AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 315-320 (1984) Discriminant Function Analysis of the Central Portion of the Innominate JAMES V. TAYLOR AND ROBERT DIBENNARDO Metropolitan Forensic Anthropology Team, Department of Anthropology, Lehman College of GUNK Bronx, New York 10468 KEY WORDS Innominate Discriminant analysis, Sex and race assessment, ABSTRACT Terry Collection innominates of 260 American whites and blacks (65 males and 65 females of each race) were analyzed by discriminant function analysis for sex assessment, race known, and for simultaneous race and sex assessment. Measurements from the preservationally favored central portion of the innominate were chosen for their potential usefulness in forensic casework and to objectify long-recognized “formal” differences between the sexes in the greater sciatic notch. These included acetabular diameter, greater sciatic notch height, and position of greatest notch depth. Accuracy of sex prediction when race is known is roughly 90% in both the base sample of 260 and a test sample of 200 (50 additional individuals for each group). For simultaneous race and sex assessment, accuracy of prediction is roughtly 60% for each group in both samples. This represents about a 45% reduction in error over random assignment by race and sex. The highest accuracy of sex prediction achieved on several populations by inspectional, osteometric, and discriminant function analyses of the innominate is based largely upon use of the pubic area (American whites and blacks; Washburn, 1948; Bantus and Bushmen; Washburn, 1949; Eskimoes; Hanna and Washburn, 1953; American blacks; Thieme, 1957; Thieme and Schull, 1957; French whites; Howells, 1964; Japanese; Hanihara et al., 1964;American blacks and whites; Phenice, 1969; DiBennardo and Taylor, 1983). When this area is poorly preserved, its measurement and use in the ischium-pubis index, or the equally effective inspectional method of Phenice, are precluded. In these cases, some investigators have turned to the central portion of the innominate bone, especially the greater sciatic notch. The compelling reason for using the central portion of the innominate for sex assessment is its higher resistance to postmortem damage. Although several successful techniques have been developed for sex assessment on this basis (Hooton, 1930; Angel, 1946; Genoves, 1959; Kelly, 19791, no similar @ 1984 ALAN R. LISS, INC method has been used for race assessment. A method for simultaneously sexing and racing the preservationally favored central portion of the innominate would, therefore, provide a useful tool for the assessment of human remains in forensic and archaeological studies. We present such a method here for application to skeletal materials that can be assumed to be either American blacks or whites, on a priori grounds. It derives from the above considerations and combines the dimensions in Kelly’s (1979) sciatic notch/ acetabular diameter index and Genoves’ (1959)greater sciatic notch index. By combining them in a multiple discriminant function analysis, however, we hoped to capitalize on the population differences in notch dimensions demonstrated by Letterman (1941) and produce functions that optimally separate the sexes and races simultaneously. As we demonstrate below, this four group separation permits individual identification at a n acceptable level of accuracy, considering the Received January 30, 1984; revised March 19, 1984; accepted March 20, 1984. 3 16 J.V. TAYLOR AND R. DIBENNARDO limited evidence provided by the body of the innominate. MATERIALS AND METHODS Our base sample consists of the innominates of 260 North American whites and blacks: 65 males and 65 females of each race. They were obtained from the Terry Collection a t the Smithsonian Institution and are of known age a t death, sex, and race. In addition, a further sample was measured to test the reliability of statistics generated by the base sample. It was also drawn from the Terry Collection (50 individuals for each group). In keeping with our stated objective, the three measurements chosen are 1) notch height-the height of the greater sciatic notch measured with sliding calipers between the points of Lazorthes: the tubercle of Bouisson (attachment of the piriformis muscle) and the tip of the ischial spine (Fig. lA,B); 2) notch position-the distance from the tip of the ischial spine to the intersection of the line of notch height and the perpendicular dropped from it to the deepest point in the notch (Fig. 1 B,C), measured with coordinate calipers; 3) acetabular diameter-diameter of the acetabular rim measured with sliding caliper parallel to the long axis of the ischiai body (Fig. 1E,F). Like Kelly (1979) and Genoves (19591, we recognize the difficulties in using the fragile ischial