2.453.8 ± 2.23 < 0.001

< 0.001 0.770.603 < 0.001

< 0.001 0.997IMPA (°)93.11 ± 6.06100.69 ± 5.2689.01 ± 6.5281.1 ± 6.44 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001 -1/FH (°)63.03 ± 6.0754.04 ± 5.8869.25 ± 6.2767.55 ± 6.2 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001 0.437-1/OP (°)^+^22.53 ± 6.6631.93 ± 6.116.62 ± 8.0416.59 ± 6.48 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001 1-1i/NB (mm)3.3 ± 1.896.47 ± 2.032.88 ± 1.833.56 ± 1.96 < 0.001

< 0.001 0.6910.845 < 0.001

< 0.001 0.301-1/NB (°)23.43 ± 5.6432.18 ± 5.4922.48 ± 6.3222.27 ± 5.41 < 0.001

< 0.001 0.5960.842 < 0.001

< 0.001 0.233One-way analysis of variance (ANOVA). p < 0.05Tukey HSD Test ^+^ Tamhane’s T2 test p < 0.05

Table 2Intercluster comparison of lateral cephalometric measurementsCluster 0 (n = 89)Cluster 1 (n = 83)Cluster 2 (n = 53)Cluster 3 (n = 63)Post-hoc comparisonLCR measurementsOrt ± SSOrt ± SSOrt ± SSOrt ± SS p 0–10–20–31–21–32–3SN/GoGn (°)^+^33.72 ± 5.3334.02 ± 5.929.11 ± 4.2639.82 ± 4.07 < 0.001 1 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 SN/OP (°)16.17 ± 4.0914.58 ± 4.3613.73 ± 5.0616.66 ± 4.73 0.001 0.102 0.01 0.9110.674 0.032

0.003 FH/MP (°)23.88 ± 4.7225.29 ± 5.5321.74 ± 4.0631.38 ± 4.28 < 0.001 0.218 0.049

< 0.001

< 0.001

< 0.001

< 0.001 ANS-Xi-PM (°)43.49 ± 4.1645.64 ± 5.1242.56 ± 3.9349.95 ± 3.95 < 0.001

0.018 0.709 < 0.001

< 0.001

< 0.001

< 0.001 SN/PP (°)10.31 ± 2.798.71 ± 3.038.74 ± 3.7110.32 ± 3.1 < 0.001

0.005

0.021 11 0.012

0.035 FH/Y (°)58.68 ± 3.4559.6 ± 4.1254.97 ± 2.8259.63 ± 4.05 < 0.001 0.363 < 0.001 0.404 < 0.001 1 < 0.001 FH/PP (°)-1.42 ± 3.3-1.84 ± 3.62-0.86 ± 3.79-0.47 ± 3.470.1030.8610.7930.3640.3860.0960.938SN/FH (°)11.72 ± 310.55 ± 3.259.6 ± 3.310.79 ± 3.27 0.002 0.077 0.001 0.2840.3170.9710.184SN/MP (°)^+^35.61 ± 5.435.83 ± 5.9631.33 ± 4.3342.17 ± 4.2 < 0.001 1 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 PP/MP (°)25.3 ± 5.1627.13 ± 5.6822.59 ± 3.9831.85 ± 5.29 < 0.001 0.095 0.014

< 0.001

< 0.001

< 0.001

< 0.001 Ba-N/Pt-Gn (°)88.24 ± 4.1587.94 ± 4.8393.26 ± 3.4787.89 ± 4.01 < 0.001 0.976 < 0.001 0.961 < 0.001 1 < 0.001 S-Go (mm)70.82 ± 5.5572.62 ± 5.5673.21 ± 6.1370.6 ± 6.44 0.021 0.1980.0810.9960.9190.180.075ANS-Me (mm) ^+^61.97 ± 5.4564.7 ± 5.7860.82 ± 4.1668.99 ± 5.76 < 0.001

0.010 0.64 < 0.001

< 0.001

< 0.001

< 0.001 N-Me (mm)110.2 ± 6.8112.43 ± 6.75110.4 ± 5.49119.32 ± 7.03 < 0.001 0.1230.998 < 0.001 0.301 < 0.001

