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Comparison of distance matrices in studies of population structure and genetic microdifferentiation Quadratic assignment.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 68:367-373 (1985)
Comparison of Distance Matrices in Studies of Population
Structure and Genetic Microdifferentiation: Quadratic Assignment
MALCOLM M . DOW AND JAMES M . CHEVERUD
Department of Anthropology and Program in Mathematical Methods in the
Social Sciences (M.M.D.) and Departments of Anthropology, Ecology and
Evolutionary Biology, and Cell Biology and Anatomy (J.M. C.),
Northwestern University, Evanston, Illinois 60201
KEY WORDS Population structure, Quadratic assignment,
Distance matrices
ABSTRACT Questions concerning the relative effects of various evolutionary forces in molding the genetic variability exhibited by groups of human
populations have typically been investigated by comparing a variety of genetic
and culturalhistorical “distance” matrices. A major methodological difficulty
has been the lack of formal testing procedures with which to assess the degree
of confirmation or disconfirmation of a n estimated measure of relationship
between such matrices. In this paper, we examine a very flexible matrix
combinatorial procedure which generates statistical significance levels for
correlational measures of pattern similarity between distance matrices. A
recent generalization of the basic procedure to the three-matrix case allows
questions concerning which of two matrices best fits a third matrix to be
formally tested. Applications of these hypothesis testing and inference procedures to two separate sets of genetic, geographic, and cultural distance matrices illustrates their potential for finally solving a long-standing problem in
anthropological genetics.
Over the past 15 years, numerous studies
have sought to assess the relative impact of
various forces in molding the genetic variability exhibited by groups of human populations. Often such studies begin by establishing relationships among a sample of communities within some subdivided human population on the basis of such factors as language
similarity, geographical distance, and migration patterns, and then attempt to ascertain
whether the observed patterns of genetic variability are reflected in these relationships
(Crawford et al., 1981; Howells, 1966; Froehlich and Giles, 1981; Kirk, 1982; Neel et al.,
1974; Friedlaender, 1975). Typically, linguistic similarity is taken as a measure of historical propinquity and related to the time span
over which differences caused by genetic drift
have accrued, whereas geographic distance
and migration patterns relate to gene flow.
However, it is also commonly recognized that
linguistic similarity can effect and be affected by current migration patterns and that
linguistic similarity, geographical distance,
0 1985 ALAN R. LISS. INC
and migration patterns may be intercorrelated due to the common effects of historical
and social processes.
A variety of formal methods have been developed to compare the resulting patterns of
variation, where the latter are usually expressed as dissimilarity matrices in which
the i j t h element is some measure of “distance” between the ith and jth subpopulations. One approach is based on generating
graphical displays of the data matrices in the
form of tree diagrams, and then looking for
correspondences between the resulting trees
(Ward, 1972; Ward and Neel, 1970). Similarities may be uncovered simply by visual inspection, or somewhat more formally by
finding where the net length of a tree generated from one data matrix falls on the empirical distribution of lengths generated from a
second data matrix (Spielman, 1973; Neel et
al., 1974). However, a major difficulty with
Received May 6, 1985; revised July 3, 1985; accepted July 16,
1985.
368
M.M. DOW AND J.M. CHEVERUD
this approach to significance testing is that
the two trees of interest are not directly compared with one another. That is, with even a
moderate number of population subdivisions
the number of “best fitting” trees in the top
5% (say) of a n empirically generated distribution may be enormous, thus indicating little about the structural similarity of the two
trees, and even less about the underlying
patterns in the data sources from which they
are generated.
A second approach to the comparison problem, the topology or map approach (Jorde,
19801, also relies on graphical representations of the data matrices. Here, the eigenstructure of a distance matrix is obtained,
typically from a principal components analysis, and the first two or three eigenvectors
are used to construct a graphical representation or map of the pattern of relations among
the sample units (Workman et al., 1975; Jorde
et al., 1982).A measure of “fit” between two
such maps can be obtained by rotating,
translating, and/or reflecting one set of points
to minimize the “residual” distances to the
second set of points (Gower, 1971; Schonemann and Carrol, 1970). These procedures
may also be used to match maps generated
from a multidimensional scaling analysis
(Nee1 et al., 1974). While this matching procedure often reveals close similarities in the
structures of two maps, the distribution of
the residual distances statistic is unknown;
hence the usual strategy for measuring the
degree of confirmation or disconfirmation of
any a priori conjecture cannot be employed.
