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Morphometrics for cephalometric diagnosis
ORIGINAL ARTICLE
Morphometrics for cephalometric diagnosis Demetrios J. Halazonetis, DDS, Dr Odont, MS Athens, Greece This article demonstrates morphometric methods by applying them to an orthodontic sample. A total of 150 pretreatment cephalograms of consecutive patients (84 female, 66 male) were traced and digitized. Fifteen points were used for the analysis. The tracings were superimposed by the Procrustes method, and shape variability was assessed by principal component analysis. Approximately 70% of the total sample variability was incorporated in the first 5 principal components. The most significant principal component, accounting for 29% of shape variability, was the divergence of skeletal pattern; the second principal component, accounting for 20% of shape variability, was the anteroposterior maxillary relationship. It is recommended that Procrustes superimposition and principal component analysis be incorporated into routine cephalometric analysis for more valid and comprehensive shape assessment. (Am J Orthod Dentofacial Orthop 2004; 125:571-81)
O
ne of the main applications of cephalometrics is as a shape descriptor. We use various linear and angular measurements to arrive at a concise and comprehensive description of the craniofacial pattern and to classify each patient, making it easier to identify treatment goals, choose treatment modalities, and predict treatment success. However, in addition to their other disadvantages,1 conventional cephalometric methods have certain inherent problems regarding their applicability as shape measures; they provide only a partial and localized description of shape and are confounded by our biases regarding the reference structures (cranial base, Frankfort horizontal, or others). For example, the conventional view states that angle SNA describes the anteroposterior position of the maxilla, because the cranial base is considered “stable.” However, 2 points that define this angle belong to the cranial base, and it would seem more logical to assume that most of the variability of the SNA measurement is due to the cranial base than to point A. The local nature of conventional measurements and our bias regarding the “stability” of the reference structures result in these problems: (1) the measurements, or rather the interpretations that we ascribe to them, are often conflicting, (2) many measurements are needed for comprehensive description and diagnosis of each patient, (3) it is not trivial to compare the craniofacial pattern between 2 patients, and (4) classification of patients is based on a limited subset of all Assistant professor, Orthodontic Department, University of Athens Dental School, Athens, Greece. Reprints requests to: Dr Demetrios Halazonetis, 6 Menandrou St, Kifissia 145 61, Greece; e-mail, [email protected]. Submitted, February 2003; revised and accepted, May 2003. 0889-5406/$30.00 Copyright © 2004 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2003.05.013
possible measurements and might therefore be biased by that particular selection. Other morphometric methods that address these problems might be more valid for describing biological shape. Methods such as Procrustes superimposition, principal component analysis (PCA), Euclidean distance matrix analysis, finite-element scaling analysis, and thin-plate splines2-10 are actively used in other branches of the biological sciences but have had, surprisingly, limited impact in orthodontics. This article demonstrates some of these methods with an orthodontic sample, in an effort to increase the orthodontic community’s awareness and familiarity with them. The results obtained on the present sample could be used as baseline data for clinical applications and future studies. MATERIAL AND METHODS
The sample consisted of initial (pretreatment) cephalometric radiographs of 150 consecutively treated patients (84 female, 66 male). Patient records were selected from a private orthodontic practice irrespective of sex, age, and type of malocclusion. Only radiographs of good quality that depicted a reference ruler on the cephalostat for exact measurement of the magnification factor were included. Patients with congenital malformations or syndromes were excluded. The radiographs were scanned at 150 dpi and digitized with Viewbox 3 software (dHAL software, Kifissia, Greece; www.dhal.com). A comprehensive set of points was digitized, but the following 15 were used in this investigation: basion (Ba), sella (S), sphenoethmoidale (Se), nasion (N), porion (Po), orbitale (O), anterior nasal spine (ANS), A point (A), posterior nasal spine (PNS), articulare (Ar), gonion (Go), antegonial 571
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Table I. Descriptive statistics for age and selected cephalometric measurements
Age (y) SNA (°) SNB (°) ANB (°) Wits (mm) SN-GoGn (°) Overjet (mm) Overbite (mm)
Mean
SD
Range
Median
12.2 80.1 75.9 4.2 1.8 34.9 5.4 3.9
3.77 3.22 3.51 2.71 3.65 5.25 2.84 2.06
6.4–33.6 71.7–87.9 66.6–86.4 ⫺5.8–10.2 ⫺13.1–10.7 21.5–46.6 ⫺4.8–12.9 ⫺1.6–9.2
11.5 80.2 75.5 4.6 2.3 35.5 5.3 4.1
SD, standard deviation.
notch (Ag), menton (Me), pogonion (Pg), and B point (B). The x and y coordinates of the points were scaled according to the magnification of each radiograph to correspond to life size. Descriptive statistics of age and selected cephalometric measurements are given in Table I. RESULTS Definition of shape and Procrustes superimposition
Shape can be defined as the geometric information that remains after we have removed any effects due to translation, rotation, and scale.5 This definition can be used as an operational definition for shape measurement: to compare 2 shapes, we adjust for size and superimpose them. Then, any differences that remain are due to shape dissimilarity. The discrepancy between the 2 shapes can be measured by assessing the distance between corresponding points of the superimposed shapes. In most cephalometric studies, the aspect of size is circumvented by the use of angular measurements. When linear measurements are used, size adjustment is usually done by scaling according to age or the size of a specific structure (eg, the cranial base). This is not a very satisfactory solution, because the structure used for size adjustment might also be the one that differs between patients. An alternative method is to scale the cephalometric tracings according to centroid size. The centroid of a shape, which is composed of landmark points, is the average of all the points (the “center of gravity” of the shape), and centroid size is the square root of the sum of the squared distances of each point to the centroid.5 After adjusting for size, we need to align the shapes, so that any effect of translation and rotation are removed. In orthodontics, we have many superimposition methods for aligning 2 cephalometric tracings. Figure 1 shows 2 tracings superimposed at an arbitrary position (Fig 1, A)
and on the anterior cranial base (Fig 1, B). The interpretation of shape differences will depend on the choice of the superimposition, a well-known fact in the orthodontic literature. Returning to the definition of shape given above, shape differences can be measured by the distance between corresponding landmarks of the superimposed tracings. It is evident that the superimposition in Figure 1, A, leads to a larger apparent shape difference than that in Figure 1, B, because the sum of the distances between corresponding points is larger. Obviously, this is because a remaining translation and rotation factor has not been adjusted for. This observation leads to a mathematic solution of the problem of the “correct” or “optimum” superimposition: translation and rotation are adjusted so as to minimize the distances between points. There are various minimization criteria, but the most widely used criterion is that which minimizes the sum of the squared distances between corresponding points. This criterion is used by the Procrustes* superimposition,5,10 a widely used morphometric method. Procrustes superimposition takes 2 shapes, resizes them to the same centroid size, and then aligns them to minimize this sum. An important distinction should be made here between comparing the tracings of the same patient at different ages and comparing the tracings of different patients. When comparing tracings of the same patient, it is valid to use biological structures that are known to remain stable. However, when comparing shape between different patients, such considerations do not apply. We cannot assume that the cranial base, or any other part, constitutes a preferred common reference structure between the patients, even though it is biologically homologous. For this reason, there is no “correct” superimposition method: each method will lead to a different interpretation of shape differences.7 Procrustes superimposition is just a method that attempts to minimize the apparent effect of translation, rotation, and scale and thus bring forward pure shape discrepancies. The Procrustes method differs from conventional cephalometric superimpositions because (1) the shapes are resized to achieve a better fit, (2) all points are treated equally instead of privileged status being given to some, and (3) there is no requirement that any corresponding points should coincide. Figure 2 shows the 2 tracings in Figure 1 superimposed by the Procrustes method. Note that the tracings seem much more similar in Figure 2 than in Figure 1, B, and it becomes apparent that the discrepancies in Figure *Procrustes was a robber in ancient Greek mythology. He laid his victims on a bed. If they were too short, he would stretch them; otherwise, he would chop off their legs. In either case, they usually died. Procrustes was killed by Theseus, who was on his way to Athens to become king.