< 0.001 S-Gn (mm)112.65 ± 5.98115.82 ± 6.8119.9 ± 6.14123.44 ± 6.91 < 0.001

0.008

< 0.001

< 0.001

0.002

< 0.001

0.019 AB distance (mm)35.32 ± 3.6936.55 ± 436.32 ± 3.2841.19 ± 3.46 < 0.001 0.1240.392 < 0.001 0.985 < 0.001

< 0.001 NB distance (mm)90.43 ± 5.8992.16 ± 5.8290.97 ± 4.5999.1 ± 5.73 < 0.001 0.1840.945 < 0.001 0.626 < 0.001

< 0.001 S-Go/N-Me (%)63.97 ± 4.2464.16 ± 4.1666.22 ± 4.0158.97 ± 3.64 < 0.001 0.989 0.008

< 0.001

0.022

< 0.001

< 0.001 Lower Face Height Ratio (%)54.14 ± 2.1154.92 ± 2.3654.71 ± 1.9956.83 ± 2.25 < 0.001 0.0910.439 < 0.001 0.946 < 0.001

< 0.001 S-N (mm)66.15 ± 3.3467.71 ± 3.4866.16 ± 3.5264.68 ± 3.38 < 0.001

0.016 1 0.048 0.052 < 0.001 0.095SN-Ba (°)132.76 ± 5.27131.49 ± 5.88129.51 ± 6.14131.76 ± 5.85 0.015 0.469 0.007 0.7180.2060.9920.154N-CC (mm)57.39 ± 3.258.56 ± 356.04 ± 3.1356.38 ± 3.34 < 0.001 0.0730.0680.212 <0.001

0.001 0.94S-Ar (mm)33.33 ± 3.1333.6 ± 3.8832.93 ± 3.3931.78 ± 4.02 0.026 0.9620.923 0.047 0.724 0.028 0.317N-S-Ar (°)126.51 ± 5.54124.1 ± 5.6122.15 ± 6.07123.53 ± 5.07 < 0.001

0.025

<0.001

0.007 0.190.9280.538S-Ar-Go (°)142.96 ± 6.6145 ± 6.5143.82 ± 7.01143.8 ± 5.830.2710.1690.8690.8590.7310.6881Björk (°)^+^395.61 ± 5.4395.83 ± 5.96391.33 ± 4.33402.17 ± 4.2 < 0.001 1 <0.001

< 0.001

<0.001

<0.001

<0.001 One-way analysis of variance (ANOVA). p < 0.05Tukey HSD Test ^+^ Tamhane’s T2 test p < 0.05

Table 3Intercluster comparison of lateral cephalometric measurementsCluster 0 (n = 89)Cluster 1 (n = 83)Cluster 2 (n = 53)Cluster 3 (n = 63)Post-hoc comparisonLCR measurementsMed. (Min-Max)Med. (Min-Max)Med. (Min-Max)Med. (Min-Max) p 0–10–20–31–21–32–3Pg/NB (mm)2.7 (-2.9-8)1.5 (-3.3-6.4)2.3 (-1.6-5.9)0.8 (-4.2-7.4) < 0.001

0.002 0.597 < 0.001 0.0120.283 0.001 ANB (°)3.9 (1.1–10)5.4 (-0.6-11.3)-1.6 (-7.5-2.5)-1.1 (-8-3.5) < 0.001 0.022 < 0.001

< 0.001

< 0.001

< 0.001 0.144NAPg (°)5.5 (-1.5-20.4)9.4 (-5-22.6)-6.7 (-21.3-1.5)-3.9 (-20.8-11.1) < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001

0.002 ABN(°)6.1 (1.6–14)8 (-0.8-14.9)-2.5 (-13.1-3.5)-1.6 (-11.6-4.9) < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001 0.095App-Bpp (mm)6.7 (0.2–17.7)8.7 (0.2–18.6)-2.2 (-11.7-5.9)-0.6 (-14.5-8) < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001 0.009AF-BF (mm)5.5 (0–17)7.5 (1.1–18)-2.5 (-8.2-0.4)-1.3 (-12.2-8.3) < 0.001

0.004

< 0.001

< 0.001

< 0.001

< 0.001 0.01AXB(FH)(°)5.4 (0-13.5)6.7 (1-15.1)-2.1 (-7.8-0.4)-1.1 (-11.3-6.5) < 0.001 0.018 < 0.001

< 0.001

< 0.001

< 0.001

0.005 AXB(SN)(°)8.4 (5-15.1)8.7 (5–16)2.2 (-2.8-5)4.5 (-2.5-8.6) < 0.001 0.28 < 0.001

< 0.001

< 0.001

< 0.001

< 0.001 Wits (mm)3.4 (-3.2-13.8)4.9 (-3.4-14.5)-5.4 (-15.9-1.5)-5.2 (-19.5-3.1) < 0.001

