Edgar 2004.3 54 55 The Mean Measure of Divergence (MMD) has become the standard statistical technique for assessing biological affinities when using frequencies of dental morphological characteristics (Scott and Turner, 1997). There are several advantages in using this statistic: It is appropriate for nominal data, it is relatively easy to compute, and it is comparable among researchers. There is however, a drawback to using the MMD; it is only appropriately used when the traits being studied are independent. The assumption of independence is weak for several dental characteristics, so inter- trait correlations must be tested, and traits that are correlated must be removed from of a MMD analysis. An alternative to MMD is the Mahalanobis’ D2 statistic, which allows correlated features to be used in affinity measures (Mahalanobis, 1936). However, as originally formulated, this statistic is useful only for metric, not nominal, data. Konigsberg (1990) used a pseudo-Mahalanobis’ D2 to determine biological affinity using non-metric data. This statistic has the potential to allow distance measures to be based on a greater variety and number of dental characteristics than the MMD. Of course, like MMD, the D2 statistic has its drawbacks. The primary problems with the application of this statistic are its limited applicability when analyzing a number of traits with little or no correlation, the need for multiple observations per individual, and its relatively more difficult computation. Because every trait must be compared to every other for each sample being studied, comparing more than a few traits at a time can become quite ABSTRACT One of the main uses of dental morphological data is to study patterns of affinities among populations. Many different approaches to this purpose are available, each one having its own strengths and weaknesses. For this study, observations were made of the morphology of 614 African American and 327 European American dentitions (n = 941). Each of these samples was divided into three groups based on the time in which they lived. Affinities among the resulting six groups were estimated based on the Dentitions, Distance, and Difficulty: A Comparison of Two Statistical Techniques for Dental Morphological Data Heather Joy Hecht Edgar* Laboratory of Human Osteology, Maxwell Museum of Anthropology, University of New Mexico, Albuquerque, New Mexioc 87131 Address for correspondence: Heather J. H. Edgar, Laboratory of Human Osteology, Maxwell Museum of Anthropology, University of New Mexico, Albuquerque, NM 87131. Email: hjhedgar@unm.edu frequencies of dental morphological characteristics, by the use of both the Mean Measure of Divergence and a Pseudo-Mahalanobis’ D2. The results of these analyses are compared using a Procrustes transformation that rotates and scales coordinates derived from distances until achieving the best fit. The two statistics produce similar, although not identical results. The appropriate use and relative value of each approach is discussed. Dental Anthropology 2004;17(2):55-62. arduous, even with a computer. Additionally, the inclusion of a new sample for analysis requires the recalculation of all measures of affinity among groups, not simply the measures of affinity of the new sample with the original groups, as with the MMD. This study presents the results of a comparison of MMD and pseudo-D2 methods for determining biological affinity among several samples. The goals are to investigate whether the two types of analysis result in similar findings, and if not, to consider why. MATERIAL The data for this study comes from the dentitions of 941 African Americans and European Americans, analyzed as part of a larger study of the microevolution of African American dental morphology. Samples come from collections temporarily or permanently housed at the National Museum of Natural History, National Museum of Health and Medicine, Cleveland Museum of Natural History, University of Tennessee Health Sciences Center, Ohio State University, and Arizona State University. The samples were divided into six groups, based on ancestry and time period. The samples sizes and time periods are listed in Table 56 57 1. For this study, a maximum of 136 observations of 32 morphological characteristics was possible per dentition. Observation procedures were based on the Arizona State University dental anthropology system (Turner et al., 1991). No significant directional asymmetry of expression or sexual dimorphism was found, so rights and lefts were combined (with the greatest trait expression being represented), as were observations from males and females. Observations were then dichotomized with guidance from Haeussler et al. (1989), Irish (1993), Irish and Turner (1990), Scott and Turner (1997), and Turner (1987). All statistics were performed using the SAS statistical package (SAS Institute Inc., 1990). Associations between traits were determined using the likelihood ratio statistic. The list of traits that was used for each analysis can be found in Table 2. Traits used in the MMD analysis are independent from each other. To invert the matrix of correlations, the