# On the assumption of equality of variance-covariance matrices in the sex and racial diagnosis of human skulls.

код для вставкиСкачатьAMERICAN JOURNAL OF PHYSICAL ANTWROF’OLOGY 60:347-367(1983) On the Assumption of Equality of Variance-covariance Matrices in the Sex and Racial Diagnosis of Human Skulls H.T. UYTTERSCHAUT AND F.W. WILMINK Department ofAnatomy and Embryology, 9713 EZ Groningen, The Netherlands KEY WORDS Equality of covariances, Likelihood ratio test, Sex diagnosis, Racial diagnosis, Crania ABSTRACT Many papers have been devoted to the assumption of equality of variance-covariance matrices (X,) with respect to the use of discriminant analysis. Most of them concentrate on the “effect” of inequality on the results, in particular on discriminant functions. In the present paper, the assumption of equality of covariances itself was investigated for measures on human skulls. Data for both sexes were compared, as well as data on several racial groups. A likelihood ratio test was used.It was concluded that the equality of X, between the sexes and among subraces was not questioned, whereas the results warrant some caution as to the equality of C, among the main racial groups. Many papers have been devoted to the problem of the equality of variance-covariance matrices ,X (Throughout this paper S, (g= 1,2,. . .I will indicate the variance-covariance matrix of sample g, and X, the variancecovariance matrix of the population from which sample g was drawn). The discussion on whether the C, are equal for variables from skeletal material started with the application of linear discriminant analysis (DA) to this kind of material. The pooling of S, of different groups in DA makes sense only if the C, are equal (Gower, 1972). When they are unequal, the Mahalanobis D2 can be calculated, but problems may arise in the interpretation of the results and the classification of individuals. Most work has concentrated on the “effect” of inequality on statistical results, in particular on discriminant functions. One group of authors suggest that heterogeneity of the X, greatly affects statistical results (Burnaby, 1966;Mardia, 1971;Lachenbruch and Goldstein, 1979; Olson, 1974; Reyment, 1962, 1966; Corruccini, 1975). Campbell, B.G. (1963)is of the opinion that the inequality of the X will affect the results only mildly. A m r a n g to Gilbert (1969)and Marks and Dunn (1974),in the case of X2 = dZ1, the effect of unequal Xg:s on the linear discriminant function OF)is very small, if d varies 0 1983 ALAN R.LISS. INC. between 0.1 and 10.0 and if the discriminatory value 0 between the two populations is sufficiently large (>2, say). The Mahalanobia can still be used as distance statistic when the X, are unwual(Chaddha and MarCUB, 1968). Cooper (1965)has shown that, subject to mild restrictions, the Mahalanobis statistic is robust, almost to the extent of being distribution-free,when used as a DF to assign an unknown observation to one of several parent populations. The main requirement is that the determinants of the X ’s of the distributions of the parent populakons &odd be equal. In this paper, we are more interested in whether the X, can be considered equal (one of the assumptions underlying linear DA) rather than in the consequences or the effect of inequality on the results, as the Fisher DF is often used without regard ta possible inequality of the X, (Marks and Dunn, 1974). Most multivariate users (in anthropology) do not check (or do so only superficially) for homogeneity of covariance matrices. Albrecht (19781,however, went to considerable effort to evaluate homogeneity among 35 populations of macaques, although he used less exacting procedures. Jolicoeur and Mosimann (19601,Reyment (19621,Corruccini Received February 16,1982; accepted September 10,1982 348 H.T. UYlTEXSCHAUT AND F. W. WILMINK k (19751,and Campbell, N.A. (1979)feel that the differences in covariance structure reN = ng; g = l flectreal biological or physical differences in the underlying variability of the populations: and k “the covariance-heterogeneity can be of taxonomic value” (Corruccini, 1975). s = ~ -g C 1 ng s,. = l It is the aim of this paper to investigate the