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Empirical comparison of distance equations using discrete traits.

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Empirical Comparison of Distance Equations Using
Discrete Traits
MICHAEL FINNEGAN AND KEVIN COOPRIDER
Osteology Laboratory, Kansas State University, Manhattan, Kansas 66506
. Statistical analysis
Inverse sine transformation
K E Y WORDS Non-metric
equations
a
-
Distance
ABSTRACT
The use of the Grewal-Smith statistic in measuring biological
distance among skeletal population samples has been questioned since it was
first applied t o human populations. Recently, in an attempt to stabilize the variance of the Grewal-Smith statistic for use with non-metric analysis, Sjmold
('73) and Green and Suchey ('76) have introduced corrections and alternative
transformations which may enhance the meaning of biological distance among
population samples. Their recommendations improve the statistics for specific
variable ranges; i.e., small sample size and low trait frequencies. Thirteen
equations representing Grewal-Smith, Freeman-Tukey, Anscombe, and Bartlett
transformations and/or corrections, were compared using rank order correlation
statistics on actual biological distances generated by real population data as
presented in existing literature. Results from testing these actual distance
models show little variation between equations based on the populational data
sets used. Based on these findings, the distance model resulting from the
Grewal-Smith statistic is not inferior to the more sophisticated models, although the latter may be superior by allowing specific improvements for small
sample size andlor low trait frequencies.
Since the 1967 article by Berry and Berry,
insight has been gained in studies of biological
distance and the statistics utilizing this approach. The most widely used statistic has
been that of Grewal-Smith, first suggested
along with the work of Grewal ('62). The basis
of the Grewal-Smith statistic is the transformation of observed frequencies of non-metric
traits by using equation [11(APPENDIX).
Once Berry and Berry ('67) used t h e
Grewal-Smith statistic on human non-metric
traits a number of researchers developed variations of this statistic or used other statistics
entirely. Most notable is the work of Lane and
Sublett ('72), Birkby ('731, Buikstra ('721,
Zegura ('73) and Finnegan ('72, '74a). Most of
the above research utilized some variation of
the Grewal-Smith statistic in an attempt to
stabilize the variance and to maximize the
information from their data, which was primarily incomplete in terms of numbers in the
sample. Although some of these attempts were
successful in reducing error in terms of samAM. J. PHYS. ANTHROP. (1978)49: 39-46.
ple size, they did not totally satisfy the theoretical problems other researchers saw in the
use of the Grewal-Smith statistic. Notably,
Zegura ('73) utilized the equation of Balakrishnan and Sanghvi's B' rather than the
Grewal-Smith equation in order to analyze his
data.
More recently, Sjsvold ('73) considered in
depth the statistical approaches used in earlier materials. He suggested modifying the
angular transformation of the frequencies to
compensate for the magnitude of the frequency. He also suggested new ways of determining the variance of the mean measure of divergence for the Grewal-Smith statistic and
finally suggested a method of determining
whether two samples from a population were
significantly different. He also recommended
a number of alternative transformations for
the frequencies to be transformed in a way
that would better stabilize the variance among others, the transformation of Anscombe ('48)and that of Freeman and Tukey
39
40
MICHAEL FINNEGAN AND KEVIN COOPRIDER
(‘50). Green and Suchey compared the various
transformations by looking a t the difference
between the actual and assumed variance for
the transformed frequencies. They utilized, in
addition to the Freeman and Tukey [131 and
Anscombe [121 methods a Barlett transformation [21 which is simply a correction to the
already existing Grewal-Smith transformation. They determined that the Freeman and
Tukey [131 transformation should be used to
transform trait frequencies in population
comparisons.
The statistical work by Green and Suchey
and Sjeivold suggests that theoretical problems now exist in the comparison of tabulated
frequencies where various transformations
and statistics have been employed in the final
Grewal-Smith statistic. This seemingly suggests that much of the previous work would
have t o be redone utilizing these new statistics.
The purpose of this paper is to test empirically the transformations and equations
which have most often been used to ascertain
whether they are similar enough to give us t h e
same relative positional outcome for the mean
measure of divergence among populations.
MATERIAL AND METHODS
By adapting a computer program designed
by Finnegan (’721, we obtained “equation
matrices” from the samples in four data sets.
These sets were chosen on the criteria of availability, on the number of samples employed,
the number of traits scored in each sample,
and the actual size of the samples concerned.