A third approach to assessing degree of
concordance between genetic and cultural
distance matrices, and probably the simplest
approach to measuring similarity, is by
means of a correlation coefficient computed
from the corresponding individual entries or
pairs of entries within two distance matrices.
The main advantage of correlation as a measure of structural similarity in this context
is that it is computed from the raw data
without the imposition of any intermediate
data reduction procedure, as in the previous
two graphical approaches. Thus none of the
original data are discarded prior to making
comparisons between data matrices. Many
parametric and nonparametric correlation
coefficients have been reported in the literature; in a recent review Jorde (1980) reports
over 50 correlations between various genetic,
geographic, migration, linguistic, and anthropometric data types. However, Jorde does
not report significance tests for any of these
correlations, stating that “significance levels
are not given for these correlations, since the
degrees of freedom for pairwise comparisons
of two matrices of intercorrelated populations cannot be specified. . . [and] no general
means of determining the degree of freedom
has yet been derived” (Jorde 1980:187). This
latter problem is also recently noted by ReIethford (1985321).
In this paper we examine a matrix combinatorial procedure which generates statistical significance levels for correlational
measures of similarity between distance
matrices. The procedure, now known as
quadratic assignment (Hubert and Schultz,
19761, was originally proposed in the biometrics literature by Mantel (19671, and various
applications in evolutionary genetics, systematics, and animal behavior have recently
appeared (Douglas and Endler, 1982; Cothran and Smith, 1983; Sokal, 1979; Schnell et
al., 1985; Smouse and Wood, 1985). Here,
however, we focus on a generalization of the
basic Mantel procedure to the three-matrix
case, thus allowing questions concerning
which of the two matrices best fits a third
matrix to be considered (Golledge et al., 1981;
Hubert and Golledge, 1981). With respect to
the present set of problems, the question of
whether blood polymorphism distances or
anthropometric distances yield a statistically
better fit to geographic distances, for example, can now be formally evaluated. Although statements about the superiority, or
similarity, of fit between various genetic and
culturalhistorical distance matrices appear
fairly frequently in the literature (Spielman,
1973; Friedlaender et al., 19711, to date there
has been no direct formal test of such conjectures. In the next section, we outline Mantel’s comparison procedure and the recent
generalization to the three-matrix case. Subsequently, we apply these test procedures to
a variety of distance matrices obtained for a
sample of 19 Yanomamo Indian (S. Venezuela) villages (Spielman, 1973) and 18 Bougainville Island (Melanesia) villages (Friedlaender, 1975).
MATRIX PERMUTATION INFERENCE
PROCEDURES
Mantel (1967) initially proposed a matrix
permutation strategy to test for disease clustering in time and space. Since this original
paper, there has been a great deal of work
done on the quadratic assignment approach
369
COMPARISON OF DISTANCE MATRICES
to inference, and many elaborations, extensions, and applications have appeared. The
references to applications given above and
the more theoretical discussions cited below
provide a n introduction to many of the details of Mantel's method; thus the following
discussion will be brief. However, the recent
extension of this approach to the three matrix case will be discussed in more depth.
As notation, suppose two numerical distance measures between n subpopulations
can be represented a s two nxn distance matrices A = {aij) and B = {bi.}, where qj and
bij refer to the two kinds of distances between
locations i and j. Usually, aii = bii = 0 for
1 5 i 5 n, that is, self-proximities are considered irrelevant to the analysis, although
this is not essential to the method and can be
relaxed if necessary (e.g., Dow 1985). Mantel's test statistic for pattern correspondence
between A and B is given by
This index may be interpreted simply as a n
unnormalized Pearson product-moment correlation between A and B. A number of strategies for obtaining a significance test of TAB
have been proposed. One approach is to use
Monte Carlo methods to generate a n empirical distribution of r A B for any two given distance matrices. That is, under the null
hypothesis that the rows and columns of one
matrix are matched at random to the other,
define (Dietz, 1983; Hubert and Schultz, 1976)
where p(*) is a permutation of the integers
(1,2, . . ., n) selected at random from the set
of n! possible permutations. For each permutation p(*) of the first n integers, the rows
and simultaneouslythe columns of matrix B
are permuted and a value of
between
A and the newly permuted B is calculated
and recorded. The significance level of the
observed index,
is taken a s the proportion of rA& ) indices as or more extreme.