assumption of equality of X for human skulls in relation to the utility oD !A in sex and/or Nagarsenker (1978)derived expressionsfor racial diagnosis. In the first part of this study, the power function of M, and Perlman (1980) the comparison of C was based on random demonstrated the unbias of the test against sets of variables. In t%e second part, the com- all alternatives. Approximations to the disparison was based on specific variable com- tribution of (functions 00M have been sumbinations for discriminating sex and race, marized by Krishnaiah (1978).We have used respectively. The term “race” will be used the X2, approximation described by Box throughout this study to designate a geo- (1949).Box has also offered a FV, approximation for the multivariate case which he graphical population, not a racial type. considered to be rather better, and which MATERIALS AND METHODS maintains a significance level close to the The material consisted of 28 samples of nominal value when at least the ng are not skulls from different populations which were too small relative to k or the number of varimeasured by W.W. Howells. The first 17 pop- ables p. In the present study, w was always ulations were previously described (Howells, large with the well-knownresult that, within 1973). These data were sent to us on tape rounding error, the computed values of v FV, along with the data of the remaining 11 pop- equaled the corresponding values xZv.Thereulations. The 28 samples included both sexes, fore, one would expect the ?” approximation except for the Anyang and the Philippines to be good enough for our purpose. Layard which included only males. The whole sam- (19741,however, has warned that deviations ple consisted of 1332 males and 1139females. from normality, especially the presence of We give here an enumeration of the different heavy tails, will increase the actual type I samples: (males, females): Norse (55,55), Za- error of the test, while the power is not serilavar (54,451,Berg (56,631,Egypt (58,53), ously affected. Since our data did not show Teita (34,49),Dogon (48,531,Zulu (55,461, such tailed distributions, we have adopted a Bushman (41,491,S-Australia (52,49),Tas- nominal significance level of 0.05 one tailed. mania (45,42), Tolai (55,55), Mokapu (51,49), S i g d c a n t results at 0.01 and 0.10have been Buriat (54,55),Eskimo (54,541,Peru (55,551, marked for the sake of comparison with the Andaman (26,281, Arikara (42,271, Ainu 0.05 level. The data were processed on a Cy(48,381, N-Japan (55,32), S-Japan (50,411, ber 74/36,using our own computer program. Hainan (45,38), Anyang (42,0),Atayal(29,18), Phillippines (50,0),Guam (30,27),Santa Cruz RESULTS (42,38),Easter Island (49,371,and Moriori In this study, the same H, was tested sev(57,531. For testing the equality of X , we refer to eral times, independent of each other. For Anderson (1958), Kullback f1959, 19671, each individual test we adopted a nominal Dempster (1964), Chaddha and Marcus significance level a. Under H,, this series of (1968),Layard (19741,Olson (1974),van Vark tests can be considered as a series of Ber(1970),and Campbell, N.A.(1979).For more nouilli experiments with a as the probability references see Corruccini (1975).We used the of success. Letting n be the number of tests, like-lihood ratio test (Wilks, 1932)modified we have considered two possibilities for the by Bartlett (1937)for testing Ho : XI = CZ = overall test of H,,: A: reject H, if at least one test is significant . . .Xk. The &t statistic used was: k (van Vark, 19701. B: let r be the observed number of signifiM = N In I S ( - g = l n,ln lSgl, cant tests. Reject H, if the probability p of observing at least r significant tests (out of where n) is less than a. k = number of samples Method B maintains the same significance ng+ 1 =number of observations in sample g X c 349 EQUALITY COVARIANCE MATRICES DIAGNOSISSKULLS level a for the individual tests and the overall test, whereas method A will have very small values of a for individual tests when n increases. For this reason, method B probably has the greater power. With H,: p = 0, where p is the expectation of a normal distribution with variance = 1,this is indeed