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Secular trends in the facial skull from the 19th century to the present, analyzed with geometric morphometrics
ORIGINAL ARTICLE
Secular trends in the facial skull from the 19th century to the present, analyzed with geometric morphometrics Erwin Jonke,a Hermann Prossinger,b Fred L. Bookstein,b,c Katrin Schaefer,b Markus Bernhard,d and Josef W. Freudenthalere Vienna, Austria, and Seattle, Wash Introduction: Over the last 100 years, Austrian facial form has changed for various reasons, including changes in growth pattern, changes in shape pattern, or a combination of these. In this study, we explored and contrasted these 2 explanations. Methods: We compared cephalograms from 54 recruits in the present-day Austrian Federal Army to those from 49 dry skulls of soldiers from the Imperial Hapsburg army. Body height was measured or acquired from military records. Forty-three landmarks were located on each lateral cephalogram. Secular change and growth allometry were analyzed with standard Procrustes methods. Results: Body height correlated only weakly with size of the facial skull in these samples, and secular change in facial size (4.5% over a century) was proportionately less than that in height. Growth allometry was nearly unchanged over the century, emphasizing the typical changes of vertical to horizontal proportions and bimaxillary prognathism. Secular changes over the century took the form of far more localized remodeling around the coronoid process and the anterior maxilla. The large-scale differences, in contrast, were opposite to those one would expect from the size change. Conclusions: The observed trends shed considerable light on secular changes in the range of dysmorphologies for clinical orthodontic correction. At the same time, the dissociation between within-century and between-century allometry is an important possibility that was hitherto typically observed only at far greater time scales than the 150 years spanned by these data. (Am J Orthod Dentofacial Orthop 2007;132:63-70)
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ecular trends in the facial skull are of considerable interest in orthodontics.1 They have been variously ascribed to assumed dietary effects,2,3 increased malocclusion,4-7 and environmental factors.8 Many studies have attempted to assess these trends geometrically, but the conclusions were conflicting or ambiguous.9-11 It is not solely the assessment of trends in size or shape that is noteworthy, but also their joint interpretation as the effects of size or secular factors. The literature has long reported changes in the facial skull across recent historical times.12 Body height and skull a
Assistant Professor, Vienna University Clinic of Dentistry, Department of Orthodontics, Medical University of Vienna, Vienna, Austria. b University Professor, Department of Anthropology, University of Vienna, Vienna, Austria. c Professor, Department of Statistics, University of Washington, Seattle. d Graduate student, Department of Anthropology, University of Vienna, Vienna, Austria. e University Professor, Vienna University Clinic of Dentistry, Department of Orthodontics, Medical University of Vienna, Vienna, Austria. Reprint requests to: Fred L. Bookstein, Department of Anthropology, University of Vienna, Althanstrasse 14, A-1091 Vienna, Austria; e-mail, fred. [email protected]. Submitted, June 2005; revised and accepted, August 2005. 0889-5406/$32.00 Copyright © 2007 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2005.08.040
dimensions both typically show increasing trends. Yet the localizations of these changes to specific facial regions and to the particular interregional relationships that are of central importance to orthodontic practice remain unclear. Although some lack of clarity is intrinsic to the subject matter, other aspects are remediable by improvements in methodology. Our previous article critiqued conventional cephalometric approaches in the context of assessing secular trends.13 That criticism appeared nearly at the same time as several other articles similarly criticizing distance-angle cephalometrics in the clinical orthodontic context: eg, McIntyre and Mossey14 and Halazonetis.15 This article extends the previous one by decomposing a secular change of size and shape into the part that is consistent between them and the part that is not. We used standard methods of geometric morphometrics to partition a century of change in the cephalogram into its allometric and nonallometric components, and interpret these separately from both theoretical and clinical points of view. MATERIAL AND METHODS
Our sample pooled 2 distinct groups: 49 cephalograms from the Weisbach collection4,16 of skulls of 63
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Table. Number
Fig 1. Landmarks used in study (Table). Figure redrawn after Riolo et al.17
ethnically white soldiers who died in the service of the Hapsburg Empire in the late 19th century and 54 cephalograms from male soldiers conscripted into the present-day Austrian Federal Army. Body heights were available for all modern subjects and for 46 of the 19th century subjects from military records. Cephalograms were represented by digitized landmark locations of 43 points from the 51-point cephalometric system of the University of Michigan University School Study.17 Figure 1 shows these on a stereotyped tracing. All subjects had all 4 first molars. All landmarks were digitized by an author (M.B.) from tracings made by the first author (E.J.). Paired landmarks were digitized as averages of left- and right-side locations. In a small replication study, 4 of the 103 films, selected at random, were digitized 4 times each by the same operator. The within-case sum of squares was only 0.13% of the between-case sum of squares for shape (in Procrustes units), for an intraclass correlation of 0.9987, which is satisfactorily high for the results to be considered robust against digitizing error. Two versions of the landmark configuration (all 43 points, or all 26 points on the bony anatomy) were analyzed by the usual methods of modern Procrustes-based geometric morphometrics.18 The mathematical theory and biological application of geometric morphometrics are well understood,18-21 and its statistical power has been proven superior to those of distance and angle-based
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Landmarks used in study (after Riolo et al17) Name
Abbreviation
Menton Gnathion Pogonion B-point Infradentale Lower incisor incisal edge Upper incisor incisal edge Supradentale A-point Anterior nasal spine L-point Upper incisor apex Upper incisor lingual bony contact point Lower incisor lingual bony contact point Lower incisor apex Lingual symphyseal point Lower molar mesial CEJ Lower molar mesial cusp tip Upper molar mesial cusp tip Upper molar mesial CEJ Upper molar distal CEJ Upper molar distal contact point Upper molar distal cusp tip Lower molar distal cusp tip Lower molar distal contact point Lower molar distal CEJ Gonion Gonial intersection Opisthion Basion Articulare, posterior Articulare, anterior Condylion Center of spheno-occipital synchondrosis Sella turcica Ethmoid registration point Glabella Nasion Frontomaxillary nasal suture Orbitale Inferior zygoma Pterygomaxillary fissure, superior Pterygomaxillary fissure, inferior Coronoid process Posterior nasal spine
ME GN PG B ID LIE UIE SD A ANS L UIA UIB LIB LIA SYMP LMJ LMT UMT UMJ UDJ UDC UDT LDT LDC LDJ GO GOI OP BA AR AA CO SOS S SE G N FMN OR IZ PTMS PTM CP PNS
CEJ, Crown enamel junction. Points 29 and 34 were missing or difficult to visualize in many 19th century skulls and were omitted from statistical analysis.
methods.22-24 Shape analysis proceeds via the configurations of Procrustes shape coordinates, which are the coordinates of all landmarks after superposition that minimizes the sum of squared distances between the landmarks of any single specimen and the landmarks of the grand mean. Centroid size is the scale factor involved in this superposition, whereas shape is the vector of all landmark locations after this superposition.