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Fig 1. Superimposition of 2 tracings: A, at arbitrary position and orientation; B, on anterior cranial base (S-N).
1, B, are due more to different cranial base orientations than to overall shape differences. Average shape and shape variability
As with any biological variable, measurement of shape entails (1) calculating the average shape of the population and (2) estimating the variability of shape. Calculation of the average is trivial and can be accomplished by calculating for each point the average x and y coordinates from the corresponding coordinates of all tracings. Figure 3 shows the 150 tracings of the sample superimposed by the Procrustes method together with the average shape. Although an estimate of average shape is important, the most important, and most difficult to estimate, is the variability of shape. The average is of little use by itself, because it does not indicate how much a specific patient differs from it. The variability of the sample around the mean is represented by the scatter of the points in Figure 3. The position of each point might vary along both the x and y directions; thus there are 2k variables to describe variability, where k is the number of points. Principal component analysis is a method to decrease the number of variables. This statistical procedure takes advantage of the fact that the points do not behave independently, because they all belong to the
Fig 2. Tracings of Fig 1 superimposed according to Procrustes method.
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Fig 3. Average shape of sample and scatter of points of 150 tracings. Superimposition of 150 tracings is according to Procrustes method.
same biological entity. Thus, for example, a patient with vertical growth pattern is expected to have N located higher than the average and Me lower than the average, whereas a patient with low vertical height will probably have the opposite arrangement. Such intercorrelations between the positions of the points are used by PCA to arrive at a different set of variables (named principal components [PCs]), each of which describes a specific shape pattern. One component, for example, might represent vertical facial height, whereas another might represent the anteroposterior relationship of the maxillae. The components of PCA are arrived at statistically, not from biological considerations. They have the following properties: (1) they are statistically unrelated to each other; (2) each component represents, in decreasing order, part of the variability of the sample (ie, the first component represents the largest part of the variability, the second component the second-largest part, and so on); and (3) each component is a linear combination of the original variables (ie, each component incorporates to some extent the variability of every point along the x and y direction). However, each point contributes a different amount. For example, if a PC
represents a vertical shape pattern, the variability of Me and N along the y direction would have a higher weight, or loading, than the variability along the x direction. The usefulness of PCA is that the PCs are sorted in order of decreasing importance. Therefore, one can retain only the PCs that describe the most significant part of the overall variability and discard the rest, thereby significantly reducing the total number of variables. In this study, 15 points were used, which would produce 30 PCs. However, because some degrees of freedom are lost because of the alignment of the shapes10 (the explanation for this is beyond the scope of this article), the total number of PCs is 26. Calculating the PCs was performed with APS 2.41 software (Xavier Penin, Caen, France; www.cpod.com/ monoweb/aps). The loadings of the first 3 PCs for each point are shown in Table II. Table III shows the percentage of total shape variability accounted for by each of the first 8 components. Approximately 70% of the variability was incorporated into the first 5 components. The average shape of the sample has, by definition,
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Table II.
Loadings of first 3 PCs for each point in x and y directions PC1
PC2
PC3
Point
x
y
x
y
x
y
Ba Se N S Po Or PNS ANS A B Pg Me Ag Go Ar
⫺0.2458 0.0549 0.1104 ⫺0.0072 ⫺0.1176 ⫺0.0060 ⫺0.1226 ⫺0.0573 ⫺0.0529 0.1646 0.3122 0.3417 ⫺0.0212 ⫺0.1392 ⫺0.2140
0.1944 ⫺0.1581 ⫺0.3148 0.0234 0.2146 ⫺0.2556 ⫺0.0502 ⫺0.2473 ⫺0.1747 0.2085 0.3281 0.2197 ⫺0.0713 ⫺0.0155 0.0985
0.0983 ⫺0.1210 ⫺0.2611 0.0264 0.2381 ⫺0.1509 ⫺0.1584 ⫺0.3554 ⫺0.3608 0.0895 0.1827 0.2398 0.1872 0.2166 0.1290
0.2087 0.1236 ⫺0.0245 0.1648 0.1598 0.0683 0.0616 ⫺0.0670 ⫺0.0657 ⫺0.1718 ⫺0.2303 ⫺0.2641 ⫺0.1172 ⫺0.0664 0.2201
⫺0.0435 ⫺0.1707 0.0925 ⫺0.2074 ⫺0.3170 ⫺0.0624 0.0151 0.0058 0.0224 0.1716 0.0617 0.0287 0.2004 0.2199 ⫺0.0172
⫺0.0570 ⫺0.0111 ⫺0.1718 ⫺0.0355 ⫺0.0615 ⫺0.1517 ⫺0.0054 ⫺0.0329 ⫺0.0481 ⫺0.1568 ⫺0.1620 ⫺0.1398 0.4615 0.5844 ⫺0.0124
Large absolute values signify that PC has relatively large effect on variability of point in corresponding direction. These values were used to arrive at warped shapes of Figure 4. Table III. Standard deviation, variance, percent variance, and cumulative variance explained by each of 8 first PCs Component
SD
% Cumulative variance Variance variance (%)
Figure 5 shows scatterplots of the first 5 PCs against each other. These plots show that the sample cannot be divided into discrete shape categories and should be regarded as a homogeneous group. Shape space
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
0.03406 0.02809 0.01922 0.01519 0.01424 0.01314 0.01227 0.01143
0.00116 0.00079 0.00037 0.00023 0.00020 0.00017 0.00015 0.00013
28.7 19.6 9.2 5.7 5.0 4.3 3.7 3.2
28.7 48.3 57.4 63.2 68.2 72.5 76.2 79.4
Percent variance was calculated on total variance of 26 components, which was equal to 0.00404.