D2 analysis requires that most variables have some tetrachoric correlation with all other variables. Several variables were eliminated from D2 analyses because they were found to have little or no correlation with other variables, and thus the tetrachoric correlation matrix was singular. Different variable combinations were used in each analysis because of the requirements of each statistics; traits should be uncorrelated for the MMD and correlated for the D2. STATISTICAL METHODS Mean Measure of Divergence The MMD statistic was developed by C. A. B. Smith, and was first used to look at changes due to inbreeding in mice (Grewal, 1962; Berry et al., 1967). Berry and Berry (1967) first applied it to the study of biological affinities or distance in humans. The MMD estimates biological distance between samples based on the degree of phenetic similarity (Irish, 1997). The statistic requires an assumption of independence of traits. Like D2, it is useful if trait expression varies in a population, when frequencies are 5-95% (de Souza and Houghton, 1977). Some major benefits of its use are its ability to work with incomplete data and its applicability to samples as small as 10-20 observations. MMD is defined as: MMD=(∑(Θ 1 - Θ 2 )2 - (1/n 1 + 1/n 2 ))/c where Θ 1 and Θ 2 are the arc sin (sin-1) transformations of the observed frequencies in the two samples being compared, n 1 and n 2 are the sample sizes, and c is the number of characters employed (Freeman and Tukey, 1950). Pseudo-Mahalanobis’ D2 The Pseudo-Mahalanobis’ D2 is defined as the sum of squares of differences between corresponding mean values of two sets of measurements, weighted by the variance/covariance matrix (Burnaby, 1966): D =2 χ χ χ χ ik jk ik jk −( ) −( )∑' where χ ik is the mean of expression for sample i for k traits, and χ jk is the same for sample j. The middle term (∑) is the pooled covariance matrix between the k traits (Manly, 1994). In this study, the means of trait expressions are the threshold values corresponding to the trait frequencies in the samples (Falconer, 1981), and the middle term is a pooled matrix of tetrachoric correlations between the traits (Brown, 1977). These transformations account for correlations between characteristics being used (Konigsburg, 1990; Mizoguchi, 1977) and the threshold nature of dental morphological traits (Scott and Turner, 1997). Early Middle Late born born born circa circa circa 1650-1850 1825-1910 1920-1960 total African American 35 414 165 614 European American 33 139 155 327 total 68 553 320 941 Table 1. Sample compositions Max MMD Mand MMD Max D2 Mand D2 DIAS LI2SS UI2SS LI1SS UCSS LCDR UCSS LI2SS UI1LC LP3LC UI1LC LP4LC UI2DS LP4LC UI2TD LM2MT UI2IG LM1AF UCTD LM1PS UM3CA LM2GP UCDR LM2PS UCTD LM1DW UP3MD LM2C5 UCDR LM1MT UP4MD LM1C6 UP3MD LM2PS UM1MC UP4MD LM2C5 UM2MC UM2MC LM1C6 UM1HC UM1HC LM2C7 UM1C5 UM2C5 UM2C5 UM2CB UM1CB UM2CB Table 2. Dental characters used in each analysis H.J.H. EDGAR 56 57STATISTICS FOR DENTAL MORPHOLOGY Procrustes’ transformation The purpose of this statistic is to rotate and scale two sets of coordinates so as to achieve the best fit between them (Gower, 1971, 1975). For this study, the coordinates come from principal coordinates analysis of four distance matrices, and represent the first two axes of each matrix. The better the fit between two sets of coordinates, the smaller the summed deviations should be. Gower (1971) refers to the statistic as R2 (for residual), but it can also be found as S2 (for sum of squares) (Goodall, 1991) and M2 (for minimum)(Jackson, 1995). R2 is defined as: R2= ∑ ∆2(P i P i *), where P i and P i * represent the corresponding points in two different sets of coordinates. The R2 statistic is the sum of squared differences after rotation and scaling. The smaller the R2, the smaller the difference is between the two sets of coordinates. For this study, a small R2 will indicate good agreement between the MMD and D2 statistics. RESULTS Before discussing the direct comparison of statistical methods, an examination of the pictures presented by each analysis is in order. Due to the difficulty in performing pseudo-Mahalanobis’ D2 with a large quantity of traits, maxillary and mandibular traits were considered separately. Measures of affinity Results for MMD analyses based on maxillary and mandibular traits can be seen in Tables 3 and 4, respectively. The maxillary traits show a separation between African Americans (AA) and European Americans (EA) at all time periods. There is a closer relationship between early and middle EA than either to late EA. Early AA is different from all groups, with middle and late AA being most like late EA. Analysis of the mandibular traits emphasizes the split between EA and AA and minimizes other details. Results for the D2 analyses are summarized in Tables 5 (maxillary traits) and 6 (mandibular traits). The results for the maxillary traits seem to emphasize the time difference between groups rather than differences in ancestry. Late and middle AA and EA cluster most closely, with early AA and EA being very distant from each other and all other groups. The results based on