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Fig 3. Overall body height against centroid size of facial cephalogram. Linear regressions: 20th century, white band; 19th century sample, black band.
Fig 2. A, Age, and B, height distributions in samples; 19th century sample is significantly older but significantly shorter.
Patterns of displacement of all shape coordinates together are often usefully visualized as thin-plate spline deformations. For expositions of all these standard tools see Bookstein,19 Rohlf and Slice,25 Rohlf and Marcus,26 or Dryden and Mardia.20 Centroid size measures were retained for further statistical use. RESULTS
The samples differed in age and height (Fig 2). Mean ages (⫾ standard deviations) were 22.2 ⫾ 1.6 years for the 19th century subjects, and 20.4 ⫾ 1.9 years for the living subjects. The corresponding average heights were 167 ⫾ 6 cm (in the reduced sample of 46) and 178 ⫾ 7 cm. The modern subsample was substantially younger than the 19th century sample and also about 7% taller, a significant difference. Body height was moderately correlated with facial
size in this data set. As Figure 3 shows, by using centroid size of the skeletal-only landmark set, the correlations were 0.492 for the 19th century data and 0.360 for the 20th century data. With size standardized, the shape coordinate distribution of the 103 specimens for all 43 landmark points is shown in Figure 4. Figure 5 shows the relationship between the 2 samples for the facial skull without molar landmarks. Molars do not contribute to shape analysis in the projection plane (conventionally called the midsagittal plane). Therefore, throughout the rest of this article, molar landmarks are not included in the analysis. In Figure 5, the corresponding means for the 26 skeletal landmarks of both samples are shown. The coordinate system for these and the following displays is the conventional sella-nasion orientation, as applied to the grand mean form, but none of the findings depends on that convention in any way. For either set of landmarks, the 43-point or the 26-point, this shape difference is significant beyond the 0.002 level by the usual permutation test on Procrustes distance.27 For each pair of mean shape coordinates in Figure 5, one can imagine a little vector of displacement linking them. These vectors deserve closer inspection. Figure 6 is a standard way of viewing these displacements as a whole: the thin-plate spline that shows the interpolated
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Fig 4. Shape coordinates for 43 landmarks on 103 specimens. Open circles, 20th century data; crosses, 19th century data. All Procrustes shape coordinates. Axes here and in subsequent figures are in dimensionless Procrustes units.
Fig 6. Thin-plate spline deformation from 19th century average shape to 20th century average. Open circles, landmark points used for warping. Outlines as in Fig 1. Deformation is exaggerated by factor of 3.5 for greater legibility (Bookstein28). Inset: opposite deformation, from later to earlier average shape, also exaggerated by 3.5.
Fig 5. Shape coordinates for 26 bony landmarks only; averages by century. Circles and crosses as in Fig 4.
deformation from the 19th century to recent, extrapolated by a factor of 3.5 for legibility (see Bookstein28). There are 3 areas of particularly striking local deformation. Anterior nasal spine projects farther anteriorly
of the occlusion in the modern men, whereas coronoidale (near the center of the diagram) has been greatly displaced from the pterygomaxillary fissure inferior in its vertical coordinate only. (Of course, these points lie at different distances from the midsagittal plane.) There is also a hint of growth of the condylar head away from the cranial base semilandmarks articulare and postarticulare anterior (again, these are evidently in different planes in the actual skull). Figure 7 is the corresponding grid for the aspects of shape covarying with centroid size—ie, size allometry. The computation is pooled within groups. The difference in these regressions between the 2 samples is not significant by permutation test. In contrast to the finding of Jonke et al,13 on 5 landmarks only, in the analysis of all 26 skeletal landmarks here, this pattern is unrelated to the group difference. Size allometry shows the expected extension along the growth axis, sella to menton, but also
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Fig 8. Scatter plot of centroid size by residual composite shape difference. Circles and crosses as in Fig 3. Linear regressions, 1 for each century, of residual mean difference as function of centroid size; 20th century, white band; 19th century, black line. Fig 7. Pooled within-group allometry: effect of change in centroid size (by an arbitrary but visually convenient factor).
shows strong local features at sphenoethmoid, in the pterygomaxillary notch itself (the displacement from PTM to PTMS), and in the remodeling at A-point and B-point. Thus, when allometry is partialled out of the group difference, the contrast at coronoidale remains, whereas large-scale mandibular retrognathism is artificially inserted. This paradoxical contrast of allometry against shape difference is clearest from a scatter plot of size against size-free shape (defined as residuals from the pooled within-group allometric regressions). In Figure 8, we see the pattern called, in biostatistics, “the ecological fallacy”: regressions of group means with the opposite sign from the within-group allometries that they might be expected to confirm. In our analysis, centroid size is correlated negatively with size-free shape in both samples separately but correlates zero with it in the pooled analysis, because of the paradoxical positive association of mean size with the mean of the shape feature indicated by the grid in Figure 7. Put most tersely, the groups differ as much in shape as they do in size, but in an entirely different pattern. In terms of the actual shape coordinate displacements (Fig 9), most landmarks are shifted oppositely between allometry and sample difference in spite of the size difference between the samples.
The group difference for just the mandibular landmarks is significant separately at about the P ⫽ .015 level by permutation test on Procrustes distance. Almost all shape changes here are local: extension of the condylar head, changes in the eruption of the molar and the incisor, and localized changes at the symphysis. None of this corresponds to the effects of within-group allometry. Analysis of just the maxilla shows that the effect in the maxilla– cranial-base complex is not affected by these changes in the mandible, nor is this interpretation affected by mandibular autorotation. DISCUSSION
In the context of ordinary clinical orthodontics, growth produces shape changes (Enlow and Hans,28 Sarnat30). Accordingly, most contemporary textbooks review the topic of normal craniofacial growth before they begin the more technical discussions of treatment planning, timing, and so on; differences of growth status, in fact, are among the most important factors entering into that planning stage. In a patient followed over time, naturally, growth changes are highly correlated with size changes in the bimaxillary and craniofacial complex. Average growth directions, which are aspects of size change in populations, are often presumed to specify a central tendency for an individual patient, for example,31 and aspects of shape difference between samples of smaller vs larger size (eg, the prognathic hypermorphosis of the stereo-
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Fig 9. Mean difference and allometry as vectors at mean shape coordinates for the 19th century sample. Solid line, group mean difference ⫻ 3; short and long dashed lines, allometries for the 19th century and the 20th century samples, arbitrary factors. Allometries are often opposite to group mean difference, even though 20th century sample was distinctly larger than the 19th century sample.