0.002

< 0.001

< 0.001

< 0.001

< 0.001 0.526Overjet (mm)3.8 (1-13.5)6.2 (1.8–12.5)1.3 (-6.4-9.3)0.1 (-9.3-7.6) < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001 0.087individualized ANB angle (mm)0.5 (-2.2-6)0.9 (-2.3-4.9)-4.8 (-9.8–2.3)-5.1 (-11.3–2.4) < 0.001 0.352 < 0.001

< 0.001

< 0.001

< 0.001 0.559Ballard correction (mm)4.1 (-0.2-13.3)3.9 (-1.4-14)-1.5 (-8.7-5.5)-0.4 (-11.1-5.6) < 0.001 0.791 < 0.001

< 0.001

< 0.001

< 0.001 0.023+ 1i/NA (mm)2.1 (-5.9-6.9)5.2 (0.4–11.1)5.9 (1.8-12.14)5.2 (-0.2-10.1) < 0.001

< 0.001

< 0.001

< 0.001 0.0830.5230.015interincisal angle (°)^+^135.02 ± 9.06114 ± 6.71129.5 ± 9.66132.1 ± 7.85 < 0.001

< 0.001

0.006 0.197 < 0.001

< 0.001 0.533Overbite (mm)3.9 (-8.5-8.5)2.6 (-4-7.6)1.2 (-5.7-7.1)-0.6 (-12.3-3.4) < 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001

< 0.001 Kruskal–Wallis test. p < 0.05Mann–Whitney test, results interpreted according to Bonferroni’s correction when necessary.p < 0.0083 (Bonferroni’s correction).

Discussion

Most studies in the literature focus on defining subphenotypes of maxillomandibular relationships, with most aiming at class 3 malocclusions [7, 12, 20–25]. Few studies focus on class 2 malocclusions [11]. Subphenotypes were evaluated using skeletal characteristics [23, 24], skeletal-dental features [26], and a combination [11, 12]. The present study did not consider soft tissue measurements. Instead, the focus was on skeletal and dental features in all skeletal malocclusions. Common intracranial reference planes affect cross-category measurement interactions. Growth in different directions may affect other groups’ measurements. In individuals with Class 1 vertical growth direction, the maxillomandibular relationship may be defined as Class 2 [27, 28]. Due to interactions, we assessed subphenotypes of people with Class 1, 2, and 3 skeletal malocclusions instead of a single malocclusion in our research. Pooling all skeletal classes allowed for the detection of latent subphenotypes that may not be apparent within isolated malocclusion categories. This strategy was particularly important considering the known interrelations among cephalometric variables—such as the tendency of vertical growth patterns to obscure sagittal discrepancies—and supports a more comprehensive phenotypic classification.

Class 2 and 3 malocclusion classification studies were performed via cluster analysis, which may be hierarchical or nonhierarchical, and “fuzzy” or K-means clustering [11, 12, 20, 25]. In addition to cluster analysis, principal component analysis was used to determine the variance explained by each component and the variables that contributed most to each principal component [11, 12, 24]. We performed X-means clustering and found four clusters that identified craniofacial types. The decision tree established the association between these 4 clusters and the variables. In the present study, pruning was performed to address the issue of excessive branches in the initial decision tree and accurately reflect the classification. A too large tree risks overfitting the data, not working well with new examples, and not providing adequate generalization. Pruning stops splitting characteristics that are not important, known as data with poor information gain. Pruning enhances the tree’s classification accuracy significantly, helping to identify and remove branches that may be oversensitive to statistical irregularities. The efficacy of the pruning process in orthodontic data directly impacts the size and accuracy of the final dentoskeletal model [29–31].

The root node in both the first and second decision trees was the Individualized ANB (°). Clusters 0 and 1 had more heterogeneous maxillomandibular relationships than Clusters 2 and 3. Cluster 0 mostly had Class 1 and partially Class 2 characteristics, and Cluster 1 mostly had Class 2 and partially Class 1.