all PCs equal to 0. To visualize the pattern of shape variability represented by each PC, the average shape can be warped by moving the points according to their loadings on the PC. Figure 4 shows the average shape of the sample and the warped shapes obtained by setting the first 3 PCs to a value equal to 3 standard deviations in the negative and positive directions. From this figure, we can interpret the first PC as representing the divergence of the skeletal pattern. Low values show a high-angle skeletal pattern, and high values show a low-angle pattern. The second component is associated with the anteroposterior relationship of the maxillae. A low value corresponds to a Class II pattern, and a high value to a Class III pattern. The third PC relates to the gonial angle. A low value is associated with a small gonial angle and an increased posterior facial height, whereas the reverse is true for a high value of the PC.
Shape is not a discrete variable but is a continuum of smoothly varying patterns. The collection of all possible shape patterns is called a “shape-space.” Each patient of the sample is located at a particular point in this shape-space, much as each person has a unique home address on the map. The PCs can be thought of as representing the coordinates that pinpoint the patient to the particular location. Because successive PCs represent ever-decreasing parts of shape variability, they can be used to describe shape in increasing detail, much as an address describes location from the global to the specific (country, town, street, number). The difference in shape between 2 patients can be measured by the distance between them in shape space. A popular measure of distance is the Procrustes distance, calculated as the square root of the sum of squared distances between corresponding points when the shapes are aligned. Application in cephalometrics
The craniofacial shape of a patient can be assessed by calculating the value of the PCs and placing the patient at the appropriate position in the population’s shape-space. The first PCs describe the shape pattern in general terms, and successive PCs concentrate on finer detail. This scheme allows us to adjust the number of
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Fig 4. Warped shapes depicting effect of varying each of first 3 PCs. Average shape (middle) is warped by applying each PC by amount equal to 3 standard deviations in negative (left) and positive (right) direction.
PCs used to describe a patient, according to the detail we are seeking. For broad classification, 2 or 3 PCs might be sufficient. In such a case, the shape space is 2or 3-dimensional, and the scatterplots of Figure 5 can be used to obtain a visual representation of the location of the patient in the shape space. As an example, Figure 6 shows the tracings of 3 patients from the present sample. Table IV shows some selected cephalometric measurements and the values of the first 3 PCs. Looking at the first PC, we can tell that
patients A and B should be roughly similar, belonging to the high-angle group, because PC1 is negative (Fig 4), whereas patient C is on the opposite side of the spectrum— definitely low-angle. The second PC shows that patients A and B are also similar in anteroposterior maxillary relationship; both are relatively Class II, whereas patient C is again on the opposite side, a Class III. The similarity of patients A and B in contrast to patient C is graphically evident by placing them in the approximate shape-space, as shown in Figure 7, A.
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Fig 5. Scatterplots of first 5 PCs against each other. Sample constitutes homogeneous group with few outliers.
However, further refinement of shape assessment with the third PC (Table IV) differentiates between patients A and B at the area of the mandibular angle. Patients A and B are relatively far apart in the scatterplot of PC2 compared with PC3 (Fig 7, B), but this does not constitute a major difference in overall shape because PC3 is less influential than PC1 and PC2. All 3 patients are at approximately the same distance from the origin of the graphs, which shows that the degree of skeletal discrepancy is similar. This can also be quantitatively assessed by calculating the Procrustes distance. DISCUSSION
Conventional cephalometric methods have existed for more than 60 years, and their limitations have been investigated and appreciated. Other methods for assessing shape are under development in biological morphometrics, and their applicability in cephalometrics should be researched. The orthodontic literature con-
tains few relevant articles (see examples in the reference list11-18), so an introductory article presenting baseline data on a general orthodontic sample was considered important. Because readers are probably unfamiliar with the methods used, extra details and explanatory material were included. Considerable debate exists concerning the theoretic validity and appropriateness of the various morphometric methods, and no consensus has been reached (for a critical, but admittedly biased, comparison, see Richtsmeier et al19). The techniques presented here were selected on the basis of their wide usage and relative conceptual ease. For more information, the reader is directed to the Morphometrics web site at http://life.bio.sunysb.edu/ morph/. The sample used in this study was not a sample of ideal or normal subjects, nor does it represent the general population, and the results cannot, therefore, be regarded as normal values for use in diagnosis. How-
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Fig 6. Tracings of 3 patients and their Procrustes superimpositions.
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Table IV.
Selected measurements and values of first 3 PCs for example patients
SNA (°) SNB (°) ANB (°) Wits (mm) SN-GoGn (°) Overjet (mm) Overbite (mm) PC1 PC2 PC3
Patient A
Patient B
Patient C
86.3 76.2 10.2 6.4 43.3 10.6 3.5 ⫺0.03827 (⫺1.12) ⫺0.03262 (⫺1.16) 0.04541 (2.36)
82.6 75.1 7.5 5 38.1 6.2 5.3 ⫺0.03170 (⫺0.93) ⫺0.03657 (⫺1.30) ⫺0.00117 (⫺0.06)
84.8 84.6 0.3 ⫺4.9 25.8 2.5 1.3 0.04475 (1.31) 0.05187 (1.85) ⫺0.01047 (⫺0.54)
Numbers in parentheses are z scores of PCs.