typed male mandible) often seem to accord with correlates of the growth process per se. This study, based on 2 samples diverging more widely than the reader is ever likely to see in clinical or research practice, shows the limitations of such inferences when extended over time intervals longer than a patient’s adolescence. The finding is known in the biometric literature as the discrepancy between growth allometry and static allometry, or, more pointedly, the ecological fallacy. Stated abstractly, in any setting in which biometric variable A truly depends linearly on causal factor B, within groups separately, then as long as that dependence is imperfect (as long as B is partly determined by other factors also, or as long as A and B are both measured with errors), the relationship between group mean values of B need not be consistent with (ie, need not even have the same sign as) those true within-group predictions. In our example, there is true prediction of shape by size (an allometric regression) in our 2 samples separately—these do not differ significantly, so it is legitimate to refer to them together as 1 regression—and also there is a true group difference of 4.5% in the average predictor (centroid size) driving those regressions. But the mean shape differ-
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ence between the groups is oblique (mainly opposite) to the contrasts implied by that size difference. Put more simply, the group with larger average size (the contemporary soldiers) shows many aspects of average shape that, if found during the 19th century, would be expected to derive from a group with smaller average size even than the 19th century skulls that were sampled here. This paradox matters for orthodontic practice when secular changes in size are invoked to explain differences in patients’ symptom prevalence. It is tempting to assert, for instance, that the reduction in the size of the jaws over recent millennia accounts for temporomandibular joint problems, dental crowding, other clinical presentations such as crossbite,7 or increasing prevalence of Class II malocclusions.4 But evolutionary and secular changes operate at different time scales. The secular size increase should then be acting to attenuate the consequences of the evolutionary size decrease, but in clinical practice we see no such attenuation. Referring this discrepancy to further factors such as tooth size is no help, because those (not measured in this study) might well show the same paradoxical allometry that we see in our bimaxillary shape analysis. The shape consequences of secular size change cannot be used to explain clinical shape phenomena when withinsample and between-sample allometry differ as radically as they do in paired samples such as ours. The contrast between Figure 6 (150 years of secular change) and Figure 7 (pooled within-sample allometry) shows this in terms of transformation grids, familiar since the time of Moorrees et al32 and Faustini et al33 as a convenient modality for the visual integration of change of the cephalogram in several regions simultaneously. The same information is given in Figure 9 as the contrast between the little vectors that move the average landmark positions around on the printed page to correspond to either the change of sample (solid lines) or the effect of size change in the sample (2 sets of dashed lines, corresponding to the 2 century-specific subsamples separately). At most of the salient points in Figure 9, the vector of size allometry is more nearly antiparallel than parallel to the corresponding vectors of size dependence (dashed lines). In these regions, the differences we observe between the 19th century sample and the 20th century sample are seen despite the observed size increase of about 5% from the earlier to the later sample. Glabella and all its neighbors show this pattern, for instance, and likewise, anterior nasal spine and the familiar trio of points on the chin (gnathion, pogonion, menton) and, surprisingly, coronoidale. At other points, such as SE, the individual displacements are much better aligned, so that (since sella is conveniently fixed
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in both processes) the discrepancy between the allometric grid (Fig 7) and the secular-change grid (Fig 6) are strongest just in the small pentahedron of S-AR-PTMCP-SE. Size allometry lengthens both the maxilla and the mandibular corpus, and increases the separation of menton and anterior nasal spine, whereas the secular shift reduces the anterior maxillary measures and leaves the mandible apparently invariant. The sample with the larger subjects has the relatively shorter mandible; evidently, any explanation of this will be paradoxical in its references to size. We have already noted 1 basis for such an explanation, or series of explanations: the soldiers from the 19th century sample are smaller yet older. This suggests systematic changes in lifestyle, nutrition, and so on. However, these factors might be expected to affect the craniofacial complex as a whole. Similarly, the sample of larger mean size has the relatively longer cranial base, even though that proportional length is typically smaller (and mandibular proportional lengths typically larger) in crania of larger size. Size allometry (dependence of shape on size) has not altered over the century (the 2 sets of dashed segments in Fig 9 do not significantly differ). Thus, these trends, the first of which directly violates the expectations associated with larger size, can be accounted for only by speculative adjustments in epigenetic or environmental conditions, such as shifts in diet (softer food of higher nutritional value) or in equally speculative hypotheses involving differences in ethnic or social origin of the 2 analogous samples. We are at a loss to account for the findings in the sense of rationalizing just these specific regional effects and no others. (For instance, the 2 samples were imaged under different biological conditions—1 group living, 1 postmortem— but how could that difference account for localized displacement just at the coronoid, without corresponding findings at the condyle and without apparent paradoxical autorotation?) The 19th century soldiers were buried in moist earth, so shrinking or geometric distortions from drying could occur only after Weisbach disinterred them and stored them in the Natural History Museum in Vienna. Even so, drying seems to contribute at most a 1% distortion to bony specimens.7 Of course, for empirical findings to be of some use to the clinician, there need not be at the same time a satisfactory theoretical explanation. The reduced jaw, together with increased body size, for instance, could easily account for the increased tooth crowding in modern populations, because tooth size is correlated with body size.1,34 The changes in condylar position might have major consequences for what in some