The selected data sets include: (1) Finnegan’s
Northwest Coast data (‘72) consisting of 15
samples, 42 cranial traits, with the sample
size varying from 12 to 107; (2) four Southwest population samples studied by Birkby
(’73) utilizing 48 cranial traits, with sample
sizes ranging from 50 to 158 individuals; (3)
Suchey’s (‘75) California samples utilizing 29
cranial traits, with 27 samples varying in size
from 20 to 135individuals; and (4) five Northwest Coast samples utilizing 30 infracranial
traits with sample sizes a t 50 individuals for
each population as reported by Finnegan
(’74b). In each of the above data sets, the N for
any particular trait may be less than the N for
the sample.
An example of an “equation matrix” appears in table 1.The groups of 13 numbers represent the distance computed between two
TABLE 1
Distance matrixgenerated by the 13equations listed in the
appendix
P1
VI (1) 0.060
(2) 0.016
(3) 0.051
(4) 0.034
(5) 0.017
(6) 0.123
(7) 0.724
(8)0.342
(9) 0.008
(10) 0.055
(11) 0.015
(12) 0.059
(13) 0.043
P1
W50 (1) 0.057
(2) 0.023
(3) 0.051
(4) 0.038
(5) 0.028
(6) 0.120
(7) 0.687
(8) 0.342
(9) 0.009
(10) 0.050
(11) 0.014
(12) 0.047
(13) 0.027
P1
W78 (1) 0.044
(2) 0.008
(3) 0.038
(4) 0.028
(5) 0.011
(6) 0.105
(7) 0.525
(8) 0.345
(9) 0.006
(10) 0.041
(11) 0.011
(12) 0.187
(13) 0.168
VI
0.084
0.012
0.072
0.05 1
0.028
0.145
1.003
0.397
0.010
0.067
0.021
0.033
0.014
VI
0.080
0.000
0.069
0.051
0.022
0.142
0.965
0.340
0.009
0.063
0.020
0.198
0.156
W50
0.059
0.010
0.051
0.037
0.015
0.122
0.710
0.350
0.008
0.052
0.015
0.202
0.174
These raw data are based on Birkby (‘73) who utilized 48 cranial
traits in four samples. Here we pooled the sexes using the trait frequency from the left side only.
corresponding samples using the 13 equations
listed in the APPENDIX. The equations include
t h e original Grewal-Smith statistic, and four
“improvements” by various authors and other
new transformations and equations (APPENDIX). The number in brackets listed with each
equation is used when reference is required.
Some equations were scaled up or down by a
constant to provide distance measures of comparable magnitude. In some equations 0.001
was substituted for 0.0 frequencies so that
division by 0.0 could be avoided.
Each equation produces a geometric model
41
DISTANCE EQUATION COMPARISONS
in N-1 dimensions, where N is the number of
samples in the data set. For instance, the four
southwest samples produced a model of the
form seen in figure 1, where the lines correspond to the distances between population
samples. Because the computed distances are
relative rather than absolute, models with
similar “shapes,” arising from two different
equations, can be said to be generating similar
genetic distances.
One method of measuring the similarity of
models is to rank the distances generated by
each equation, and then employ a nonparametric test on each pair of equations to determine the correlation between the ranks. High
correlation indicates model-similarity. We
have chosen Spearman’s rho as a statistic of
comparison. By including data sets differing
with respect to sample size, number of traits
considered, and the number of samples involved, we should detect the effect of any or all
of these variables on the resultant distances.
RESULTS
The results of the correlation analysis indicate a high degree of similarity among virtually all equations in all major data sets
(tables 2, 3). The correlations are significant
a t the 0.001 level in 100%of the comparisons
utilizing Finnegan’s cranial data and Suchey’s data (table 2). The Finnegan infracranial
subgrouping, where sides and sexes were
pooled, produced correlations significant a t
the 0.001 level in 100%of the pairings (table
3). Birkby’s data, comprised of sexes and sides
combined, produced correlations significant a t
W50
W50
pi
.060
W5 0
VI
PI
,051
VI
W50
W 78
Fig, 1 Four models representing the distance displays produced by the equations [11, [31, [81, and 1121, for
Birkby’s populations P1,V1,W50, and W78.These correspond with distances produced in table 1. Each model
was constructed in three dimensions and vertically reduced to the two dimensional plane defined by the
points P1, V l and W50.The scale for model [Sl is reduced by a power of 10.