Fig 7. Position of 3 patients in shape space defined by first 3 PCs. This position gives rough estimate of shape of tracing and of distance from average (center of graphs). A, PC1 vs PC2. B, PC2 vs PC3.
ever, the size of the sample and the method of patient selection (consecutive patients) lend reasonable support to the belief that the sample uniformly covers most of the shape space, excluding syndromic patients. Thus, the calculated average shape should not be far from the population average. Conventional cephalometric measures of the calculated average form were within normal limits. Furthermore, the average form was very similar to the templates of the Michigan Growth Study.20 Data from the general population are therefore expected to be similar but, probably, have smaller standard deviations. PCA is a statistical technique for reducing the number of variables when a significant correlation between the variables is present. The application of PCA in this sample resulted in 5 PCs explaining
approximately 70% of the total shape variability. This small number of variables is sufficient to reconstruct the approximate shape of a patient’s craniofacial pattern (cephalometric tracing), in contrast to conventional cephalometric measurements, with which this inverse function, from measurements to tracing, is, in general, not possible. Therefore, the use of PCs as adjunct variables of a mixed-mode (conventional/morphometric) cephalometric analysis should be seriously considered. This analysis could provide the following advantages: 1. Comprehensive description of the overall craniofacial shape with a small number of measurements, which are not conflicting because they are unrelated statistically. More detailed description is possible by
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increasing the number of reported variables (PCs). The approximate tracing of a patient can be recreated from the values of the PCs and the average tracing of the population. Relative insensitivity to errors in landmark identification. This is a major problem in conventional analyses,21-25 because a small error in the location of the points that are used for the reference plane (eg, Po in defining the Frankfort horizontal) can have a significant impact on all measurements that use that particular reference plane (eg, Frankfort-mandibular plane angle, Frankfort-mandibular incisor angle, facial angle).23 In contrast, Procrustes superimposition treats all points equally, and an error in the location of 1 point will not have as significant an effect. Easy assessment of the degree of variation from the average. This is computed as the Procrustes distance and consists of a single variable that shows how far from the average is the overall skeletal pattern. Easy comparison of shape between patients (again by the Procrustes distance). This might be particularly helpful when planning the treatment of a new patient, because it allows retrieval of data for previously treated patients of a similar pattern to consider their responses to treatment. More valid selection of patients for research projects regarding sample homogeneity.
The first PC explained approximately 29% of the shape variability, whereas the second PC, related to anteroposterior maxillary discrepancy, accounted for 20% of the total sample variability. The second PC describes the variability of the anteroposterior position of the mandible relative to the combined craniomaxillary structure. The maxilla and anterior cranial base seem to remain relatively stable to each other when the second PC takes values from ⫺3 to ⫹3 standard deviations (Fig 4). A significant hindrance to applying morphometric methods to cephalometrics is that the points have to be digitized and special software used to arrive at the results. The ubiquitous use of computers in orthodontic offices and the development of new software should alleviate these problems in the near future.
A significant disadvantage of PCA is that the resulting components, because they are derived statistically, do not necessarily have a clear biological interpretation. Usually, only the first, or the first few, can be described satisfactorily, and this was true for the results of this study. Furthermore, the components might be influenced by the number and spatial distribution of cephalometric points used to describe the craniofacial shape. An uneven density of points might introduce bias in both the Procrustes superimposition and the PCA results,9 and, for this reason, an effort was made to select points evenly distributed over the whole shape. Analyses with fewer points (7 total points) were also conducted, and the results were very similar to those presented here (results available on request). The PCA showed that the largest PC was related to variability of the craniofacial complex in the vertical direction. This was unexpected, considering the orthodontic community’s traditional preoccupation with the anteroposterior direction, although notable exceptions exist (eg, the work by Shudy26). It seems that, although malocclusions in the anteroposterior direction might be more apparent, vertical discrepancies of the skeletal component are more pronounced.
Morphometric methods might provide useful adjunctive information for assessing skeletal pattern.
2.
3.
4.
5.
CONCLUSIONS
Procrustes superimposition and PCA of a widespectrum orthodontic sample showed the following: 1. Seventy percent of shape variability can be described by the first 5 PCs. 2. The first PC, explaining the largest percentage of shape variability (29%), described vertical components, and the second PC (20%) described anteroposterior discrepancies. 3. The sample was a homogeneous group that could not be differentiated into discrete subgroups.
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morphometrics. Int J Adult Orthod Orthognath Surg 2002;17: 116-32. Franchi L, Baccetti T, Cameron CG, Kutcipal EA, McNamara JA Jr. Thin-plate spline analysis of the short- and long-term effects of rapid maxillary expansion. Eur J Orthod 2002;24:143-50. Singh GD, Hodge MR. Bimaxillary morphometry of patients with Class II Division 1 malocclusion treated with Twin-block appliances. Angle Orthod 2002;72:402-9. Richtsmeier JT, Deleon VB, Lele SR. The promise of geometric morphometrics. Yrbk Phys Anthropol 2002;45:63-91. Johnston LE. Template analysis. In: Jacobson A, Caufield PW, editors. Introduction to radiographic cephalometry. Philadelphia: Lea & Febiger; 1985. Baumrind S, Frantz RC. The reliability of head film measurements. 1. Landmark identification. Am J Orthod 1971;60:111-27. Baumrind S, Frantz RC. The reliability of head film measurements. 2. Conventional angular and linear measures. Am J Orthod 1971;60:505-17. Baumrind S, Miller DM, Molthen R. The reliability of head film measurements. 3. Tracing superimpositions. Am J Orthod 1976; 70:617-44. Broch J, Slagsvold O, Ro¨ sler M. Error in landmark identification in lateral radiographic headplates. Eur J Orthod 1981;3:9-13. Houston WJB, Maher RE, McElroy D, Sherriff M. Sources of error in measurements from cephalometric radiographs. Eur J Orthod 1986;8:149-51. Schudy FF. Vertical growth versus anteroposterior growth as related to function and treatment. Angle Orthod 1964;34:75-93.
