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

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AMERICAN 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. Differences in
variance-covariance structure among human
populations seem to increase if populations
are less related. These results throw some
doubt on the use of DA for the discrimination
of races, whereas its use is not challenged for
discrimination of sexes and local populations.
ACKNOWLEDGMENTS
The authors are indebted to Dr.W.W. Howells for sending us his data. We also wish to
thank Dr. G.N.van Vark and Dr. A.G. de
Wilde for critically reviewing the manuscript, and Mrs. J.G.Benjamins for her secretarial assistance.
356
H.T.UYTIERSCHAUT AND F. W.WILMINK
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