42
MICHAEL FINNEGAN AND KEVIN COOPRIDER
TABLE 2
Correlation matrices of distancrsgenerated by the 13equations listed in the appendix
Eouation
1
2
3
4
5
6
7
8
9
10
11
12
13
1
3
2
0.7908
0.9997
0.9747
0.8499
1.0000
1.0000
0.8262
0.9569
0.9855
0.9993
0.5880
0.5063
0.7028 0.9998
0.7131
0.7948
0.8372 0.9771
0.9801 0.8531
0.7908 0.9998
0.7908 0.9998
0.5391 0.8236
0.8467 0.9568
0.8014 0.9849
0.7852 0.9988
0.4976 0.5833
0.5252 0.5020
6
5
4
8
9
10
11
12
13
0.8524
0.5937
0.8515
0.7296
0.7129
0.8524
0.8524
0.9319
0.7560
0.9324
0.8479
0.8286
0.9319
0.9319
0.8078
0.9873
0.7423
0.9877
0.8796
0.8603
0.9873
0.9873
0.8624
0.9714
0.9999
0.7024
0.9997
0.8693
0.8505
0.9999
0.9999
0.8546
0.9319
0.9876
0.8644
0.9454
0.8723
0.9986
0.9995
0.8644
0.8644
07234
0.8392
0.8725
0.8641
0.6888
0.9957
0.6994
0.9408
0.9532
0.6888
0.6888
0.5851
0.7292
0.7229
0.6884
0.9454
7
0.8695 0.8509 1.0000 1.0000
0.9425 0.9525 0.7028 0.7028
0.8772 0.8591 0.9998 0.9998
0.9977 0.8696 0.8696
0.8509 0.8509
0.8862
1.0000
0.9747 0.8499
0.9747 0.8499 1.0000
0.7807 0.6310 0.8263 0.8263
0.9337 0.8845 0.9570 0.9569
0.9541 0.8550 0.9855 0.9855
0.9721 0.8460 0.9993 0.9993
0.5257 0.5207 0.5880 0.5880
0.4519 0.5156 0.5063 0.5063
0.7708
0.8522
0.8385
0.6116
0.5049
0.9739
0.9555 0.9875
0.5797 0.5999 0.5950
0.5306 0.5284 0.5112 0.9710
Correlations from Suchey's ('75) data are presented above the diagonal with correlations from Fmnegan's 1'72) cranial data below. In each
case correlations are based on sexes combined using the left slde only. and all correlations are significant at the 0.001 level
TABLE 3
Correlation matrices of distances generated by the 13 equations listed in the appendix
Equation
1
1
2
3
4
5
6
7
8
9
10
11
12
13
0.9879
1.0000
1.0000
1.0000
1.0000
1.0000
0.9394
0.9758
0.9879
1,0000
1.0000
0.9879
2
3
1.0000 1.0000
1.0000
0.9879
0.9879 1.0000
0.9879 1.0000
0.9879 1.0000
0.9879 1.0000
0.9273 0.9394
0.9879 0.9758
1,0000 0.9879
0.9879 1,0000
0.9879 1.0000
1.0000 0.9879
4
5
6
7
8
1.0000 0.9429 1.0000 1.0000
1,0000 0.9429 1.0000 1.0000
1.0000 0.9429 1.0000 1.0000
0.9429 1.0000 1.0000
1.0000
0.9429 0.9429
1.0000 1.0000
1,0000
1.0000 1.0000 1.0000
0.9394 0.9394 0.9394 0.9394
0.9758 0.9758 0.9758 0.9758
0.9879 0.9879 0.9879 0.9879
1,0000 1.0000 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000
0.9879 0.9879 0.9879 0.9879
9
0.8857
0.8857
0.8857
0.8857
0.9429
0.8857
0.8857
0.9394
0.9273
0.9394
0.9394
0.9273
1
0.9429
0.9429
0.9429
0.9429
0.8286
0.9429
0.9429
0.7714
0
1
1.0000
1.0000
1.0000
1.0000
0.9429
1.0000
1.0000
0.8857
0.9429
1
12
13
1.0000
1.0000
1.0000
1.0000
0.9429
1.0000
1.0000
0.8857
0.9429
1.0000
- 0.0286
-0.0286
-0.0286
-0.0286
- 0.0286
- 0.0857
-0,0286
- 0.0286
-0.3714
0.1429
-0.0286
- 0.0286
1.oooo
0.9879
0.9758 0.9879
0.9758 0.9879 1.0000
0.9879 1.0000 0.9879
-0,0286
-0.0286
-0.0286
-0.0857
-0,0286
- 0.0286
-0,3714
0.1429
-0.0286
-0.0286
0.9879
Correlations below the diagonal are based on Fmnegan's ('74b) infracranial data, while those above are based on Birkby's ('731 cranial data.
sides and sexes combined in each case. See text for a discussion of the significance levels.
the 0.001 level in 38.46%of the cases, an additional 32.05% were significant a t the 0.01
level and an additional 2.56%were significant
a t the 0.05 level (table 3). The most aberrant
subgrouping of the Birkby data - sexes combined, using the left side only - produced the
correlations presented in table 4. Here, 19.23%
of the correlations were significant a t the
0.001 level, an additional 5.13%were significant at the 0.01 level, and an additional
14.10% were significant at the 0.05 level.