spine as a n osteometric point. However, the alternative use of the poorly defined base of the spine consistently posed a greater operational problem than did extrapolating the position of the tip of the spine, where this was necessary. The statistical analysis was carried out in two parts. First, we elaborated on Kelly’s protocol by adding notch position to his two variables and using the discriminant function technique, instead of a n index, to discriminate sex when race is known. This adds a “formal” component to Kelly’s simple linear approach and permits the calculation of posterior probabilities of group membership for individual cases (e.g., in forensic case work). Second, we used the same three variables in a multiple discriminant function analysis for simultaneously discriminating race and sex. The DISCRIMINANT procedure of SPSS, METHOD=DLRECT, was used in both analyses (Nie et al., 1975). The coefficients generated by both analyses were then used to classify the test samples. RESULTS Table I provides the simple descriptive statistics for our data. Tables 2 and 3 give the results for our analysis of sexing when race is known, and Tables 4 and 5 give results for the simultaneous discrimination of sex and race. Fig. 1. Illustration of the innominate showing the measurements used in the study. See text for details. Sex, race known The structure coefficients in both whites (Table 2) and blacks (Table 3) indicate that males tend to have larger acetabular diameters, smaller notch heights, and the position of maximum depth situated more superiorly than females. On the original samples, accuracy of prediction averages 94.6% for whites (Table 2) and 90.8% for blacks (Table 3). On the test samples, accuracy declines to a n average of 89% for whites, but remains about 91% for blacks (Tables 2 and 3). Overall (i.e., weighted across samples), whites average 92.1% and blacks 90.9%. In short, both races can be correctly classified about 91% of the time. 317 DISCRIMINANT FUNCTION ANALYSIS OF THE INNOMINATE TABLE 1. Simple descriptive statistics for the base sample analyzed in the text. N each erouu and all dimensions are i n millimeters Variables = 65 i n White males White females Black males Black females SD 56.5 3.2 50.3 2.6 55.1 2.8 50.0 2.6 Notch height Mean SD 48.6 4.6 51.7 5.7 44.7 5.2 49.3 5.3 Notch position Mean 39.5 4.4 33.7 4.9 37.0 5.2 33.5 4.7 Acetabular diameter Mean SD TABLE 2. Discriminant function for sex assessment when race is known to be white Variable Raw coefficients Acetabular diameter 0.24674 Notch height - Notch position Constant Group White males White females Standardized coefficients 0.15865 1.15406 ~ - Structure coefficients 0.71313 0.89 0.81941 --0.35 0.71512 0.64 10.85599 Group means - 1.46 1.46 96 Accuracy base sample 95.4 93.8 % Accuracy test sample 88.0 90.0 All coefficients are for the base sample: raw coefficients are for computing discriminant scores from raw measurements, standardized coefficients are for computing scores from standardized data, and structure coefficients are correlations between discriminant scores and original variables, over the total sample. Discriminant scores are standardized within groups so that the pooled within-groups dispersion matrix is a n identity matrix. TABLE 3. Discriminant function for sex assessment when race is known to be black Variable Raw coefficients Acetabular diameter Notch height 0.21627 - Notch position Constant Group Black males Black females Standardized coefficients 0.19787 0.18268 - 0.58603 - 1.04304 0.90882 Structure coefficients 0.86 -0.50 0.42 8.50729 Group means 1.34 - 1.34 % Accuracy base sample 90.8 90.8 % Accuracy test sample 92.0 90.0 All coefiicients are for the base sample: raw coefficients are for computing discriminant scores from raw measurements, standardized Coefficients are for computing scores from standardized data, and structure coefficients are correlations between discriminant scores and original variables, over the total sample. Discriminant scores are standardized within groups so that the pooled within-groups dispersion matrix is a n identity matrix. 