countries is called a contemporary epidemic of temporomandibular joint problems. Additionally, and still without theoretical rationale, our findings suggest more generalized investigations. We can certainly look for secular trends, with a special interest in the counterallometric ones, in the posteroanterior view, relying on a different system of landmarks for shape coordinates that incorporate such features as width profiles. A more sophisticated approach would look at data in 3 dimensions, not 2 dimensions, assuming that craniofacial orthopedics will sooner or later rely on low-dose computed tomography instead of cephalograms. An author (F.L.B.) recently argued35 that such an analysis will almost certainly fail if limited solely to landmarks but, according to the principles of every source of wisdom from Martin12 on, needs to incorporate information from curves and surfaces also. Fortunately, the software that accommodates that extension is already in widespread use in anthropology36,37 and needs only to be brought to the attention of the dentofacial orthopedist. CONCLUSIONS
What was intended as a routine comparison of 2 samples of skulls of men separated by only a century in time turned out to be paradoxical in that, even though the samples differed in average size in the expected way, the difference between their average shapes was not aligned with what would be expected from the size difference of 4.5% over the century. Rather, those differences either ran directly opposite to what would have been expected from the size difference (ie, mandibular retrognathism in the sample of larger average size—robustness of the 19th century specimens!) or dealt with local features of remodeling at the upper alveolar ridge and coronoid that had little size dependency. We have no theory to explain these findings, which are statistically significant, but we suggest only that they must arise from some combination of ethnic changes, conditions of imaging living vs postmortem materials, or changes in diet over the last hundred years. Nevertheless, rationalized or not, contrasts like this are likely to be of interest to practicing orthodontists, especially those with a sense of the history of their clientele and their typical malocclusions. Note, for instance, that the calendar span between these 2 samples is only about twice the distance of contemporary samples from the Broadbent-Bolton sample used for a popular set of clinical norms. We thank Maria Teschler-Nicola, Natural History Museum, Vienna, for permission to access the skulls under her care and Otto Vyslozil, Klosterneuburg, for
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help with collecting and x-raying the skulls. Philipp Gunz and Philipp Mitteroecker, Institute for Anthropology, University of Vienna, supplied us with software routines programmed in Mathematica. REFERENCES 1. Smith BH, Garn SM, Hunter WS. Secular trends in face size. Angle Orthod 1986;56;196-204. 2. Kiliaridis S, Engström C, Thilander B. The relationship between masticatory function and craniofacial morphology. Part I: a cephalometric longitudinal analysis in the growing rat fed a soft diet. Eur J Orthod 1985;7:273-83. 3. Yom-Tov Y, Yom-Tov S. Climatic change and body size in two species of Japanese rodents. Biol J Linn Soc 2004;82:263-7. 4. Vyslozil O, Jonke E. Kieferorthopädisch-anthropometrische vergleichsuntersuchung an 100 jahre alten menschlichen schädeln und österreichischen bundesheersoldaten. Informationen 1994;4: 409-36. 5. Weiland FJ, Jonke E, Bantleon HP. Secular trends in malocclusion in Austrian men. Eur J Orthod 1997;19:355-9. 6. Varrela J. Occurrence of malocclusion in attritive environments: a study of a skull sample from southwest Finland. Scand J Dent Res 1990;98:242-7. 7. Lindsten R, Ögaard B, Larsson E, Bjerklin K. Transverse dental and dental arch depth dimensions in the mixed dentition in a skeletal sample from the 14th to the 19th century and Norwegian children and Norwegian Sami children of today. Angle Orthod 2002;72:439-48. 8. Jaeger U, Bruchhaus H, Finke L, Kronemeyer-Hauschild K, Zellner K. Säkularer trend bei der körperhöhe seit dem neolithikum. Anthrop Anz 1998;56:117-30. 9. Anderson DL, Thompson GW, Popovich F. Evolutionary dental changes. Am J Phys Anthropol 1975;43:95-102. 10. Key P, Jantz RL. A multivariate analysis of temporal change in Arikara craniometrics: a methodological approach. Am J Phys Anthropol 1981;55:247-59. 11. Dibbets JMH, Nolte K. Comparison of linear cephalometric dimensions in Americans of European descent (Ann Arbor, Cleveland and Philadelphia) and Americans of African descent (Nashville). Angle Orthod 2004;72:324-30. 12. Martin R. Lehrbuch der anthropologie in systematischer darstellung. Jena, Germany: Gustav Fischer; 1914. 13. Jonke E, Schaefer K, Freudenthaler JW, Prossinger H, Bookstein FL. A cephalometric comparison of skulls from different time periods—the Bronze Age, the 19th century and the present. Coll Antropol 2003;27:789-801. 14. McIntyre GT, Mossey PA. Size and shape measurement in contemporary cephalometrics. Eur J Orthod 2003;25:231-42. 15. Halazonetis DJ. Morphometrics for cephalometric diagnosis. Am J Orthod Dentofacial Orthop 2004;125:571-81.
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16. Weisbach A. Deutsche männer aus niederösterreich. Wien: Verlag Hölzl; 1892. 17. Riolo ML, Moyers RE, McNamara JA, Hunter WS. An atlas of craniofacial growth. Craniaofacial Growth Series. Monograph no. 2. Ann Arbor: Center for Human Growth and Development; University of Michigan; 1974. 18. Marcus LF, Corti M, Loy A, Naylor G, Slice DE, editors. Advances in morphometrics. New York: Plenum Press; 1996. 19. Bookstein FL. Morphometric tools for landmark data: geometry and biology. New York: Cambridge University Press; 1991. 20. Dryden IL, Mardia KV. Statistical shape analysis. Chichester, United Kingdom: Wiley; 1998. 21. Slice DE, editor. Modern morphometrics in physical anthropology. New York: Kluwer; 2005. 22. Rohlf FJ. On the use of shape spaces to compare morphometric methods. Hystrix, Ital Journ Mamm 2000;11:9-25. 23. Rohlf FJ. Statistical power comparisons among alternative morphometric methods. Am J Phys Anthropol 2000;111:463-78. 24. Rohlf FJ. Bias and error estimates of mean shape in geometric morphometrics. J Hum Evol 2003;44:665-83. 25. Rohlf FJ, Slice DE. Extensions of the Procrustes method for the optimal superimposition of landmarks. Syst Zool 1990;39:40-59. 26. Rohlf FJ, Marcus LF. A revolution in morphometrics. Trends in Ecology and Evolution 1993;8:129-32. 27. Good P. Permutation tests. 2nd ed. Berlin, Heidelberg, New York: Springer Verlag; 2000. 28. Bookstein FL. Creases as local features of deformation grids. Med Image Anal 2000;4:93-110. 29. Enlow DH, Hans MG. Essentials of facial growth. Philadelphia: Saunders; 1996. 30. Sarnat BG. Postnatal growth of the nasomaxillary complex. In: Dixon AD, Hoyte DA, Rönning O, editors. Fundamentals of craniofacial growth. New York: CRC Press; 1997. 31. Bookstein FL, Moyers RE. Do horizontal growers grow horizontally? Swed Dent J Suppl 1982;15:179-87. 32. Moorrees CFA, van Venrooij ME, Lebret TLML, Glatky CB, Kent RL, Reed RB. New norms for the mesh diagram analysis. Am J Orthod 1976;69:57-71. 33. Faustini MM, Hale C, Cisneros GJ. Mesh diagram analysis: developing a norm for African Americans. Angle Orthod 1997; 67:121-8. 34. Garn SM. Human growth. Ann Rev Anthropol 1980;9:275-92. 35. Bookstein FL, Schaefer K, Mitteroecker P, Gunz P, Seidler H. The geometry of anthropometrics: a new typology of landmarks [abstract]. Am J Phys Anthropol 2004;123(Suppl 38):66. 36. Mitteroecker P, Gunz P, Bernhard M, Schaefer K, Bookstein FL. Comparison of cranial ontogenetic trajectories among hominoids. J Hum Evol 2004;46:679-97. 37. Gunz P, Mitteroecker P, Bookstein FL. Semilandmarks in three dimensions. In: Slice DE, editor. Modern morphometrics in physical anthropology. New York: Kluwer; 2005. p. 73-98.