Cluster 0 represented heterogeneous maxillomandibular malocclusion with a retrusive maxilla, a retrusive mandible, slightly retrusive maxillary incisors, normally or protrusive mandibular incisors, and normal vertical dimensions. Cluster 1 represented heterogeneous maxillomandibular malocclusion with a normal or slightly protrusive maxilla, a retrusive mandible, protrusive maxillary and mandibular incisors, and normal vertical dimensions. The sagittal direction parameters of the maxilla influenced the distinction in the maxillomandibular relationship between Cluster 0 and Cluster 1. Except for the Pg/NB, Clusters 0 and 1 displayed similar characteristics in the mandible’s position and vertical morphology. Pg/NB demonstrated that Cluster 0 had a significantly more prominent symphysis than Cluster 1. Clusters 0 and 1 had similar cranial base characteristics, except for the anterior cranial base length and sellar angle. Their main differences were maxillary position and the maxillary and mandibular incisor inclinations. Cluster 1 demonstrated a significant decrease in the interincisal angle compared with the other clusters. Saltaji reported that individuals with Class 2 Division 1 malocclusions had greater overjet and exhibited a reduced interincisal angle. These findings in Cluster 1, which mainly consisted of individuals who had Class 2 Division 1 malocclusions, were compatible with Saltaji et al.‘s findings [32].

The literature mainly focused on one maxillomandibular relationship and compared clusters, which restricted the comparability of our results, particularly for Clusters 0 and (1) Limited studies have been conducted using the cluster method in subphenotype evaluations of Class (2) Our research showed that Cluster 1 differed from the five cluster results obtained by Uribe et al. [11] due to the heterogeneity of our sample.

Cluster analysis was used to identify subphenotypes within the class 3 maxillomandibular relationship, with an overall number ranging from 3 to 14 [25]. However, 14 subclusters may not be practical for identifying common phenotypes associated with Class 3 malocclusions because some clusters within 14 clusters consisted of a single patient. Clusters consisting of a single patient cannot provide a classification of the phenotype [33]. Optimizing the number of subclusters could improve classification and orthodontic treatments [25].

The mandibular plane, maxillary deficiency and/or retrusion, and mandibular prognathism and/or hyperplasia were the most prevalent criteria to define Class 3 subphenotypes [25]. In two studies with different samples, Auconi et al. found Cluster 1 to be a Class 3 phenotype with increased mandibular dimensions, Cluster 2 with increased maxilla-mandibular divergence, and Cluster 3 with intermediate characteristics between Clusters 1 and 2 [9, 21]. Abu Alhaija and Richardson observed Class 3 subtypes in Cluster 1 with a prominent mandible, Cluster 2 with decreased maxillary length, and Cluster 3 with vertical Class 3 [34]. According to Moreno Uribe et al., class 3 subtypes had five cluster types. Clusters 1 and 2 are characterized by bimaxillary origin and flat/normal mandibular plane angulation. Cluster 3 had vertical class 3 malocclusion. Despite significant mandibular prognathism, Cluster 4 had a normal maxilla and mandibular plane. Cluster 5 had normal mandible, mandibular plane, and severe maxillary retrognathia [12]. Li et al. identified four skeletal and dental mild mandibular prognathism, steep plane, prognathic/retrusive maxillary, and flat/normal mandibular plane, severe prognathism and normal plane, and mild maxillary deficiency and severe prognathism with lowest mandibular plane angle [35]. The mandibular plane was a significant factor in all investigations. However, there was variation in the definition of the mandibular plane. Hong and Yi used the mandibular plane determined using antegonial points (Ag) and menton points (Me) [36]. In contrast, most investigations used the mandibular plane produced using gonial (Go) and Me points [25]. In our study, both the Go-Me and Go-Gn planes indicated mandibular plane inclination. However, because we included all maxillomandibular relationships in our research, we observed only two identifiable clusters demonstrating Class 3. Cluster 2 represented Class 3, characterized by a normal maxillary position, a prognathic mandible, protrusive maxillary incisors, retrusive mandibular incisors, and reduced vertical dimensions. Cluster 3 represented another cluster representing Class 3, which exhibited a normal or retrusive maxilla, a normally or slightly prognathic mandible, protrusive maxillary, and retrusive mandibular incisors, and increased vertical dimensions. However, the most significant difference between these clusters was the N-Go-Gn (°), primarily influenced by the mandibular plane. Clusters 2 and 3 had the same cranial base characteristics except for the sum of the Bjork angle. The difference in Bjork angle was based on the N-Go-Gn (°).