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Morphometrics for cephalometric diagnosis Demetrios J. Halazonetis, DDS, Dr Odont, MS Athens, Greece This article demonstrates morphometric methods by applying them to an orthodontic sample. A total of 150 pretreatment cephalograms of consecutive patients (84 female, 66 male) were traced and digitized. Fifteen points were used for the analysis. The tracings were superimposed by the Procrustes method, and shape variability was assessed by principal component analysis. Approximately 70% of the total sample variability was incorporated in the first 5 principal components. The most significant principal component, accounting for 29% of shape variability, was the divergence of skeletal pattern; the second principal component, accounting for 20% of shape variability, was the anteroposterior maxillary relationship. It is recommended that Procrustes superimposition and principal component analysis be incorporated into routine cephalometric analysis for more valid and comprehensive shape assessment. (Am J Orthod Dentofacial Orthop 2004; 125:571-81)
O
ne of the main applications of cephalometrics is as a shape descriptor. We use various linear and angular measurements to arrive at a concise and comprehensive description of the craniofacial pattern and to classify each patient, making it easier to identify treatment goals, choose treatment modalities, and predict treatment success. However, in addition to their other disadvantages,1 conventional cephalometric methods have certain inherent problems regarding their applicability as shape measures; they provide only a partial and localized description of shape and are confounded by our biases regarding the reference structures (cranial base, Frankfort horizontal, or others). For example, the conventional view states that angle SNA describes the anteroposterior position of the maxilla, because the cranial base is considered “stable.” However, 2 points that define this angle belong to the cranial base, and it would seem more logical to assume that most of the variability of the SNA measurement is due to the cranial base than to point A. The local nature of conventional measurements and our bias regarding the “stability” of the reference structures result in these problems: (1) the measurements, or rather the interpretations that we ascribe to them, are often conflicting, (2) many measurements are needed for comprehensive description and diagnosis of each patient, (3) it is not trivial to compare the craniofacial pattern between 2 patients, and (4) classification of patients is based on a limited subset of all Assistant professor, Orthodontic Department, University of Athens Dental School, Athens, Greece. Reprints requests to: Dr Demetrios Halazonetis, 6 Menandrou St, Kifissia 145 61, Greece; e-mail, [email protected]. Submitted, February 2003; revised and accepted, May 2003. 0889-5406/$30.00 Copyright © 2004 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2003.05.013
possible measurements and might therefore be biased by that particular selection. Other morphometric methods that address these problems might be more valid for describing biological shape. Methods such as Procrustes superimposition, principal component analysis (PCA), Euclidean distance matrix analysis, finite-element scaling analysis, and thin-plate splines2-10 are actively used in other branches of the biological sciences but have had, surprisingly, limited impact in orthodontics. This article demonstrates some of these methods with an orthodontic sample, in an effort to increase the orthodontic community’s awareness and familiarity with them. The results obtained on the present sample could be used as baseline data for clinical applications and future studies. MATERIAL AND METHODS
The sample consisted of initial (pretreatment) cephalometric radiographs of 150 consecutively treated patients (84 female, 66 male). Patient records were selected from a private orthodontic practice irrespective of sex, age, and type of malocclusion. Only radiographs of good quality that depicted a reference ruler on the cephalostat for exact measurement of the magnification factor were included. Patients with congenital malformations or syndromes were excluded. The radiographs were scanned at 150 dpi and digitized with Viewbox 3 software (dHAL software, Kifissia, Greece; www.dhal.com). A comprehensive set of points was digitized, but the following 15 were used in this investigation: basion (Ba), sella (S), sphenoethmoidale (Se), nasion (N), porion (Po), orbitale (O), anterior nasal spine (ANS), A point (A), posterior nasal spine (PNS), articulare (Ar), gonion (Go), antegonial 571
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Table I. Descriptive statistics for age and selected cephalometric measurements
Age (y) SNA (°) SNB (°) ANB (°) Wits (mm) SN-GoGn (°) Overjet (mm) Overbite (mm)
Mean
SD
Range
Median
12.2 80.1 75.9 4.2 1.8 34.9 5.4 3.9
3.77 3.22 3.51 2.71 3.65 5.25 2.84 2.06
6.4–33.6 71.7–87.9 66.6–86.4 ⫺5.8–10.2 ⫺13.1–10.7 21.5–46.6 ⫺4.8–12.9 ⫺1.6–9.2
11.5 80.2 75.5 4.6 2.3 35.5 5.3 4.1
SD, standard deviation.
notch (Ag), menton (Me), pogonion (Pg), and B point (B). The x and y coordinates of the points were scaled according to the magnification of each radiograph to correspond to life size. Descriptive statistics of age and selected cephalometric measurements are given in Table I. RESULTS Definition of shape and Procrustes superimposition
Shape can be defined as the geometric information that remains after we have removed any effects due to translation, rotation, and scale.5 This definition can be used as an operational definition for shape measurement: to compare 2 shapes, we adjust for size and superimpose them. Then, any differences that remain are due to shape dissimilarity. The discrepancy between the 2 shapes can be measured by assessing the distance between corresponding points of the superimposed shapes. In most cephalometric studies, the aspect of size is circumvented by the use of angular measurements. When linear measurements are used, size adjustment is usually done by scaling according to age or the size of a specific structure (eg, the cranial base). This is not a very satisfactory solution, because the structure used for size adjustment might also be the one that differs between patients. An alternative method is to scale the cephalometric tracings according to centroid size. The centroid of a shape, which is composed of landmark points, is the average of all the points (the “center of gravity” of the shape), and centroid size is the square root of the sum of the squared distances of each point to the centroid.5 After adjusting for size, we need to align the shapes, so that any effect of translation and rotation are removed. In orthodontics, we have many superimposition methods for aligning 2 cephalometric tracings. Figure 1 shows 2 tracings superimposed at an arbitrary position (Fig 1, A)
and on the anterior cranial base (Fig 1, B). The interpretation of shape differences will depend on the choice of the superimposition, a well-known fact in the orthodontic literature. Returning to the definition of shape given above, shape differences can be measured by the distance between corresponding landmarks of the superimposed tracings. It is evident that the superimposition in Figure 1, A, leads to a larger apparent shape difference than that in Figure 1, B, because the sum of the distances between corresponding points is larger. Obviously, this is because a remaining translation and rotation factor has not been adjusted for. This observation leads to a mathematic solution of the problem of the “correct” or “optimum” superimposition: translation and rotation are adjusted so as to minimize the distances between points. There are various minimization criteria, but the most widely used criterion is that which minimizes the sum of the squared distances between corresponding points. This criterion is used by the Procrustes* superimposition,5,10 a widely used morphometric method. Procrustes superimposition takes 2 shapes, resizes them to the same centroid size, and then aligns them to minimize this sum. An important distinction should be made here between comparing the tracings of the same patient at different ages and comparing the tracings of different patients. When comparing tracings of the same patient, it is valid to use biological structures that are known to remain stable. However, when comparing shape between different patients, such considerations do not apply. We cannot assume that the cranial base, or any other part, constitutes a preferred common reference structure between the patients, even though it is biologically homologous. For this reason, there is no “correct” superimposition method: each method will lead to a different interpretation of shape differences.7 Procrustes superimposition is just a method that attempts to minimize the apparent effect of translation, rotation, and scale and thus bring forward pure shape discrepancies. The Procrustes method differs from conventional cephalometric superimpositions because (1) the shapes are resized to achieve a better fit, (2) all points are treated equally instead of privileged status being given to some, and (3) there is no requirement that any corresponding points should coincide. Figure 2 shows the 2 tracings in Figure 1 superimposed by the Procrustes method. Note that the tracings seem much more similar in Figure 2 than in Figure 1, B, and it becomes apparent that the discrepancies in Figure *Procrustes was a robber in ancient Greek mythology. He laid his victims on a bed. If they were too short, he would stretch them; otherwise, he would chop off their legs. In either case, they usually died. Procrustes was killed by Theseus, who was on his way to Athens to become king.