the mandibular trait D2 are the most difficult to characterize. There is a large difference between early and middle AA, and a relatively small difference between middle and late AA. While the indication that change in the African American gene pool slowed down after the Civil War reflects known historical patterns of admixture (Davis, 1991), it does not explain the apparent similarity of early EA and middle AA, the smallest distance in the matrix. This information is graphically presented in Figure 1, which shows the principal coordinates of the relationships among the six groups resulting from MMD analyses, Late AA Late EA Middle AA Middle EA Early AA Early EA Late AA 0 0.113 0.074 0.443 0.244 0.402 Late EA 0.113 0 0.113 0.231 0.395 0.239 Middle AA 0.074 0.113 0 0.222 0.187 0.247 Middle EA 0.443 0.231 0.222 0 0.292 0 Early AA 0.244 0.395 0.187 0.292 0 0.218 Early EA 0.402 0.239 0.247 0.000 0.218 0 Table 3. MMD distances, maxillary traits Late AA Late EA Middle AA Middle EA Early AA Early EA Late AA 0 0.507 0.094 0.471 0.122 0.488 Late EA 0.507 0 0.525 0.119 0.601 0.148 Middle AA 0.094 0.525 0 0.401 0.122 0.374 Middle EA 0.471 0.119 0.401 0 0.449 0.000 Early AA 0.122 0.601 0.122 0.449 0 0.410 Early EA 0.488 0.148 0.374 0 0.410 0 Table 4. MMD distances, mandibular traits 58 59 Late AA Late EA Middle AA Middle EA Early AA Early EA Late AA 0 4.175 7.692 7.755 6.676 17.243 Late EA 4.175 0 4.472 4.563 10.015 10.769 Middle AA 7.692 4.472 0 3.184 7.982 8.698 Middle EA 7.755 4.563 3.184 0 8.303 6.499 Early AA 6.676 10.015 7.982 8.303 0 10.295 Early EA 17.243 10.763 8.698 6.499 10.295 0 Table 5. D2 distances, maxillary traits Late AA Late EA Middle AA Middle EA Early AA Early EA Late AA 0 1.473 8.630 3.593 8.725 8.302 Late EA 1.473 0 4.598 4.714 6.300 5.243 Middle AA 8.630 4.598 0 8.442 7.281 2.442 Middle EA 3.593 4.714 8.442 0 5.040 7.459 Early AA 8.725 6.300 7.281 5.040 0 8.800 Early EA 8.302 5.243 2.448 7.459 8.800 0 Table 6. D2 distances, mandibular traits �� ����������������������������� ���� ���� ���� � ��� ��� ��� ��� ���� ���� ���� ���� � ��� ��� ��� ��� ������ � � �� �� ������������� �������������� ������� ������� ������ ������ �������� �������� ������� ������� ������ ������ �������� �������� Fig. 1. Principal coordinates for MMD analyses. H.J.H. EDGAR 58 59STATISTICS FOR DENTAL MORPHOLOGY �� ���� �� ���� � ��� � ��� � ��� ���� �� ���� �� ���� � ��� � ��� � ��� ������ � � �� �� ������� ������������� ������������ � ������� ������������ �������� �������� ������� ������� ������ ������ �������� �������� Fig. 2. Principal coordinates for D2 analyses. �� ���� ���� ���� ���� � ��� ��� ��� ��� ���� ���� ���� ���� � ��� ��� ��� ��� ������ � � �� �� ������� ������� ������ �������� �������� ������ ������������� �������������� ������� ������� ������ ������ �������� �������� Fig. 3. MMD Principal coordinates after procrustes transformation. 60 61 Fig. 5. Principal coordinates of residuals. Fig. 4. D2 Principal coordinates after procrustes transformation. �� ���� ���� � ��� ��� ��� ��� ���� ���� ���� ���� ���� � ��� ��� ��� ��� ������ � � �� �� ������� ������� ������ �������� �������� ������ ������������� �������������� ������� ������� ������ ������ �������� �������� �� ���� ���� ���� ���� � ��� ��� ��� ��� �� ���� �� ���� � ��� � ��� � ������ � � �� �� ������� �������� ������� ������ H.J.H. EDGAR 60 61STATISTICS FOR DENTAL MORPHOLOGY and Figure 2, which shows the same relationships for D2 analyses. Procrustes analysis Figures 3 and 4 show the relationships between the six samples after rotation and scaling of the principal coordinates for MMD and D2, respectively. The coordinates for maxillary MMD results acting as a baseline for both tables. Each of the other groups has been redrawn to its best fit, meaning the one that yields the smallest residual. The residuals between all the groups are summarized in Table 7. There is no test of significance for R2, but it can be seen that all the values are relatively small except for between the D2 for maxillary and mandibular characteristics. It is possible to simplify this table by performing a principal coordinates analysis for this R2 matrix and display the relationships in the simplest geometric space. A graph of these coordinates shows relationship between the four methods of determining affinity. Figure 5 shows that the two MMD matrices are in nearly perfect agreement. The two D2 matrices are quite different from each other, but neither is more different from the MMD matrices than the other. It remains to be explained why the D2 matrices are so different from each other. One possible explanation is a lack of differences between the samples being studied in these particular traits. In fact, among the traits used for the mandibular D2 analysis, there is half the average difference in expression between groups as there is in the maxillary D2 and MMD, and one quarter as much difference as in mandibular MMD. CONCLUSIONS Overall, there is very good agreement between the biological distance matrices generated using MMD and pseudo-Mahalanobis’ D2 statistics. Both statistics have their place in the analysis of biological distance, especially