the case as can be seen from Table 1.Therefore, we have adopted method B. Also, from power considerations it was felt that cn = 0.01 would be too conservative, and we have adopted the 5% level throughout the paper. Although we are interested in the X, of combinations of variables that discriminate sex or race well, it is just not feasible to investigate all variable combinations which have ever been used or described by physical anthropologists. For this reason, we will first test H, with several random sets (section 1). In section 2, we will discuss H, with variable combinationsthat are purposively chosen because of their sex- or race-discriminating power. subrace of the Europidi (Negrids, Mongolids) is equal to the E, of the females of the same subrace. We have computed the test statistic M for six Merent populations (Norse, Zalavar, Teita, Dogon, Hainan, S-Japan). So for each main race we have 18 tests. Results are presented in Tables 2a and 2b. H, is accepted for the Europids and the Mongolids, but rejected for the Negrids. This throws some doubt on the assumption of equal covariance matrices for Negrids. Equality of Cg among differentraces First, we tested the equality of C, among three main races: Europids-Negrids-Mongolids. The null hypothesis is defined as: the C, of a population (only males) from some main race is equal to the C, of a population (only males) from another main race. The same holds for the females. Two samples were used. The first consisted of Norse, Teita, and Hainan; the second of Zalavar, Dogon, and SJapan. Tables 3a and 3b give the test results of the equality of Cg among the three main Equality of Zg based on random sets We tested H, with nine randomly chosen races. In addition, we compared the C i s of all 28 sets (sets 1-91. Sets 1-3 each include five variables; sets 4-6 each include 10 variables, populations of males and of 26 populations of and sets 7-9 each 15 variables. These sizes females (see Materials and Methods). The rewere thought to reflect actual practice. Tests sults are represented in Tables 4a and 4b. From Tables 3b and 4b, it can be seen that with these random sets were considered to be the number of significant tests is clearly beindependent. yond what could be expected by chance alone under H,, except for the females in one case. Equality of Xg among the sexes of one This suggests that the researcher who wishes population to compare populations from different main The €&, is defined as follows: for any given races should do well to first test for equality set of variables the Xg of the males of one of covariance matrices. TABLE 1: Cakulation o f p w e r acwding to method A and ocwrding to method B' p = a = 0.01 a = 0.05 a = 0.10 'For p = n A 9 18 36 78 9 18 36 78 9 18 36 78 232 232 232 232 1127 1129 1130 1130 2170 2177 2181 2183 0.5 (0.5)2.0 (1o000). = 0'5 B 310 479 649 1162 1935 2826 4107 6540 3616 5283 7010 9128 A 534 535 535 2421 2435 2441 2445 4300 4341 4362 4373 1.0 p = 1.5 p = 2.0 B A B A B 1017 1922 3190 6002 5284 7664 9406 9985 7754 9518 9965 10000 1200 1204 1206 1207 4705 4759 4789 4802 7175 7288 7346 7378 3068 5456 8098 9835 8778 9881 9999 10000 9768 9997 10000 10000 2565 2582 2591 2596 7589 7716 7782 7818 9341 9459 9517 9548 6753 9105 9953 10000 9910 loo00 10000 10000 9994 10000 10000 10000 350 H.T. WTIERSCHAUT AND F.W.WILMINK TABLE 2a 2 values for the equality of Zg between the sexes, fir sir different populations, based on nine mndom sets Variable combination set 1 set2 set3 set4 set5 set6 set7 set8 set9 (df= 15) 15) 15) 55) 55) 55) (df= 120) (df= 120) (df= 120) (df= (df= (df= (df= (df= Norse 550,559 Zalavar 540,459 18.29 18.13 9.47 81.00** 58.90 68.21 144.31* 115.55 124.93 20.82 10.29 15.57 59.01 53.29 67.37 127.41 121.87 123.43 = significant at 0.10 level, onetailed; Teita 34O,499 13.67 9.08 23.20* 74.45* * 42.78 71.93* 110.58 117.79 140.42* Dogon 480,53Q Hainan 450,389 9.59 19.73 27.02** 79.40** 65.85 82.09** 103.21 168.73*** 143.10* 15.93 18.68 10.26 55.61 45.85 84.63*** 113.25 122.54 121.87 Mapan 500,410 6.98 8.65 14.66 65.52 43.64 69.57* 117.61 142.94* 143.77* ** = significant at 0.05 level, one-tailed; *** = significantat 0.01 level, one-tailed. TABLE 2b. Number of significant tests for the equality of X, between the sexes, based on nine random Sets' Europida Negrida Mongolids c0.01 c0.05 0 l(1655) l(1655) 1(6028)* 5(<0.01) l(6028) n ~0.10 26497) 18 9 ( ~ 0 . 