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Fig 1. Superimposition of 2 tracings: A, at arbitrary position and orientation; B, on anterior cranial base (S-N).
1, B, are due more to different cranial base orientations than to overall shape differences. Average shape and shape variability
As with any biological variable, measurement of shape entails (1) calculating the average shape of the population and (2) estimating the variability of shape. Calculation of the average is trivial and can be accomplished by calculating for each point the average x and y coordinates from the corresponding coordinates of all tracings. Figure 3 shows the 150 tracings of the sample superimposed by the Procrustes method together with the average shape. Although an estimate of average shape is important, the most important, and most difficult to estimate, is the variability of shape. The average is of little use by itself, because it does not indicate how much a specific patient differs from it. The variability of the sample around the mean is represented by the scatter of the points in Figure 3. The position of each point might vary along both the x and y directions; thus there are 2k variables to describe variability, where k is the number of points. Principal component analysis is a method to decrease the number of variables. This statistical procedure takes advantage of the fact that the points do not behave independently, because they all belong to the
Fig 2. Tracings of Fig 1 superimposed according to Procrustes method.
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Fig 3. Average shape of sample and scatter of points of 150 tracings. Superimposition of 150 tracings is according to Procrustes method.
same biological entity. Thus, for example, a patient with vertical growth pattern is expected to have N located higher than the average and Me lower than the average, whereas a patient with low vertical height will probably have the opposite arrangement. Such intercorrelations between the positions of the points are used by PCA to arrive at a different set of variables (named principal components [PCs]), each of which describes a specific shape pattern. One component, for example, might represent vertical facial height, whereas another might represent the anteroposterior relationship of the maxillae. The components of PCA are arrived at statistically, not from biological considerations. They have the following properties: (1) they are statistically unrelated to each other; (2) each component represents, in decreasing order, part of the variability of the sample (ie, the first component represents the largest part of the variability, the second component the second-largest part, and so on); and (3) each component is a linear combination of the original variables (ie, each component incorporates to some extent the variability of every point along the x and y direction). However, each point contributes a different amount. For example, if a PC
represents a vertical shape pattern, the variability of Me and N along the y direction would have a higher weight, or loading, than the variability along the x direction. The usefulness of PCA is that the PCs are sorted in order of decreasing importance. Therefore, one can retain only the PCs that describe the most significant part of the overall variability and discard the rest, thereby significantly reducing the total number of variables. In this study, 15 points were used, which would produce 30 PCs. However, because some degrees of freedom are lost because of the alignment of the shapes10 (the explanation for this is beyond the scope of this article), the total number of PCs is 26. Calculating the PCs was performed with APS 2.41 software (Xavier Penin, Caen, France; www.cpod.com/ monoweb/aps). The loadings of the first 3 PCs for each point are shown in Table II. Table III shows the percentage of total shape variability accounted for by each of the first 8 components. Approximately 70% of the variability was incorporated into the first 5 components. The average shape of the sample has, by definition,
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Table II.
Loadings of first 3 PCs for each point in x and y directions PC1
PC2
PC3
Point
x
y
x
y
x
y
Ba Se N S Po Or PNS ANS A B Pg Me Ag Go Ar
⫺0.2458 0.0549 0.1104 ⫺0.0072 ⫺0.1176 ⫺0.0060 ⫺0.1226 ⫺0.0573 ⫺0.0529 0.1646 0.3122 0.3417 ⫺0.0212 ⫺0.1392 ⫺0.2140
0.1944 ⫺0.1581 ⫺0.3148 0.0234 0.2146 ⫺0.2556 ⫺0.0502 ⫺0.2473 ⫺0.1747 0.2085 0.3281 0.2197 ⫺0.0713 ⫺0.0155 0.0985
0.0983 ⫺0.1210 ⫺0.2611 0.0264 0.2381 ⫺0.1509 ⫺0.1584 ⫺0.3554 ⫺0.3608 0.0895 0.1827 0.2398 0.1872 0.2166 0.1290
0.2087 0.1236 ⫺0.0245 0.1648 0.1598 0.0683 0.0616 ⫺0.0670 ⫺0.0657 ⫺0.1718 ⫺0.2303 ⫺0.2641 ⫺0.1172 ⫺0.0664 0.2201
⫺0.0435 ⫺0.1707 0.0925 ⫺0.2074 ⫺0.3170 ⫺0.0624 0.0151 0.0058 0.0224 0.1716 0.0617 0.0287 0.2004 0.2199 ⫺0.0172
⫺0.0570 ⫺0.0111 ⫺0.1718 ⫺0.0355 ⫺0.0615 ⫺0.1517 ⫺0.0054 ⫺0.0329 ⫺0.0481 ⫺0.1568 ⫺0.1620 ⫺0.1398 0.4615 0.5844 ⫺0.0124
Large absolute values signify that PC has relatively large effect on variability of point in corresponding direction. These values were used to arrive at warped shapes of Figure 4. Table III. Standard deviation, variance, percent variance, and cumulative variance explained by each of 8 first PCs Component
SD
% Cumulative variance Variance variance (%)
Figure 5 shows scatterplots of the first 5 PCs against each other. These plots show that the sample cannot be divided into discrete shape categories and should be regarded as a homogeneous group. Shape space
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
0.03406 0.02809 0.01922 0.01519 0.01424 0.01314 0.01227 0.01143
0.00116 0.00079 0.00037 0.00023 0.00020 0.00017 0.00015 0.00013
28.7 19.6 9.2 5.7 5.0 4.3 3.7 3.2
28.7 48.3 57.4 63.2 68.2 72.5 76.2 79.4
Percent variance was calculated on total variance of 26 components, which was equal to 0.00404.