Valla et al. [24] identified six Class 3 subtypes with different characteristics. As the number of clusters increased, clusters indicating the features of the maxilla, mandible, and mandibular plane and clusters revealing the features of facial height and symphysis morphology were formed. Post-hoc comparisons demonstrated that Cluster 2 had a more pronounced symphysis morphology than Cluster 3. Berlanga et al. [37] stated that the symphysis morphology was wide in Class 3 with short facial height and that the symphysis lengthened and narrowed with a compensation mechanism in Class 3 with long facial height. Based on their study, we can say that the Pog point can be moved further back due to the increased vertical dimension and narrowed symphysis morphology. The results of Berlanga et al. [37] were consistent with our findings.

The present study found that Cluster 2 had higher upper incisor protrusion and lower incisor retrusion than Cluster 3. Dentoalveolar compensation influenced by the type of malocclusion also significantly affected lower incisor retrusion, especially in Cluster 3, which represents hyperdivergent individuals, confirming previous findings [37, 38].

In contrast to previous studies focusing solely on Class 3 patients [9, 12, 21, 24, 25, 34–36], our analysis incorporated all skeletal malocclusion classes, allowing for a more comprehensive clustering outcome. Notably, two distinct Class 3 sub-phenotypes Cluster 2, which corresponded to a classical skeletal Class 3 profile with a prognathic mandible and reduced vertical dimension, and Cluster 3, which demonstrated a more compensated Class 3 pattern with increased vertical proportions and retrusive mandibular incisors. This dual representation reflects the internal variation within Class 3 cases, which is often overlooked in studies that rely on strict sagittal classifications. Our findings differ from earlier clustering studies, in that our approach avoided over-fragmentation of the dataset and produced clinically interpretable phenotypes. In particular, vertical and dentoalveolar variables such as N-Go-Gn° and incisor inclination played a key role in differentiating Class 3 subtypes. These aspects are critical in orthodontic treatment planning but may be often underrepresented in phenotype-based classification models.

The clinical implications of the identified craniofacial sub-phenotypes lie in their ability to improve diagnostic precision and assist orthodontists in tailoring treatment strategies. For example, distinguishing between compact and compensated skeletal Class 3 patterns may guide clinicians in choosing between orthodontic camouflage and surgical approaches. Moreover, understanding the underlying phenotypic variation can support early identification of complex cases and provide better patient-specific treatment planning. These applications reinforce the value of incorporating unsupervised learning techniques in orthodontic diagnostics.

Limitations and Clusters that can include pruned branches can be created by increasing the number of samples. To mitigate perplexity, the number of measurements used in LCR evaluation can be decreased because numerous measurements yield similar results. Analyses for classification on massive datasets can broaden perspectives regarding treatment planning and diagnosis. Including soft tissue measurements in the research may enhance the results. Sex-related distinctions can be uncovered through further research by conducting distinct assessments for male and female individuals within a more extensive dataset because the X-means algorithm we implemented solely depends on numerical data.

Conclusion

This study effectively used X-means clustering and C4.5 decision tree algorithms to find different craniofacial subphenotypes in skeletal Class 1, 2, and 3 malocclusions. The most important measurements that helped identify these subphenotypes were the personalized ANB (°), interincisal angle (°), and N-Go-Gn (°), which were the key factors in the decision-making process of the hierarchical tree. Cluster 0 included individuals with relatively average sagittal and vertical proportions, reflecting mild Class 1–2 features. Cluster 1 exhibited pronounced Class 2 skeletal features, such as increased ANB angles and proclined incisors. Cluster 2 represented a Class 3 skeletal profile with a prognathic mandible, reduced vertical facial height, and dentoalveolar compensation. Cluster 3 also reflected Class 3 characteristics but was characterized by greater vertical dimension, retrusive mandibular incisors, and a more compensated profile. Significant differences were found among clusters for nearly all cephalometric parameters, highlighting meaningful variation in skeletal and dentoalveolar structure across malocclusion groups. These findings suggest that using unsupervised learning methods could improve how we diagnose and plan treatments based on different patient characteristics in orthodontics. Future studies with larger and more diverse samples may refine these phenotypic distinctions further and contribute to personalized treatment strategies.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary Material 1

Supplementary Material 2

Supplementary Material 3