Cumulatively, over all data sets presented in
the tables, 71.28% of all correlations were significant at the 0.001 level, an additional 7.44%
met the 0.01 level and an additional 3.85%
were significant at the 0.05 level. This suggests t h a t any of the equations may be used
with the assurance of obtaining very reasonable information with the following possible
exception.
Analysis of Birkby's subset data ('73) (table
4) revealed that equations [21,@I, t121 and [131
give somewhat different information. The
Birkby populations are all relatively large, as
is the number of traits scored, although the
number of populations is small. It is not
known a t this time whether one of the above
factors, a combination of these factors, or
some other variable(s) are affecting these correlations. The chance that this is a particularly unusual set of data is quite small, since
each correlation pair computed from the
Birkby sample (combined sex, combined sides,
etc.) showed these equations to be somewhat
divergent from the data sets of Finnegan and
Suchey. The lack of significance may be due to
the small number of samples in his study and
the fact that Birkby utilized about 30%more
traits than Finnegan ('72) used in his cranial
study.
43
DISTANCE EQUATION COMPARISONS
DISCUSSION
w
-r
o!
0
o-tot-w
ON?
olom
100
4
Indo
0 0 0
0 0 0
I 1
v v v
wwm-r
!&!&La
rnrnu??
0 0 0 0
1
I
II
II
mmm+
fXc4-t-
I I
1
1
The primary purpose of this paper has been
to test whether one or another equation appears to be more productive in terms of the
empirical results which can be expected from
one of the equations utilized. Little real difference is found, in terms of actual data utilizing the above equations, even though there is
a range of difference in the absolute numerical value between population sample distances depending on the equation used. It
must be remembered that these distances are
only relative to the other populations with
which these distances were generated. The
analysis of biological difference must then
come from these distances in terms of ordering
but not in terms of absolute value.
It is interesting that although theoretical
statistical arguments favor certain equations
based on ratios of the assumed variance to the
actual variance, in practice little difference is
found. From this we conclude that (1) it does
not matter which statistical equation is utilized when sample sizes are as large as the
population samples utilized in this discussion;
(2)the frequency of each trait can either be
large or small, corrected or uncorrected, and
not make any real difference in between population comparisons, at least t o the extent of
the frequency magnitude and sample sizes in
this analysis, (3) any of these statistical equations can be used with confidence when a relatively large number of traits are scored for
each skeleton, i.e., 25-45 traits; (4) caution
should be exercised when choosing a transformation for analysis of a few population samples, as seen in the Birkby data, although we
have not defined limits based on the above
data sets.
We can now adequately show that the population sample separation seen in this and previous studies indeed reflects something about
biological separation and not numerical separation based on the type of statistic used.
This has been alluded to by a number of researchers, and biological separation fits well
with linguistic separation using glottochronology and lexicostatistics (Finnegan, '72),
and with other population studies, primarily
metric studies of the skeleton (Cybulski, '73;
Corruccini, '73, '74; Ortner and Corruccini,
'76, etc.). Dental traits have been utilized,
both metric and non-metric, t o produce results
similar to both skeletal metrics and skeletal non-metrics (Ortner and Corruccini, '76;
44
MICHAEL FINNEGAN AND KEVIN COOPRIDER
Greene and Armelagos, '72). Blood studies
have also shown separation similar t o nonmetric trait variation of the skeleton (Finnegan, '72; Hulse, '55). The distribution of material culture remains has also suggested divisions of population samples which correlate
highly with both non-metric cranial traits and
linguistic differences (Finnegan, '72, '74a;
Hirsch, '54).
From the above we conclude the following:
(1)various forms of the Grewal-Smith statistic are quite similar and seem not to vary
greatly by empirical testing, where angular
transformations of the frequency data are utilized. No dependency is suggested on either
the number of non-metric traits or the range
in frequency of those non-metric traits when
utilizing one of the above equations. (2) The
argument for standardizing the variance in
each of t h e equations seems not to be particularly important when applied to real populations. (3) The equation which seems most
preferable to us is equation [51 for the following reason: not only does this statistic give a
biological measure in numerical form between
populations which are highly comparable by
correlation with all other examples, but it
presently seems to be the most widely used
equation.