Secular trends in the facial skull from the 19th century to the present, analyzed with geometric morphometrics Erwin Jonke,a Hermann Prossinger,b Fred L. Bookstein,b,c Katrin Schaefer,b Markus Bernhard,d and Josef W. Freudenthalere Vienna, Austria, and Seattle, Wash Introduction: Over the last 100 years, Austrian facial form has changed for various reasons, including changes in growth pattern, changes in shape pattern, or a combination of these. In this study, we explored and contrasted these 2 explanations. Methods: We compared cephalograms from 54 recruits in the present-day Austrian Federal Army to those from 49 dry skulls of soldiers from the Imperial Hapsburg army. Body height was measured or acquired from military records. Forty-three landmarks were located on each lateral cephalogram. Secular change and growth allometry were analyzed with standard Procrustes methods. Results: Body height correlated only weakly with size of the facial skull in these samples, and secular change in facial size (4.5% over a century) was proportionately less than that in height. Growth allometry was nearly unchanged over the century, emphasizing the typical changes of vertical to horizontal proportions and bimaxillary prognathism. Secular changes over the century took the form of far more localized remodeling around the coronoid process and the anterior maxilla. The large-scale differences, in contrast, were opposite to those one would expect from the size change. Conclusions: The observed trends shed considerable light on secular changes in the range of dysmorphologies for clinical orthodontic correction. At the same time, the dissociation between within-century and between-century allometry is an important possibility that was hitherto typically observed only at far greater time scales than the 150 years spanned by these data. (Am J Orthod Dentofacial Orthop 2007;132:63-70)
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ecular trends in the facial skull are of considerable interest in orthodontics.1 They have been variously ascribed to assumed dietary effects,2,3 increased malocclusion,4-7 and environmental factors.8 Many studies have attempted to assess these trends geometrically, but the conclusions were conflicting or ambiguous.9-11 It is not solely the assessment of trends in size or shape that is noteworthy, but also their joint interpretation as the effects of size or secular factors. The literature has long reported changes in the facial skull across recent historical times.12 Body height and skull a
Assistant Professor, Vienna University Clinic of Dentistry, Department of Orthodontics, Medical University of Vienna, Vienna, Austria. b University Professor, Department of Anthropology, University of Vienna, Vienna, Austria. c Professor, Department of Statistics, University of Washington, Seattle. d Graduate student, Department of Anthropology, University of Vienna, Vienna, Austria. e University Professor, Vienna University Clinic of Dentistry, Department of Orthodontics, Medical University of Vienna, Vienna, Austria. Reprint requests to: Fred L. Bookstein, Department of Anthropology, University of Vienna, Althanstrasse 14, A-1091 Vienna, Austria; e-mail, fred. [email protected]. Submitted, June 2005; revised and accepted, August 2005. 0889-5406/$32.00 Copyright © 2007 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2005.08.040
dimensions both typically show increasing trends. Yet the localizations of these changes to specific facial regions and to the particular interregional relationships that are of central importance to orthodontic practice remain unclear. Although some lack of clarity is intrinsic to the subject matter, other aspects are remediable by improvements in methodology. Our previous article critiqued conventional cephalometric approaches in the context of assessing secular trends.13 That criticism appeared nearly at the same time as several other articles similarly criticizing distance-angle cephalometrics in the clinical orthodontic context: eg, McIntyre and Mossey14 and Halazonetis.15 This article extends the previous one by decomposing a secular change of size and shape into the part that is consistent between them and the part that is not. We used standard methods of geometric morphometrics to partition a century of change in the cephalogram into its allometric and nonallometric components, and interpret these separately from both theoretical and clinical points of view. MATERIAL AND METHODS
Our sample pooled 2 distinct groups: 49 cephalograms from the Weisbach collection4,16 of skulls of 63
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Table. Number
Fig 1. Landmarks used in study (Table). Figure redrawn after Riolo et al.17
ethnically white soldiers who died in the service of the Hapsburg Empire in the late 19th century and 54 cephalograms from male soldiers conscripted into the present-day Austrian Federal Army. Body heights were available for all modern subjects and for 46 of the 19th century subjects from military records. Cephalograms were represented by digitized landmark locations of 43 points from the 51-point cephalometric system of the University of Michigan University School Study.17 Figure 1 shows these on a stereotyped tracing. All subjects had all 4 first molars. All landmarks were digitized by an author (M.B.) from tracings made by the first author (E.J.). Paired landmarks were digitized as averages of left- and right-side locations. In a small replication study, 4 of the 103 films, selected at random, were digitized 4 times each by the same operator. The within-case sum of squares was only 0.13% of the between-case sum of squares for shape (in Procrustes units), for an intraclass correlation of 0.9987, which is satisfactorily high for the results to be considered robust against digitizing error. Two versions of the landmark configuration (all 43 points, or all 26 points on the bony anatomy) were analyzed by the usual methods of modern Procrustes-based geometric morphometrics.18 The mathematical theory and biological application of geometric morphometrics are well understood,18-21 and its statistical power has been proven superior to those of distance and angle-based
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Landmarks used in study (after Riolo et al17) Name
Abbreviation
Menton Gnathion Pogonion B-point Infradentale Lower incisor incisal edge Upper incisor incisal edge Supradentale A-point Anterior nasal spine L-point Upper incisor apex Upper incisor lingual bony contact point Lower incisor lingual bony contact point Lower incisor apex Lingual symphyseal point Lower molar mesial CEJ Lower molar mesial cusp tip Upper molar mesial cusp tip Upper molar mesial CEJ Upper molar distal CEJ Upper molar distal contact point Upper molar distal cusp tip Lower molar distal cusp tip Lower molar distal contact point Lower molar distal CEJ Gonion Gonial intersection Opisthion Basion Articulare, posterior Articulare, anterior Condylion Center of spheno-occipital synchondrosis Sella turcica Ethmoid registration point Glabella Nasion Frontomaxillary nasal suture Orbitale Inferior zygoma Pterygomaxillary fissure, superior Pterygomaxillary fissure, inferior Coronoid process Posterior nasal spine
ME GN PG B ID LIE UIE SD A ANS L UIA UIB LIB LIA SYMP LMJ LMT UMT UMJ UDJ UDC UDT LDT LDC LDJ GO GOI OP BA AR AA CO SOS S SE G N FMN OR IZ PTMS PTM CP PNS
CEJ, Crown enamel junction. Points 29 and 34 were missing or difficult to visualize in many 19th century skulls and were omitted from statistical analysis.
methods.22-24 Shape analysis proceeds via the configurations of Procrustes shape coordinates, which are the coordinates of all landmarks after superposition that minimizes the sum of squared distances between the landmarks of any single specimen and the landmarks of the grand mean. Centroid size is the scale factor involved in this superposition, whereas shape is the vector of all landmark locations after this superposition.