when utilizing characteristics of dental morphology. As with all statistics, the MMD and D2 are limited by the data they analyze. If there is little difference between samples for the characteristics in question, the results will show small distances; if the differences are large for those particular characteristics, the distances will be large as well. A careful evaluation of the data should be made before attempting any measure of affinity. When there are many traits available for analysis and they have little inter-trait correlation, MMD is appropriate. When the data consist of a relatively few, correlated traits, a pseudo-Mahalanobis’ D2 is more accurately applied, as it makes no assumption about a lack of correlation between traits. In a large study, the use of both statistics may allow analysis of more of the collected data. If all things are equal and either statistic is applicable, MMD is simpler to use and more widely comparable. ACKNOWLEDGEMENTS This research was supported by a grant from the Graduate Student Alumni Research Award of Ohio State University and by National Science Foundation grant #0087400. Thanks go to Dr. Paul Sciulli and all the people and institutions that allowed access to their collections for this project. LITERATURE CITED Berry AC, Berry RJ. 1967. Epigenetic variation in the hu- man cranium. J Anat 101:361-379. Berry AC, Berry RJ, Ucko PJ. 1967. Genetical change in Ancient Egypt. 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PROTEST: a PROcrustean Random- ization TEST of community environment concor- dance. Ecoscience 2:297-303. Konigsberg LW. 1990. Analysis of prehistoric biological variation under a model of isolation by geographic and temporal distance. Hum Biol 62:49-70. Mahalanobis PC. 1936. On the generalized distance in statistics. Proc Nat Inst Sci India 12:49-55. Manly BFJ. 1994. Multivariate statistical methods: a primer. London: Chapman and Hall. Mizoguchi Y. 1977. Genetic variability in tooth crown characters: Analysis by the tetrachoric correlation method. Bull Nat Sci Mus, Series D 3:37-62. SAS Institute Inc. 1990. SAS/STAT Users guide, Version 6, 4th ed. Cary, NC: SAS Institute, Inc. Scott GR, Turner CG II. 1997. The Anthropology of modern human teeth: dental morphology and its variation in recent human populations. Cambridge: Cambridge University Press. Turner CG, Nichol CR, Scott GR. 1991. Scoring proce- dures for key morphological traits of the permanent dentition: the Arizona State University dental an- thropology system. In: M Kelley, Larsen CS, editors. Advances in dental anthropology. New York: Wiley- Liss, p 13-31. Turner CG. 1987. Late Pleistocene and Holocene popula- tion history of East Asia based on dental variation. Am J Phys Anthropol 73:305-321. H.J.H. EDGAR 13th International Symposium on Dental Morphology First Announcement The 13th International Symposium on Dental Morphology is taking place from Wednesday 24 to Saturday 27 August 2005, hosted by the University of Lódz, Poland. The conference web-site is at: http:// www.biol.uni.lodz.pl/antropolog/conference/index.html Files can be downloaded from the web-site for 1) Symposium Registration, 2) Presenter’s Information, and 3) Guideline for manuscript preparation with presenter’s instructions. Documents should be completed and return by the 28th February 2005. Scientific Program: The Scientific Programme will be held in the conference facilities at the University of Lódz and will follow the general pattern of previous meetings, with single oral and poster sessions. Abstracts: We welcome abstract submission, with the deadline of 28th February 2005. An abstract submission form and a presenter’s form is available from the organizers, with the choice of preferred option of poster or oral communication. Symposium Proceedings: The Symposium proceedings will consist of the presentations as short papers. Our proposed deadline for manuscripts will be 31th May 2005. See information on the web-site for submission formats. The Symposium and the accommodation are organized in the University Conference Centrum. The Centrum is set in the University District in very pleasant grounds, close to the city center (Piotrkowska street). We will be using all the conference facilities on site. The accommodation includes single and double rooms. Travel from this venue to our social events and return is included in the fee. Symposium Costs: Participant 130 Euro by 28.02.2005 180 Euro by 31.05.2005 Accompanying person 50 Euro by 28.02.2005 80 Euro by 31.05.2005 Please note: We regret that any cancellation after 01.07.2005 will not be refundable. Conference Fee covers: book of abstracts, the Symposium Proceedings, attendance to all sessions, refreshments during the meeting, conference facilities, the Welcome Reception, sightseeing of Lódz, grill party and the Gala Dinner. Much more information is available on the website. Editor’s Note: This information is abstracted from a detailed e-mail sent in early October. Be certain to refer to the web-site for specifics. http://www.biol.uni.lodz.pl/antropolog/conference/index.html http://www.biol.uni.lodz.pl/antropolog/conference/index.html