all PCs equal to 0. To visualize the pattern of shape variability represented by each PC, the average shape can be warped by moving the points according to their loadings on the PC. Figure 4 shows the average shape of the sample and the warped shapes obtained by setting the first 3 PCs to a value equal to 3 standard deviations in the negative and positive directions. From this figure, we can interpret the first PC as representing the divergence of the skeletal pattern. Low values show a high-angle skeletal pattern, and high values show a low-angle pattern. The second component is associated with the anteroposterior relationship of the maxillae. A low value corresponds to a Class II pattern, and a high value to a Class III pattern. The third PC relates to the gonial angle. A low value is associated with a small gonial angle and an increased posterior facial height, whereas the reverse is true for a high value of the PC.
Shape is not a discrete variable but is a continuum of smoothly varying patterns. The collection of all possible shape patterns is called a “shape-space.” Each patient of the sample is located at a particular point in this shape-space, much as each person has a unique home address on the map. The PCs can be thought of as representing the coordinates that pinpoint the patient to the particular location. Because successive PCs represent ever-decreasing parts of shape variability, they can be used to describe shape in increasing detail, much as an address describes location from the global to the specific (country, town, street, number). The difference in shape between 2 patients can be measured by the distance between them in shape space. A popular measure of distance is the Procrustes distance, calculated as the square root of the sum of squared distances between corresponding points when the shapes are aligned. Application in cephalometrics
The craniofacial shape of a patient can be assessed by calculating the value of the PCs and placing the patient at the appropriate position in the population’s shape-space. The first PCs describe the shape pattern in general terms, and successive PCs concentrate on finer detail. This scheme allows us to adjust the number of
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Fig 4. Warped shapes depicting effect of varying each of first 3 PCs. Average shape (middle) is warped by applying each PC by amount equal to 3 standard deviations in negative (left) and positive (right) direction.
PCs used to describe a patient, according to the detail we are seeking. For broad classification, 2 or 3 PCs might be sufficient. In such a case, the shape space is 2or 3-dimensional, and the scatterplots of Figure 5 can be used to obtain a visual representation of the location of the patient in the shape space. As an example, Figure 6 shows the tracings of 3 patients from the present sample. Table IV shows some selected cephalometric measurements and the values of the first 3 PCs. Looking at the first PC, we can tell that
patients A and B should be roughly similar, belonging to the high-angle group, because PC1 is negative (Fig 4), whereas patient C is on the opposite side of the spectrum— definitely low-angle. The second PC shows that patients A and B are also similar in anteroposterior maxillary relationship; both are relatively Class II, whereas patient C is again on the opposite side, a Class III. The similarity of patients A and B in contrast to patient C is graphically evident by placing them in the approximate shape-space, as shown in Figure 7, A.
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Fig 5. Scatterplots of first 5 PCs against each other. Sample constitutes homogeneous group with few outliers.
However, further refinement of shape assessment with the third PC (Table IV) differentiates between patients A and B at the area of the mandibular angle. Patients A and B are relatively far apart in the scatterplot of PC2 compared with PC3 (Fig 7, B), but this does not constitute a major difference in overall shape because PC3 is less influential than PC1 and PC2. All 3 patients are at approximately the same distance from the origin of the graphs, which shows that the degree of skeletal discrepancy is similar. This can also be quantitatively assessed by calculating the Procrustes distance. DISCUSSION
Conventional cephalometric methods have existed for more than 60 years, and their limitations have been investigated and appreciated. Other methods for assessing shape are under development in biological morphometrics, and their applicability in cephalometrics should be researched. The orthodontic literature con-
tains few relevant articles (see examples in the reference list11-18), so an introductory article presenting baseline data on a general orthodontic sample was considered important. Because readers are probably unfamiliar with the methods used, extra details and explanatory material were included. Considerable debate exists concerning the theoretic validity and appropriateness of the various morphometric methods, and no consensus has been reached (for a critical, but admittedly biased, comparison, see Richtsmeier et al19). The techniques presented here were selected on the basis of their wide usage and relative conceptual ease. For more information, the reader is directed to the Morphometrics web site at http://life.bio.sunysb.edu/ morph/. The sample used in this study was not a sample of ideal or normal subjects, nor does it represent the general population, and the results cannot, therefore, be regarded as normal values for use in diagnosis. How-
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Fig 6. Tracings of 3 patients and their Procrustes superimpositions.
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Table IV.
Selected measurements and values of first 3 PCs for example patients
SNA (°) SNB (°) ANB (°) Wits (mm) SN-GoGn (°) Overjet (mm) Overbite (mm) PC1 PC2 PC3
Patient A
Patient B
Patient C
86.3 76.2 10.2 6.4 43.3 10.6 3.5 ⫺0.03827 (⫺1.12) ⫺0.03262 (⫺1.16) 0.04541 (2.36)
82.6 75.1 7.5 5 38.1 6.2 5.3 ⫺0.03170 (⫺0.93) ⫺0.03657 (⫺1.30) ⫺0.00117 (⫺0.06)
84.8 84.6 0.3 ⫺4.9 25.8 2.5 1.3 0.04475 (1.31) 0.05187 (1.85) ⫺0.01047 (⫺0.54)
Numbers in parentheses are z scores of PCs.