0 1 ) 18 4(0982) 18 'Calculated from Table 2s. *In parentheses: probability p of observing at least r significant teats (out of n), multiplied by 1OOOO. TABLE 3 a Variable combination set1 (df= 30) set2 -~~ (df = 30) st3 set4 set5 set 6 set 7 set 8 set9 (df (df (df (df (df (df ~ iZ= Soi = 110) = 110) = 110) = 240) = 240) = 240) 2 values for the equality of Xfl among the three main races, based on nine random sets' Norse-Teita-Hainan 99 (55.34.45) (55,49,38) Zalavar-Dogon-SJapan 0.0 99 (54,48,50) (55,49,38) 00 34.01*** 24.48 .. ~33.54 154.00*** 98.89 126.50 278.90** 279.29** 284.54** 35.52 39.83 26.64 122.47 117.74 159.33*** 277.14** 317.90*** 291.80** 19.79 25.15 -~ ~. 19.73 128.37 86.75 116.08 247.46 265.53 259.65 'As representative populations were uaed: Norse-Teita-Hainanand Zalavar-Dogon-S-Japan. 28.53 25.50 46.63** 128.16 86.04 132.49* 286.53** 274.99* 315.28*** * = significant at 0.10 level, one-tailed; ** = significantat 0.05 level, one-tailed, *** = significantat 0.01 level, one-tailed. TABLE 3b. Number of significant tests for the equality of 4among the three main races,based on nine random sets' co.01 ow 99 3(0007) l(1655) 4(<0.01) c 0.05 8 ( c 0.01)* 3 (0581) 11( ~ 0 . 0 1 ) CO.10 n 8 ( c 0.01) 18 18 5(0282) 13 ( ~ 0 . 0 1 ) 36 'Calculated from Table 3b. *=inparenthesea: probability p ofobsening at least r significant testa (out of n), multiplied by 1oooO. TABLE 4a ?values for the equality of Eg between 28 populations (males), and between 26populations (females), based on nine random sets Variable combination set 1 set2 set3 set 4 set 5 set 6 set 7 set 8 set 9 Males (28 groups) (df= 405) (df= 405) (df= 405) (df= 1485) (df=1485) (df=1485) (df=3240) (df=3240) (df=3240) Females (26 groups) 498*** 532*** 457** 1813*** 1768*** 1824** * 3824*** 3891*** 3943*** set 1 set2 set3 set4 set 5 set 6 set 7 set8 set 9 (df= 375) (df= 375) (df= 375) (df=1375) (df=1375) (df= 1375) (df=3000) (df=3000) (df=3000) 461*** 436** 435** 1662** * 1529** * 1548*** 3481* * * 3479*** 3678*** *=significant at 0.10 level, one-tailed; **=significant at 0.06 level, one-tailed; ***=significant at 0.01 level, one-tailed. TABLE 4b. Number of significant tests fir the equality of L' between 28 populrrtions (males), and between 26 populationa (females).respectively, based on nine random tests' ww Q9 c 0.01 c 0.05 c0.10 n 8(<0.01)* 7(<0.01) 1McO.01) 9(<0.01) 9(<0.01) 18(<0.01) 9(<0.01) 9(<0.01) 18(<0.01) 9 9 la 'Calculated from Table 4a. =in parenprobability p of observing at lead r sienifieant tests(out of n), multiplied by 1oooO. TABLE 5a 2 values for the equality of C, betweeen two groups fhm Europe (Norse and Zalavar), two Negrid groups (Teita and Dogonl, and two Mongolid groups (Hainun and SJapan), based on nine random sets Variable combination set 1 set2 set3 set4 set5 set6 set7 set8 set9 (df= 15) (df= 15) (df= 15) (df= 55) (df= 55) (df= 55) (df=120) (df=120) (df=120) Norse-Zalavar 55a,54w 559,459 16.22 22.28 10.13 63.50 54.14 51.82 122.24 140.46 122.63 22.92* 11.86 19.45 76.28+* 55.92 58.92 126.91 124.03 149.58** Teita-Dogon 34w,48W 499,539 10.13 4.72 17.20 73.77** 49.93 63.20 127.06 136.72 149.56** Hainan-SJapan 45w,50w 380,419 13.40 14.62 19.78 61.21 42.93 56.64 101.03 119.02 132.99 12.20 21.46 18.37 95.63* 55.45 67.70 144.19* 144.15' 125.39 16.61 13.91 16.41 59.36 42.08 62.46 138.40 131.72 135.60 *=significant at 0.10 level, one-tailed; **=significant at 0.06 level, onetailed; ***=significant at 0.01 level, one-tailed. TABU 56. Number of significant tests fir the equality of C, among subrace6 b a e d on nine mndom sets1 Europids(w + Q ) Negrids(w + Q ) Mongolids (a+ Q co.01 <0.05 co.10 n 0 0 l(1655) 2(2265)* 2(2266) l(6028) 32662) 26497) 32662) 18 la 18 'Calculated from Table Sa. =in parentheses: probability p of obaewing at least r significantteeta (out of n). multiplied by 1oooO. H.T. W?TERSCHAUT AND F.W.WILMINK 352 I 1 I I I Equality ofX, based on specific sex- or racediscriminating variable combinations Equality of X, among the sexes of one