ACKNOWLEDGMENTS
The authors wish to acknowledge the program assistance of S. A. McGuire and Ms.
Lorraine Douglas for typing various drafts of
this article. The infracranial data was collected with the support of a grant from the
Smithsonian Research Foundation Fellowship
SFC-3-0875and COAA Grant 4F0875.
LITERATURE CITED
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- 1974a A Migration Model for Northwest North
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717-722.
45
DISTANCE EQUATION COMPARISONS
Sanghvi, L. D. 1953 Comparison of genetical and morphological methods for a study of biological differences.
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Sjovold, T. 1973 The occurrence of minor non-metrical
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Adaptation by Finnegan of the CAB Smith
equation in which the average number of individuals over all traits accounts for the variance factor (Finnegan, "72: p. 30).
R
0
In each of the following equations:
trait
i = trait no. under summation
j
=
phenotype
(:czeeIi)
under summation
N1 = total skulls sample 1
Nli = skulls of sample 1with observable trait i
Ki
pli = % of trait i in sample 1i-4
Nli
plij = % of trait i, observationj in sample 1
R = no. of traits for particular data set
K1,= count of positive observations for trait i
R
z
(ell - 82i)Z
Original Grewal-Smith formula which they
claimed allocated too much of the measure to
random sampling error (Grewal, '62: pp. 229230).
sin-' (1-2plii
R
APPENDIX
H l i and @lI= transformation angles of first sample ith
=
Grewal's adaptation with variance factor
based on total population rather than individual trait population (Grewal, '62: pp. 229230).
@lj =
sin-' (1-2~1,)
Constandse-Westermann formula (from CAB
Smith) basing the variance factor on individual trait population. This appears to be the
most widely accepted formula a t this time,
though experiments with other transformations may show it deficient (ConstandseWestermann, '72: p. 119).
Oliver and Howell's use of the Fisher transformation which Constandse-Westermann had
some trouble deciphering from their original
publication. Fisher transformation is apparently well-known and useful in other
situations (Oliver and Howells, '60: pp. 498500; Fisher, '25).
R
2
t
[cos-1
L
1=1
4-
(7)
j=l
8 = sin-' (1-2p1J
with Bartlett's b r r .
of p1, = 1/4nl, when pll =
0.000
pi1 = 1-1/4n1, pi1 = 1.000
Bartlett's correction for stabilizing the variance of observations (traits) whose percentage positive recordings were either 0,000 or
1.000 (Sjevold, '73: pp. 224-226).
R
z
loll -82,P
Malyutov et al. formula for evolutionary tree
building, designed for situations of multiple
alleles a t a single locus, but perhaps suitable
for non-metrics (Malyutov et al., '72: p. 50;
Constandse-Westermann, '72: p. 105).
10,000
R
~
i=l
Pli - ~ 2 i
pHi-pLi
)p
Hiernaux's Ag distance utilized by Gaherty.
We have followed the example of Gaherty in
utilizing the range of the particular data set,
rather than a world-wide range (Hiernaux,
'65: pp. 1748-1750.
46
MICHAEL FINNEGAN AND KEVIN COOPRIDER
Gaherty's use of the very simple unmodified
Euclidean distance (Gaherty, '74: p. 4).
R
011 = sin-' (l-Z(K1, + 3/8)/(N1, + 3/41)
Sanghvi's formula derived from his previous
x2 work. This equation is again devised for
multiple allele use (Sanghvi, '53).
R
2 . 1 (11=I
This is the same basic equation as GrewalSmith 12) with substitution of the Anscombe
transform and its variance (Sjovold, '73: pp.
212-213; Anscombe, '48).
R
L
i=l
2
t 1j=l
(11)
R
Cavalli-Sforza et al. distance equation designed for multiple alleles and binomial cases
(Cavalli-Sforza et al., '69; Spuhler, '72).
81, =
[(o - 0
1,
1
2,
i'
-i
1
+ 1/2 + -N z l + 1/2
Nil
R
U2sin-l (l.ZKli/(Nli + 1)) +
1/2sin-' (1-2 (Klk
ij
(13)
+ l ) / ( N ] , + 1))
This is similar to Grewal-Smith with insertion of the Freeman and Tukey transformation and its corresponding variance (Sjavold,
'73: p. 312; Freeman and Tukey, '50).
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