Fig 7. Position of 3 patients in shape space defined by first 3 PCs. This position gives rough estimate of shape of tracing and of distance from average (center of graphs). A, PC1 vs PC2. B, PC2 vs PC3.
ever, the size of the sample and the method of patient selection (consecutive patients) lend reasonable support to the belief that the sample uniformly covers most of the shape space, excluding syndromic patients. Thus, the calculated average shape should not be far from the population average. Conventional cephalometric measures of the calculated average form were within normal limits. Furthermore, the average form was very similar to the templates of the Michigan Growth Study.20 Data from the general population are therefore expected to be similar but, probably, have smaller standard deviations. PCA is a statistical technique for reducing the number of variables when a significant correlation between the variables is present. The application of PCA in this sample resulted in 5 PCs explaining
approximately 70% of the total shape variability. This small number of variables is sufficient to reconstruct the approximate shape of a patient’s craniofacial pattern (cephalometric tracing), in contrast to conventional cephalometric measurements, with which this inverse function, from measurements to tracing, is, in general, not possible. Therefore, the use of PCs as adjunct variables of a mixed-mode (conventional/morphometric) cephalometric analysis should be seriously considered. This analysis could provide the following advantages: 1. Comprehensive description of the overall craniofacial shape with a small number of measurements, which are not conflicting because they are unrelated statistically. More detailed description is possible by
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increasing the number of reported variables (PCs). The approximate tracing of a patient can be recreated from the values of the PCs and the average tracing of the population. Relative insensitivity to errors in landmark identification. This is a major problem in conventional analyses,21-25 because a small error in the location of the points that are used for the reference plane (eg, Po in defining the Frankfort horizontal) can have a significant impact on all measurements that use that particular reference plane (eg, Frankfort-mandibular plane angle, Frankfort-mandibular incisor angle, facial angle).23 In contrast, Procrustes superimposition treats all points equally, and an error in the location of 1 point will not have as significant an effect. Easy assessment of the degree of variation from the average. This is computed as the Procrustes distance and consists of a single variable that shows how far from the average is the overall skeletal pattern. Easy comparison of shape between patients (again by the Procrustes distance). This might be particularly helpful when planning the treatment of a new patient, because it allows retrieval of data for previously treated patients of a similar pattern to consider their responses to treatment. More valid selection of patients for research projects regarding sample homogeneity.
The first PC explained approximately 29% of the shape variability, whereas the second PC, related to anteroposterior maxillary discrepancy, accounted for 20% of the total sample variability. The second PC describes the variability of the anteroposterior position of the mandible relative to the combined craniomaxillary structure. The maxilla and anterior cranial base seem to remain relatively stable to each other when the second PC takes values from ⫺3 to ⫹3 standard deviations (Fig 4). A significant hindrance to applying morphometric methods to cephalometrics is that the points have to be digitized and special software used to arrive at the results. The ubiquitous use of computers in orthodontic offices and the development of new software should alleviate these problems in the near future.
A significant disadvantage of PCA is that the resulting components, because they are derived statistically, do not necessarily have a clear biological interpretation. Usually, only the first, or the first few, can be described satisfactorily, and this was true for the results of this study. Furthermore, the components might be influenced by the number and spatial distribution of cephalometric points used to describe the craniofacial shape. An uneven density of points might introduce bias in both the Procrustes superimposition and the PCA results,9 and, for this reason, an effort was made to select points evenly distributed over the whole shape. Analyses with fewer points (7 total points) were also conducted, and the results were very similar to those presented here (results available on request). The PCA showed that the largest PC was related to variability of the craniofacial complex in the vertical direction. This was unexpected, considering the orthodontic community’s traditional preoccupation with the anteroposterior direction, although notable exceptions exist (eg, the work by Shudy26). It seems that, although malocclusions in the anteroposterior direction might be more apparent, vertical discrepancies of the skeletal component are more pronounced.
Morphometric methods might provide useful adjunctive information for assessing skeletal pattern.
2.
3.
4.
5.
CONCLUSIONS
Procrustes superimposition and PCA of a widespectrum orthodontic sample showed the following: 1. Seventy percent of shape variability can be described by the first 5 PCs. 2. The first PC, explaining the largest percentage of shape variability (29%), described vertical components, and the second PC (20%) described anteroposterior discrepancies. 3. The sample was a homogeneous group that could not be differentiated into discrete subgroups.
REFERENCES 1. Moyers RE, Bookstein FL. The inappropriateness of conventional cephalometrics. Am J Orthod 1979;75:599-617. 2. Bookstein F. Morphometric tools for landmark data: geometry and biology. Cambridge: Cambridge University Press; 1991. 3. Lele S, Richtsmeier JT. Euclidean distance matrix analysis: a coordinate-free approach for comparing biological shapes using landmark data. Am J Phys Anthropol 1991;86:415-27. 4. Richtsmeier JT, Cheverud JM, Lele S. Advances in anthropological morphometrics. Annu Rev Anthropol 1992;21:283-305. 5. Dryden IL, Mardia KV. Statistical shape analysis. New York: John Wiley & Sons; 1998. 6. O’Higgins P. Ontogeny and phylogeny: some morphometric approaches to skeletal growth and evolution. In: Chaplain MAJ, Singh GD, McLachlan JC, editors. On growth and form Spatiotemporal pattern formation in biology. New York: John Wiley & Sons; 1999. 373-93. 7. Lele S. Invariance and morphometrics: a critical appraisal of statistical techniques for landmark data. In: Chaplain MAJ, Singh GD, McLachlan JC, editors. On growth and form Spatiotemporal pattern formation in biology. New York: John Wiley & Sons; 1999. 325-36. 8. Mardia KV. Statistical shape analysis and its applications. In: Chaplain MAJ, Singh GD, McLachlan JC, editors. On growth
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and form Spatio-temporal pattern formation in biology. New York: John Wiley & Sons; 1999. 337-55. Lele SR, Richtsmeier JT. An invariant approach to statistical analysis of shapes. Boca Raton, Fla: Chapman and Hall/CRC; 2001. Stegmann MB, Gomez DD. A brief introduction to statistical shape analysis [lecture notes]. Department of Informatics and Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark; March 2002. Singh GD, McNamara JA Jr, Lozanoff S. Craniofacial heterogeneity of prepubertal Korean and European-American subjects with Class III malocclusions: Procrustes, EDMA, and cephalometric analyses. Int J Adult Orthod Orthognath Surg 1998;13: 227-40. Singh GD, Clark WJ. Localization of mandibular changes in patients with Class II Division 1 malocclusions treated with Twin-block appliances: finite element scaling analysis. Am J Orthod Dentofacial Orthop 2001;119:419-25. Hennessy RJ, Moss JP. Facial growth: separating shape from size. Eur J Orthod 2001;23:275-85. Lux CJ, Rubel J, Starke J, Conradt C, Stellzig PA, Komposch PG. Effects of early activator treatment in patients with Class II malocclusion evaluated by thin-plate spline analysis. Angle Orthod 2001;71:120-6. Franchi L, Baccetti T, McNamara JA Jr. Thin-plate spline analysis of mandibular growth. Angle Orthod 2001;71:83-9. Cakirer B, Dean D, Palomo JM, Hans MG. Orthognathic surgery outcome analysis: 3-dimensional landmark geometric
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