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Fig 3. Overall body height against centroid size of facial cephalogram. Linear regressions: 20th century, white band; 19th century sample, black band.
Fig 2. A, Age, and B, height distributions in samples; 19th century sample is significantly older but significantly shorter.
Patterns of displacement of all shape coordinates together are often usefully visualized as thin-plate spline deformations. For expositions of all these standard tools see Bookstein,19 Rohlf and Slice,25 Rohlf and Marcus,26 or Dryden and Mardia.20 Centroid size measures were retained for further statistical use. RESULTS
The samples differed in age and height (Fig 2). Mean ages (⫾ standard deviations) were 22.2 ⫾ 1.6 years for the 19th century subjects, and 20.4 ⫾ 1.9 years for the living subjects. The corresponding average heights were 167 ⫾ 6 cm (in the reduced sample of 46) and 178 ⫾ 7 cm. The modern subsample was substantially younger than the 19th century sample and also about 7% taller, a significant difference. Body height was moderately correlated with facial
size in this data set. As Figure 3 shows, by using centroid size of the skeletal-only landmark set, the correlations were 0.492 for the 19th century data and 0.360 for the 20th century data. With size standardized, the shape coordinate distribution of the 103 specimens for all 43 landmark points is shown in Figure 4. Figure 5 shows the relationship between the 2 samples for the facial skull without molar landmarks. Molars do not contribute to shape analysis in the projection plane (conventionally called the midsagittal plane). Therefore, throughout the rest of this article, molar landmarks are not included in the analysis. In Figure 5, the corresponding means for the 26 skeletal landmarks of both samples are shown. The coordinate system for these and the following displays is the conventional sella-nasion orientation, as applied to the grand mean form, but none of the findings depends on that convention in any way. For either set of landmarks, the 43-point or the 26-point, this shape difference is significant beyond the 0.002 level by the usual permutation test on Procrustes distance.27 For each pair of mean shape coordinates in Figure 5, one can imagine a little vector of displacement linking them. These vectors deserve closer inspection. Figure 6 is a standard way of viewing these displacements as a whole: the thin-plate spline that shows the interpolated
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Fig 4. Shape coordinates for 43 landmarks on 103 specimens. Open circles, 20th century data; crosses, 19th century data. All Procrustes shape coordinates. Axes here and in subsequent figures are in dimensionless Procrustes units.
Fig 6. Thin-plate spline deformation from 19th century average shape to 20th century average. Open circles, landmark points used for warping. Outlines as in Fig 1. Deformation is exaggerated by factor of 3.5 for greater legibility (Bookstein28). Inset: opposite deformation, from later to earlier average shape, also exaggerated by 3.5.
Fig 5. Shape coordinates for 26 bony landmarks only; averages by century. Circles and crosses as in Fig 4.
deformation from the 19th century to recent, extrapolated by a factor of 3.5 for legibility (see Bookstein28). There are 3 areas of particularly striking local deformation. Anterior nasal spine projects farther anteriorly
of the occlusion in the modern men, whereas coronoidale (near the center of the diagram) has been greatly displaced from the pterygomaxillary fissure inferior in its vertical coordinate only. (Of course, these points lie at different distances from the midsagittal plane.) There is also a hint of growth of the condylar head away from the cranial base semilandmarks articulare and postarticulare anterior (again, these are evidently in different planes in the actual skull). Figure 7 is the corresponding grid for the aspects of shape covarying with centroid size—ie, size allometry. The computation is pooled within groups. The difference in these regressions between the 2 samples is not significant by permutation test. In contrast to the finding of Jonke et al,13 on 5 landmarks only, in the analysis of all 26 skeletal landmarks here, this pattern is unrelated to the group difference. Size allometry shows the expected extension along the growth axis, sella to menton, but also
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Fig 8. Scatter plot of centroid size by residual composite shape difference. Circles and crosses as in Fig 3. Linear regressions, 1 for each century, of residual mean difference as function of centroid size; 20th century, white band; 19th century, black line. Fig 7. Pooled within-group allometry: effect of change in centroid size (by an arbitrary but visually convenient factor).
shows strong local features at sphenoethmoid, in the pterygomaxillary notch itself (the displacement from PTM to PTMS), and in the remodeling at A-point and B-point. Thus, when allometry is partialled out of the group difference, the contrast at coronoidale remains, whereas large-scale mandibular retrognathism is artificially inserted. This paradoxical contrast of allometry against shape difference is clearest from a scatter plot of size against size-free shape (defined as residuals from the pooled within-group allometric regressions). In Figure 8, we see the pattern called, in biostatistics, “the ecological fallacy”: regressions of group means with the opposite sign from the within-group allometries that they might be expected to confirm. In our analysis, centroid size is correlated negatively with size-free shape in both samples separately but correlates zero with it in the pooled analysis, because of the paradoxical positive association of mean size with the mean of the shape feature indicated by the grid in Figure 7. Put most tersely, the groups differ as much in shape as they do in size, but in an entirely different pattern. In terms of the actual shape coordinate displacements (Fig 9), most landmarks are shifted oppositely between allometry and sample difference in spite of the size difference between the samples.
The group difference for just the mandibular landmarks is significant separately at about the P ⫽ .015 level by permutation test on Procrustes distance. Almost all shape changes here are local: extension of the condylar head, changes in the eruption of the molar and the incisor, and localized changes at the symphysis. None of this corresponds to the effects of within-group allometry. Analysis of just the maxilla shows that the effect in the maxilla– cranial-base complex is not affected by these changes in the mandible, nor is this interpretation affected by mandibular autorotation. DISCUSSION
In the context of ordinary clinical orthodontics, growth produces shape changes (Enlow and Hans,28 Sarnat30). Accordingly, most contemporary textbooks review the topic of normal craniofacial growth before they begin the more technical discussions of treatment planning, timing, and so on; differences of growth status, in fact, are among the most important factors entering into that planning stage. In a patient followed over time, naturally, growth changes are highly correlated with size changes in the bimaxillary and craniofacial complex. Average growth directions, which are aspects of size change in populations, are often presumed to specify a central tendency for an individual patient, for example,31 and aspects of shape difference between samples of smaller vs larger size (eg, the prognathic hypermorphosis of the stereo-
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Fig 9. Mean difference and allometry as vectors at mean shape coordinates for the 19th century sample. Solid line, group mean difference ⫻ 3; short and long dashed lines, allometries for the 19th century and the 20th century samples, arbitrary factors. Allometries are often opposite to group mean difference, even though 20th century sample was distinctly larger than the 19th century sample.