318 J.V. TAYLOR AND R. DIBENNARDO TABLE 4. Discriminant functions for simultaneous race and sex assessment o f whites and blacks Variable Function 1 Function 2 Function 3 Raw coefficients Acetabular diameter Notch height Notch position Constant 0.23417 0.17650 0.16645 - 9.81200 0.18120 0.18891 - 0.06092 - 16.57900 - 0.22282 0.04641 0.23183 5.72817 0.65602 0.50763 0.98578 - 0.29310 - 0.62423 0.24217 1.11543 - 0.25 0.38 0.70 Standardized coefficient Acetabular diameter Notch weight Notch position - 0.92102 Structure coefficient Acetabular diameter Notch height Notch position - 0.42 Group means on functions White male group White female group Black male group Black female group - 0.80085 0.87 - 0.53 0.43 0.82 0.48 1.41 1.55 1.37 1.24 0.43 0.24 - 0.41 - 0.26 0.04 0.06 - 0.04 0.07 - All coefficients are for the base sample described in the text. Structure coefficients are correlations between discriminant scores and original variables, over the total sample. Discriminant scores are standardized within group so that the pooled within-groups dispersion matrix is a n identity matrix. TABLE 5. Accuracy ofsimultaneous race and sex assessment for the discriminant function analysis giuen in Table 4 Group White males White females Black males Black females % Accuracy base samule % Accuracy 63.1 58.5 64.6 58.5 62.0 58.0 56.0 52.0 test samule Race and sex, simultaneously Only the first two canonical discriminant functions are statistically significant at the 0.05 level (although the third is retained for classificatory purposes). From the group means (Table 4) it is clear that function 1 separates the sexes (males have positive scores, females negative ones). Function 2, on the other hand, separates the races (positive for whites, negative for blacks). The structure coeficients (Table 4). suggest that the bases for these separations differ. For the sex function, males have larger acetabular diameters, smaller notch heights, and more superiorly situated positions of maximum depth. This is consistent with our analysis of sex with race known. The race function, however, shows all positive coefficients, suggesting that racial separation is based primarily on size, with whites generally larger than blacks. For the original sample, accuracy of prediction over all four groups averages 61.2%. On the test sample, this declines to 57%. Random assignment of race and sex would produce a n expected accuracy of only 25%. To express the improvement achieved by these discriminant functions, we used a proportional reduction in error statistic, tau, given in Klecka (1980, pp. 50-51). For our original sample, tau = 48.2%, and for the test sample, 42.7%. The functions, therefore, represent a 48.2% and 42.7% reduction in error over chance, for these two samples, respectively. DISCUSSION In the inspectional approach to sex assesment, the form of the greater sciatic notch is described as usually wide and shallower in the female (Hooton, 1930; Angel, 1946; Washburn, 1948). But though HrdliEka considered notch form as the single most effective sexing criterion of the pelvis (in Stewart, 1952),only Hooton and Angel have employed it in a purely inspectional approach to sexing the innominate bone (see Genovks, 1959, p. 49). Both of these investigators use notch form with either the subpubic angle (Hooton, 1930) or subpubic angle and the degree of excavation of the preauricular sulcus (Angel, 1946). As noted above, use of the subpubic angle may be compromised by the circumstances of preservation. However, when the notch form DISCRIMINANT FUNCTION ANALYSIS OF THE INNOMINATE has been used alone, the stated accwacy of sex prediction ranges from 79 to 81% (Hooton, 1930; Genoves, 1959). These are tolerably good results, roughly comparable to those achieved with the skull (70 to 80% according to Stewart, cited in Kelly, 1979). Admittedly, it is difficult to assess the validity of such estimates because of the subjective component in inspectional assessments. Stewart (1954) has summarized the strength of the inspectional method in his remark that it may be rather a waste of effort to measure simply to verify what can be so quickly seen and integrated into a total morphological pattern by the trained eye. Nonetheless, some objective criteria of sex assessment are clearly desirable for court testimony in forensic cases, as Stewart himself acknowledges (1979). Such objective quantitative assessments of predictive accuracy have been customarily given in metric approaches to sexing and racing the greater sciatic notch. The use of a metric approach does not, of course, always insure success. Letterman (19411, for example, could not race and sex American whites and blacks using three measurements of the notch. He found that the degree of overlap, especially in notch height (cf. Washburn, 1949; and Genoves, 19591, made classification of race and sex uncertain in a high percentage of cases. This is not surprising, however, since his simple measure by measure approach to morphology does not, in fact, provide a n analysis of “form.” What are needed are indices or other compound-derived measures. For example, Genoves’ (1959) greater sciatic notch index uses two of Letterman’s measurements to demonstrate the formal fact that the perpendicular from the maximum height line to the deepest point in the notch divides the height into roughly equal chords in females; in males the upper chord is typically smaller. His accuracy of sex prediction with this index is 79.9% (Bruxelles and