population: We have computed the test statistic M for 26 different populations (i.e., 26 populations from Howells which included as well male as female individuals) and for three different variable cornbinations (S1, S,, S3). The measures which belong to each combination have been listed in Table 6. A more detailed description of these measures is given by Howells (1973).From the literature (Giles and Elliot, 1962, 1963; Giles, 1970; Larnach and Freedman, 1964; Kajanoja, 1966; Birkby, 1966; Howells, 1966, 1973; Ghosh, 1967; Drummond, 1968; Boulinier, 1968; Schwidetzky, 1969;Ducros et al., 1973; Ferembach et al., 1979;Calcagno, 1980)and in our experience we have found that combinations S1 and Sz form in general good sex-discriminating functions. Especially for combination S1, we found that the sex-discriminatory value was high for nearly all racial groups. We shall discuss this topic elsewhere (uytterschaut, 1983).Combination S3 was constructed by performing a stepwise DA on 1220 males versus 1040females, all of them deriving from the Howells' series. Test results are presented in Table 7a.Unlike the random sets, the specific sex variable combinations S1-Ss are mutually highly correlated (having some measures in common). Therefore, in Table 7b each function was treated separately in the calculation of the number of significant tests. The null hypothesis was accepted. Equality of C, among different races: First, we tested the equality of X among the three main races: Europidsdegrids-Mongolids. The same samples as in a previous section were used: Norse-Teita-Hainanand ZalavarDogon-S-Japan. The variable combinations R1-& were chosen as race discriminating combinations. Combination R1 was derived from a survey of the literature (Giles and Elliot, 1962; Schulter, 1976;Sanghvi, 1953;Pietrusewsky, 1974; Drummond, 1968; Saksena, 1974; Schwidetzky, 1971, 1975; Howells, 1966, 1969,1972,1973;Crichton, 1966;Rightmire, 1970; Goldstein, 1979; Szathmary, 1979; Casey et al., 1979). Rz had a high race-discriminating power, according to our experience. R3 and R5 are combinations with high F-ratios for males and females, respectively, concomputed by Howells (1973).R4 and sist of the first 12 variables of the DF found 353 EQUALITY COVARIANCE MATRICES DIAGNOSIS SKULLS TABLE 7a f values for the eguality of Cg between the sexen for 26 different populations, based on S1,S , and S, Variable combination s1(df= 10) S, (df= 15) 53 (df=78) s1(df= 10) S, (df=15) s3 (df= 78) s1(df=10) S, (df= 15) s3 (df=78) s, (df=lO) (af= 15) s3(df=78) 6 Norse 9.85 14.17 74.48 Bushmen 4.26 19.71 74.97 Peru 10.62 12.15 78.39 Atayal 14.71 28.77** 76.23 Zalavar 8.09 9.95 83.36 SAustralia 13.33 23.87** 93.95 Andaman 13.12 20.39 86.37 Berg 8.75 13.15 89.79 Tasmania 9.54 8.13 103.05** Arikara 9.76 24.15* 95.62* Guam SantaCruz 10.97 34.35 103.51** 12.39 26.07** 77.45 *=significant at 0.10 level, one-tailed, **=signif,,, Emt 20.86** 16.97 84.26 Tolai 26.43*** 28.02** 86.24 Ainu 13.99 18.61 76.16 Easter Island 14.53 19.72 66.65 Teita 3.80 14.80 85.22 Mokapu 13.23 11.38 87.54 N-Japan 7.44 31.31*** 70.47 Dogon 13.68 14.83 108.39** Buriat 9.13 19.35 87.50 mapan 3.31 11.95 83.74 Zulu 4.15 6.60 70.72 Eskimo 10.77 14.84 84.30 Hainan 12.53 9.64 60.05 Moriori 10.88 17.33 65.74 at 0.06 level, onetailed; ***=nignifieant at 0.01 level, onetailed. TABLZ 7b. Number of signifkant tests for the equality of C , betweeen the sexes, based on SI,S, and Sal 51 S, 5 c 0.01 <0.05 co.10 n 1(2299)* l(2299) la2991 a37951 4(0487) 3(1386) 2(7487) M0499) 4(2581) 26 26 26 'Calculated from Table 7a. *=in parentheses:probabilityp ofobserving at least r significant testa (out of n) multiplied by 10000. by Howells (1973) for males and females, respectively. Test results are represented in Table 8a. The specific race variable combinations (R1R4 for the males; R1, &, R5, for the females) have few measures in common and were treated as independent. We considered the four functions (Rl-R4 for the males; R1, RP, R5, and & for the females) together in the calculation of the number of significant tests (see Table 8b). In addition to the comparison of the X, of Norse-Teita-Hainanand of Zalavar-Dogon-SJapan, we also tested the