typed male mandible) often seem to accord with correlates of the growth process per se. This study, based on 2 samples diverging more widely than the reader is ever likely to see in clinical or research practice, shows the limitations of such inferences when extended over time intervals longer than a patient’s adolescence. The finding is known in the biometric literature as the discrepancy between growth allometry and static allometry, or, more pointedly, the ecological fallacy. Stated abstractly, in any setting in which biometric variable A truly depends linearly on causal factor B, within groups separately, then as long as that dependence is imperfect (as long as B is partly determined by other factors also, or as long as A and B are both measured with errors), the relationship between group mean values of B need not be consistent with (ie, need not even have the same sign as) those true within-group predictions. In our example, there is true prediction of shape by size (an allometric regression) in our 2 samples separately—these do not differ significantly, so it is legitimate to refer to them together as 1 regression—and also there is a true group difference of 4.5% in the average predictor (centroid size) driving those regressions. But the mean shape differ-
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ence between the groups is oblique (mainly opposite) to the contrasts implied by that size difference. Put more simply, the group with larger average size (the contemporary soldiers) shows many aspects of average shape that, if found during the 19th century, would be expected to derive from a group with smaller average size even than the 19th century skulls that were sampled here. This paradox matters for orthodontic practice when secular changes in size are invoked to explain differences in patients’ symptom prevalence. It is tempting to assert, for instance, that the reduction in the size of the jaws over recent millennia accounts for temporomandibular joint problems, dental crowding, other clinical presentations such as crossbite,7 or increasing prevalence of Class II malocclusions.4 But evolutionary and secular changes operate at different time scales. The secular size increase should then be acting to attenuate the consequences of the evolutionary size decrease, but in clinical practice we see no such attenuation. Referring this discrepancy to further factors such as tooth size is no help, because those (not measured in this study) might well show the same paradoxical allometry that we see in our bimaxillary shape analysis. The shape consequences of secular size change cannot be used to explain clinical shape phenomena when withinsample and between-sample allometry differ as radically as they do in paired samples such as ours. The contrast between Figure 6 (150 years of secular change) and Figure 7 (pooled within-sample allometry) shows this in terms of transformation grids, familiar since the time of Moorrees et al32 and Faustini et al33 as a convenient modality for the visual integration of change of the cephalogram in several regions simultaneously. The same information is given in Figure 9 as the contrast between the little vectors that move the average landmark positions around on the printed page to correspond to either the change of sample (solid lines) or the effect of size change in the sample (2 sets of dashed lines, corresponding to the 2 century-specific subsamples separately). At most of the salient points in Figure 9, the vector of size allometry is more nearly antiparallel than parallel to the corresponding vectors of size dependence (dashed lines). In these regions, the differences we observe between the 19th century sample and the 20th century sample are seen despite the observed size increase of about 5% from the earlier to the later sample. Glabella and all its neighbors show this pattern, for instance, and likewise, anterior nasal spine and the familiar trio of points on the chin (gnathion, pogonion, menton) and, surprisingly, coronoidale. At other points, such as SE, the individual displacements are much better aligned, so that (since sella is conveniently fixed
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in both processes) the discrepancy between the allometric grid (Fig 7) and the secular-change grid (Fig 6) are strongest just in the small pentahedron of S-AR-PTMCP-SE. Size allometry lengthens both the maxilla and the mandibular corpus, and increases the separation of menton and anterior nasal spine, whereas the secular shift reduces the anterior maxillary measures and leaves the mandible apparently invariant. The sample with the larger subjects has the relatively shorter mandible; evidently, any explanation of this will be paradoxical in its references to size. We have already noted 1 basis for such an explanation, or series of explanations: the soldiers from the 19th century sample are smaller yet older. This suggests systematic changes in lifestyle, nutrition, and so on. However, these factors might be expected to affect the craniofacial complex as a whole. Similarly, the sample of larger mean size has the relatively longer cranial base, even though that proportional length is typically smaller (and mandibular proportional lengths typically larger) in crania of larger size. Size allometry (dependence of shape on size) has not altered over the century (the 2 sets of dashed segments in Fig 9 do not significantly differ). Thus, these trends, the first of which directly violates the expectations associated with larger size, can be accounted for only by speculative adjustments in epigenetic or environmental conditions, such as shifts in diet (softer food of higher nutritional value) or in equally speculative hypotheses involving differences in ethnic or social origin of the 2 analogous samples. We are at a loss to account for the findings in the sense of rationalizing just these specific regional effects and no others. (For instance, the 2 samples were imaged under different biological conditions—1 group living, 1 postmortem— but how could that difference account for localized displacement just at the coronoid, without corresponding findings at the condyle and without apparent paradoxical autorotation?) The 19th century soldiers were buried in moist earth, so shrinking or geometric distortions from drying could occur only after Weisbach disinterred them and stored them in the Natural History Museum in Vienna. Even so, drying seems to contribute at most a 1% distortion to bony specimens.7 Of course, for empirical findings to be of some use to the clinician, there need not be at the same time a satisfactory theoretical explanation. The reduced jaw, together with increased body size, for instance, could easily account for the increased tooth crowding in modern populations, because tooth size is correlated with body size.1,34 The changes in condylar position might have major consequences for what in some