Saint Bride series combined). Kelly improved on this accuracy of prediction by adding the acetabular diameter to the sciatic notch height-ie, his sciatic notchlacetabular index; however, this index omits any formal description of the sciatic notch. Our own results, whether by the functions for sex alone or for race and sex together, confirm and objectify the observations of HrdliEka concerning notch form and ratify the metrical conclusions of Genoves. Namely, in males the notch has a n inverted “J” shape, 319 and in females, a n open “C” shape. Addition of the acetabular diameter sharpens the contrast between the sexes by emphasizing the greater acetabular size in males in conjunction with their smaller notch heights. This corroborates Kelly’s findings with regard to his index. As compared with Kelly (1979), our accuracy of prediction for the sexing functions is slightly higher, a t least as regards our base sample (92.7 vs. 90.0%). This might be accounted for by the additional variable in our study (notch position) andor the discriminant function method. However, the difference in accuracy of prediction between our base sample and test series (for whites, a 5% difference) indicates that such percentages must be accepted with caution. Rather than a rigid figure, they suggest a domain of accuracy, which in this case is quite high. Finally, it would appear from the structure coefficients for the racing function (Table 4, function 2), that “size,” especially with regard to notch height, is the primary discriminator between the races. Thus, whites are generally larger in this central portion of the innominate, as they also are in measurements of the entire innominate (see Di Bennardo and Taylor, 1983). In conclusion, our discriminant functions using notch height, notch depth position, and acetabular diameter express long, recognized formal distinctions between the sexes and allow high accuracy of sex prediction. Additionally, they express the size differences between blacks and whites we have reported elsewhere and provide a substantial improvement in race prediction over chance assignment-especially considering the limited evidence given by the central portion of the innominate. ACKNOWLEDGMENTS We thank Dr. J.L. Angel and the support staff of the NMNH at the Smithsonian Institution for their valuable assistance. We also thank Dean F.C. Shaw for his continuing support of the Metropolitan Forensic Anthropology Team at Lehman College. This study was made possible by PSC-CUNY grants #13456 and #6-62088. LITERATURE CITED Angel, JL (1946) Skeletal change in ancient Greece. Am. J. Phys. Anthropol. I3:489. DiBennardo, R, and Taylor, JV (1983)Multiple discriminant function analysis of sex and race in the postcranial skeleton. Am. J. Phys. Anthropol. 61.305-314. 320 J.V. TAYLOR AND R. DIBENNARDO Genoves, S (1959) L’estimation des differences sexuelles dans 1’0s coxal: Differences metriques et differences morphologiques. Bull. Mem. Soc. Anthropol. (Paris) IO(X):3-95. Hanihara, K, Kimura, K, and Miaamidate, T (19641 The sexing of Japanese skeletons by means of discriminant functions. Nihon Hogaku Zassi (Japanese Journal of Legal Medicine) 18t107-114. Hanna, RE, and Washburn, SL (1953) The determination of sex of skeletons, as illustrated by a study of the Eskimo pelvis. Hum. Biol. 2 5 2 - 2 7 . Hooton, EA (1930) The Indians of Pecos Pueblo. A study of Their Skeletal Remains. New Haven: Yale University Press. Howells, WW (1964) Determination du sex du basin par fonction discriminante: Etude de materiel du Docteur Gaillard. Bull. Mem. Soc. Anthropol. Paris 7(11):95105. Kelly, MA (1979) Parturition and pelvic changes. Am. J. Phys. Anthropol. 51:541-545. Klecka, WR (1980) Discriminant Analysis. Beverly Hills: Sage Publications. Letterman, G (1941) The greater sciatic notch in Ameri- can whites and Negroes. Am. J. Phys. Anthropol. 28:99-116. Nie, NN, Hull, CH, Jenkins, JG, Steinbrenner, K, and Bent, DH (1975). Statistical Package for the Social Sciences. New York McGraw-Hill. Phenice, TW (1969) A newly developed method of sexing the 0s pubis. Am. J . Phys. Anthropol. 30:297-301. Stewart, TD (ed.) (1952) HrdliEka’s Practical Anthropometry. Fourth Edition. Philadelphia: Wistar. Stewart, TD (1954) Sex determination of the skeleton by guess and by measurement. Am. J. Phys Anthropol. 12t385-392. Stewart, TD (1979) Essentials of Forensic Anthropology. Springfield C.C. Thomas. Thieme, F P (1957) Sex in Negro skeletons. J. Forenslc Med. 432-81. Thieme, FP, and &hull, WJ (1957) Sex determination of the skeleton. Hum. Biol. 29:242-273. Washburn, SL (1946) Sex differences in the pelvic bone. Am. J. Phys. Anthropol. 6:199-208. Washburn, SL (1949) Sex differences in the pubic bone of Bantu and Bushman. Am. J. Phys. 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