X of all 28 populations of males and of all $6 populations of females. We did not consider the combinations R3 and R5, because we learned from the previous samples that the multivariate combinations R4 and generally have higher discriminatory power between races than R3 and R5. We considered a new combination R7, which, according to the literature, discriminates well between local populations (Salzano et al., 1980; Marcellino et al., 1978; Howells, 1970; Pietrusewsky, 1973; Rightmire, 1972). The results are represented in Table 9a. The variable combinations are fairly independent, so we considered the four functions together in the calculation of the number of significant tests (see Table 9b). Table 8a shows that there is sufficient evidence to reject Ho for the male individuals. For the female individuals, however, there is no sufficient evidence to reject the null hypothesis. The results of Tables 9a and 9b are clear. H, was rejected for both sexes. Considering the results of Tables 8a and 8b and 9a and 9b together, we conclude that our results warrant some caution as to assuming equality of covariances among races. Equality of Zg among subraces: We considered the same samples as in a previous section: two groups from Europe (Norse and Zalavar), two Negrid groups (Teita and Dogod, and two Mongolid groups (S-Japan and Hainan). For the male individuals we used the variable combinationsR1, &,R4, and &.For the female individuals: R1, &,Rg, and R7. Tables 10a and lob give the results. The null hypothesis was accepted. About the equality of Z, based on specific H.T. UylTERScHAUT AND F.W.WILMINK 354 TABLE 8a 2 values for the equality of Xg among the three main races, based on RI-RG' Norse-Teita-Hainan Variable combination 55,34,45 Rl(df= 72 Rz(df= 42) RS (df= 132) R4 (df=132) Rs (df= 132) Re (df= 132) Zalavar-Dogon-SJapan 0.0 QQ 54,48,50 55,49,38 99.12** 58.54** 152.96 145.09 83.93 46.18 QQ 55,49,38 W W 68.93 68.92 * * * 142.92 133.86 57.10 60.06** - - -- - 145.61 106.97 152.64 142.73 - 'As representative populations were used Norse-Teita-Hainanand Zalavar-Dogon-SJapan. * = significant at 0.10 level, one-tailed; ** = significant at 0.05 level, one-tailed; *** = significant at 0.01 level, onetailed. TABLE 86. Number of s i g n i m n t tests for the equality of C , among the three main races, based on RI-Rs' c 0.01 <0.05 c0.10 n ww 1(0773)* 99 0 X0058) l(3366) 4(0070) 3(0381) l(5695) 4(0684) 8 8 16 l(1485) 'Calculated from Table 8a. =in parentheses:probability p of observing at least r m@icant testa (out ofn), multiplied by 10000. TABLE 9a x2 values for the equality of Eg between 28 populations of males, and between 26 populations of females, respectively, based on RI, Rz. R4 (Rd, and R7 TABLE 9b. Number of significant tests for the equality of CE among the three main races, based on mce discriminating variable combinations' Variable Males combination (28 groups) aa Rl(df= 972) Rz(df= 567) (df= 1782) R ~ ( d f = 756) 1264*** 813*** 2197*** 1010*** Females (26groups) Rl(df= 900) Rz(df= 525) R6 (df= 1650) RT(df= 700) 00 1121*** 746*** 1935*** 801*** co.01 c0.05 c0.10 n 4(<0.01)* NcO.01) NcO.01) NCO.01) Nc0.01) NcO.01) NcO.01) Nc0.01) 4 4 8 8(<0.01) 'Calculated from Table 9a. = in parentheses: probability p of observing at least r by Iooo0. aiH'mt -(Out Of d l * = significant at 0.10 level, one-tailed; ** = significant at 0.05 level, one-tailed; *** = significant at 0.01 level, one-tailed. TABLE 10a xa values for the equality of L, between two groups from Europe (Norse and Zalavar), two Negrid p u p s (Teita and Dogon), and two Mongolid groups (Hainan and SJapanJ based on RI, Ra R4 (Rs) Variable combination Rq 6 Idf= 36) (g=Zij & (Rp) (df= 66) R-7 (df=28) * = aign&ant Norse-Zalavar 55w,54a 550,459 51.54** 26.35 66.48 30.82 at 0.10 level, one-tailed; Teita-Dogon 34a,480 490,530 49.20* 14.60 76.85 42.19. * 36.68 23.49 71.23 19.74 Hainan-SJapan 45a30W 38 9 A1 0 32.04 19.47 51.66 41.83** 35.95 35.96** 73.10 20.47 33.78 28.78 56.37 35.59 ** = significant a t 0.05 level, one-tailed; *** = significant a t 0.01 level, one-tailed. TABLE lob. Number of crignifiant testa for the equality of Xs among subrace4 based on racediscriminating variable