countries is called a contemporary epidemic of temporomandibular joint problems. Additionally, and still without theoretical rationale, our findings suggest more generalized investigations. We can certainly look for secular trends, with a special interest in the counterallometric ones, in the posteroanterior view, relying on a different system of landmarks for shape coordinates that incorporate such features as width profiles. A more sophisticated approach would look at data in 3 dimensions, not 2 dimensions, assuming that craniofacial orthopedics will sooner or later rely on low-dose computed tomography instead of cephalograms. An author (F.L.B.) recently argued35 that such an analysis will almost certainly fail if limited solely to landmarks but, according to the principles of every source of wisdom from Martin12 on, needs to incorporate information from curves and surfaces also. Fortunately, the software that accommodates that extension is already in widespread use in anthropology36,37 and needs only to be brought to the attention of the dentofacial orthopedist. CONCLUSIONS
What was intended as a routine comparison of 2 samples of skulls of men separated by only a century in time turned out to be paradoxical in that, even though the samples differed in average size in the expected way, the difference between their average shapes was not aligned with what would be expected from the size difference of 4.5% over the century. Rather, those differences either ran directly opposite to what would have been expected from the size difference (ie, mandibular retrognathism in the sample of larger average size—robustness of the 19th century specimens!) or dealt with local features of remodeling at the upper alveolar ridge and coronoid that had little size dependency. We have no theory to explain these findings, which are statistically significant, but we suggest only that they must arise from some combination of ethnic changes, conditions of imaging living vs postmortem materials, or changes in diet over the last hundred years. Nevertheless, rationalized or not, contrasts like this are likely to be of interest to practicing orthodontists, especially those with a sense of the history of their clientele and their typical malocclusions. Note, for instance, that the calendar span between these 2 samples is only about twice the distance of contemporary samples from the Broadbent-Bolton sample used for a popular set of clinical norms. We thank Maria Teschler-Nicola, Natural History Museum, Vienna, for permission to access the skulls under her care and Otto Vyslozil, Klosterneuburg, for
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help with collecting and x-raying the skulls. Philipp Gunz and Philipp Mitteroecker, Institute for Anthropology, University of Vienna, supplied us with software routines programmed in Mathematica. REFERENCES 1. Smith BH, Garn SM, Hunter WS. Secular trends in face size. Angle Orthod 1986;56;196-204. 2. Kiliaridis S, Engström C, Thilander B. The relationship between masticatory function and craniofacial morphology. Part I: a cephalometric longitudinal analysis in the growing rat fed a soft diet. Eur J Orthod 1985;7:273-83. 3. Yom-Tov Y, Yom-Tov S. Climatic change and body size in two species of Japanese rodents. Biol J Linn Soc 2004;82:263-7. 4. Vyslozil O, Jonke E. Kieferorthopädisch-anthropometrische vergleichsuntersuchung an 100 jahre alten menschlichen schädeln und österreichischen bundesheersoldaten. Informationen 1994;4: 409-36. 5. Weiland FJ, Jonke E, Bantleon HP. Secular trends in malocclusion in Austrian men. Eur J Orthod 1997;19:355-9. 6. Varrela J. Occurrence of malocclusion in attritive environments: a study of a skull sample from southwest Finland. Scand J Dent Res 1990;98:242-7. 7. Lindsten R, Ögaard B, Larsson E, Bjerklin K. Transverse dental and dental arch depth dimensions in the mixed dentition in a skeletal sample from the 14th to the 19th century and Norwegian children and Norwegian Sami children of today. Angle Orthod 2002;72:439-48. 8. Jaeger U, Bruchhaus H, Finke L, Kronemeyer-Hauschild K, Zellner K. Säkularer trend bei der körperhöhe seit dem neolithikum. Anthrop Anz 1998;56:117-30. 9. Anderson DL, Thompson GW, Popovich F. Evolutionary dental changes. Am J Phys Anthropol 1975;43:95-102. 10. Key P, Jantz RL. A multivariate analysis of temporal change in Arikara craniometrics: a methodological approach. Am J Phys Anthropol 1981;55:247-59. 11. Dibbets JMH, Nolte K. Comparison of linear cephalometric dimensions in Americans of European descent (Ann Arbor, Cleveland and Philadelphia) and Americans of African descent (Nashville). Angle Orthod 2004;72:324-30. 12. Martin R. Lehrbuch der anthropologie in systematischer darstellung. Jena, Germany: Gustav Fischer; 1914. 13. Jonke E, Schaefer K, Freudenthaler JW, Prossinger H, Bookstein FL. A cephalometric comparison of skulls from different time periods—the Bronze Age, the 19th century and the present. Coll Antropol 2003;27:789-801. 14. McIntyre GT, Mossey PA. Size and shape measurement in contemporary cephalometrics. Eur J Orthod 2003;25:231-42. 15. Halazonetis DJ. Morphometrics for cephalometric diagnosis. Am J Orthod Dentofacial Orthop 2004;125:571-81.
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16. Weisbach A. Deutsche männer aus niederösterreich. Wien: Verlag Hölzl; 1892. 17. Riolo ML, Moyers RE, McNamara JA, Hunter WS. An atlas of craniofacial growth. Craniaofacial Growth Series. Monograph no. 2. Ann Arbor: Center for Human Growth and Development; University of Michigan; 1974. 18. Marcus LF, Corti M, Loy A, Naylor G, Slice DE, editors. Advances in morphometrics. New York: Plenum Press; 1996. 19. Bookstein FL. Morphometric tools for landmark data: geometry and biology. New York: Cambridge University Press; 1991. 20. Dryden IL, Mardia KV. Statistical shape analysis. Chichester, United Kingdom: Wiley; 1998. 21. Slice DE, editor. Modern morphometrics in physical anthropology. New York: Kluwer; 2005. 22. Rohlf FJ. On the use of shape spaces to compare morphometric methods. Hystrix, Ital Journ Mamm 2000;11:9-25. 23. Rohlf FJ. Statistical power comparisons among alternative morphometric methods. Am J Phys Anthropol 2000;111:463-78. 24. Rohlf FJ. Bias and error estimates of mean shape in geometric morphometrics. J Hum Evol 2003;44:665-83. 25. Rohlf FJ, Slice DE. Extensions of the Procrustes method for the optimal superimposition of landmarks. Syst Zool 1990;39:40-59. 26. Rohlf FJ, Marcus LF. A revolution in morphometrics. Trends in Ecology and Evolution 1993;8:129-32. 27. Good P. Permutation tests. 2nd ed. Berlin, Heidelberg, New York: Springer Verlag; 2000. 28. Bookstein FL. Creases as local features of deformation grids. Med Image Anal 2000;4:93-110. 29. Enlow DH, Hans MG. Essentials of facial growth. Philadelphia: Saunders; 1996. 30. Sarnat BG. Postnatal growth of the nasomaxillary complex. In: Dixon AD, Hoyte DA, Rönning O, editors. Fundamentals of craniofacial growth. New York: CRC Press; 1997. 31. Bookstein FL, Moyers RE. Do horizontal growers grow horizontally? Swed Dent J Suppl 1982;15:179-87. 32. Moorrees CFA, van Venrooij ME, Lebret TLML, Glatky CB, Kent RL, Reed RB. New norms for the mesh diagram analysis. Am J Orthod 1976;69:57-71. 33. Faustini MM, Hale C, Cisneros GJ. Mesh diagram analysis: developing a norm for African Americans. Angle Orthod 1997; 67:121-8. 34. Garn SM. Human growth. Ann Rev Anthropol 1980;9:275-92. 35. Bookstein FL, Schaefer K, Mitteroecker P, Gunz P, Seidler H. The geometry of anthropometrics: a new typology of landmarks [abstract]. Am J Phys Anthropol 2004;123(Suppl 38):66. 36. Mitteroecker P, Gunz P, Bernhard M, Schaefer K, Bookstein FL. Comparison of cranial ontogenetic trajectories among hominoids. J Hum Evol 2004;46:679-97. 37. Gunz P, Mitteroecker P, Bookstein FL. Semilandmarks in three dimensions. In: Slice DE, editor. Modern morphometrics in physical anthropology. New York: Kluwer; 2005. p. 73-98.