combinations' Europida(a + Q) Negride ( a + 9 ) Mongolida(a + 9) <0.01 c0.05 0 2(0572)* l(3366) l(3366) 0 0 c0.10 n 3(0381) 8 "698) 8 l(5698) 8 'Calculated from Tsble 1Oa. = in parenthenem: probabilitjr p of observing at least r signifkauttests (out of n), multiplied by 1ooo0. mLhU”YCOVARIANCEMATFUCES DIAGNOSIS SKULLS sex- or racediscriminating variable combinations, we conclude that the X, of the sexes of one population, and also those of subraces can be considered as equal, whereas among main races the equality of Xg is questionable, maybe more so for males than for females. DISCUSSION The conclusions of this study were based on two kinds of variable sets. In the first part we used random variable sets; in the second part we used specific sex- or race-discriminating variable combinations. For the second part, we carefully selected several combinations, the remarks of Burnaby11966)and Campbell (1977) kept in mind that differences in covariance structure may often be due to differences in the correlation patterns between variables which do not discriminate between groups. Perhaps this is the reamn why the combination & led to a s i g d k a n t 2 value (unequal Z,) when comparing main raw,possibly its r a d s c r i m i nating power was not as high as for the other combinations (Table 3). Elimination of variables which are not discriminating between groups will oRen lead to a marked improvement in the equality of the covariance structure (Campbell, N.A.,19771. We tried to find aome explanations for the high significance of the inequality of X, between 28 race groups. First,by adding more groups, thereby increasing racial diversity, we would expect the probability of finding significant differences to increase. Second, the pooled within-variance matrix is based on a large number of individuals, and from some minor numerical examples we found that the addition of individuals increases the power of the test even if the quotient between the number of individuals and the number of variables remains unchanged. One might also consider an alternative explanation which is perhaps more intriguing since it p i n t s to a possible circularity in the customary procedure of variable selection and DA. Usually, we felt that the “best” variables are the ones that produce large differences between the samples under study, whereas these differences are determined by means of DA. From this study, however, it can be seen that the best combinations for racial differentiation have highly different Xg,and it is tempting to consider the possibility that the “best” combinations are those which have Zg as different as possible (the produced differences being an artefact of the DA plrocedurex 355 For combination %, we have computed the spearman rank correlation between the 2 value of the test for equality of covariances and the Pcvalue’ based on the same variables, for all 15 pairs of six populations (Norse, Zalavar, Dogon, ‘Eta, &Japan, and Hainan). We found this correlation significant at the 15% level (one-tailed).It should be noted that sample sizes have been accounted for in the observed 2 and pcvalues. Further investigation of this possibility is, however, beyond the mpe of this paper. Because of the apparent inequality of 8, between racial groups, for classification purposes it is perhaps preferable to apply other procedures for racial diagnosis such as methods which do not start from the assumption of equal Z, e.g., cluster analytical procedures. Anderson and Bahadur (1962),Chernoff (19731, and Campbell, N.A. (1979) gave test statistics and algorithms which may be used for discrimination of and classification into multivariate normal distributions with unequal X, (Corruccini, 1975). Of course, the results cannot be immediately generalized to other populations. It would be interesting to continue this study with more data and to compute the amount of difference in covariances between races. CONCLUSIONS We conclude that the Zg)s for comparisons of craniometric data between the sexes and among subraces can be considered as equal, but the equality of Z, among the three main race groups is rather doubtful. 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