For even moderate n, however, complete enumeration of all n! possible reorderings of B
is computationally extremely burdensome;
hence, a n approximate distribution is usually obtained using random subsampling
(with replacement) from the set of n! possible
permutations. That is, if M random permu-
tations are generated and T of the resulting
indices are a t least as extreme as the
observed rAB, then the significance level
is taken as (T l)/(M l), where the observed index is also taken a s a random draw
under the null. Note that it makes no difference which matrix is randomized and which
is held fixed, since the same distribution is
generated in either case.
An alternative to empirically generating
a n approximate distribution is to compute
the exact mean and variance of the permutation distribution and estimate a significance level using a n assumed probability
distribution. Mantel (1967) derived estimating equations for the first two moments and
suggested that these permutation distributions were approximately normally distributed. Thus, Z = ( r A B - E(rAB))/(Var(rAB)Yh
converts the observed gamma to a standardized variate which can be referred to the
normal distribution tables. However, Mielke
(1979) and Ascher and Bailar (1981) have
shown that even asymptotic normality does
not hold for certain not unreasonable forms
of proximity matrices, and they suggest curve
fitting procedures using a Pearson type I11
distribution? which incorporates a n exact
skewness parameter (third moment). Mielke
(1979) provides a computationally efficient
algorithm to calculate the skewness parameter. A comparison of the Monte Carlo, Normal, and Pearson type 111 distributions is
given in Costanzo et al. (1983), who demonstrate the superiority of the type I11 approximation over the Normal approximation to a n
empirically generated distribution. The type
I11 approximation is reported in the examples discussed below.
Hubert and Golledge (1981) and Golledge
et al. (1981) have developed a n extension of
the above Mantel inference strategy which
permits a test of fit between a n original data
matrix and one or more reconstructions based
on different mathematical models or a priori
theoretical conjectures. Although the concerns of these latter authors are primarily
with evaluating the fit of alternative clustering or scaling outputs to the original data
matrix, their approach is also applicable to
three original data matrices, since exactly
the same statistical problem arises: the comparison of two dependent correlations. Given
three distance matrices A, B, and C, the
problem can be phrased in terms of finding a
test for, say, the equality of r A B and rAC, i.e.,
Ho: PAB = PAC,against a n appropriate alter-
+
+
370
M.M. DOW AND J.M. CHEVERUD
native hypothesis. While Fisher’s Z-transformation is the commonly employed test of
equality of two independent correlations, it
is inappropriate when the correlations are
dependent. However, Wolfe (1976, 1977) has
observed that for three variables (XI, Xz, X3)
having a trivariate distribution where
uij = pij ui uj is the i, jth element of the corresponding covariance matrix with pii = 1,
(i,j = 1,2,3), then the correlation between
variable X1 and variable Z = X3 - X2 is
If the restriction that a$ = ug is imposed,
then (3) simplifies to
So, in the case that the variances of X2 and
X3 are equal, Ho: p i 3 = p12 can be tested by
testing the equivalent hypothesis Hd:
p1z = 0. That is, a test for the equality of two
dependent correlations is equivalent to a test
of the correlation between the common (dependent) variable and the difference between
the other two variables after the latter have
been standardized to have equal variances.
A matrix extension of this result was initially suggested by Hubert and Golledge
(1981) and Golledge et al. (19811, who also
provide various applications. More recent examples are given in Nakao and KimballRomney (1984) and Dow (1985). In the matrix
generalization, two matrices B and C are
made commensurable by transforming their
elements, e.g., to ranks or Z-scores, after
which the difference matrix B-C is compared
to matrix A using the permutation strategy
outlined above. As in Wolfe’s three variable
case, a statistically significant positive
rA,B-C index implies that matrix B is a better
fit to matrix A than is matrix C, while a
significant negative index implies the converse.
With respect to the question of standardizing matrices B and C to have equal variances, it should be noted that in the Mantel
paradigm the reference distribution is conditional on the two matrices being compared.
No underlying population model is assumed,
and the null hypothesis of random association is defined only in terms of the empirically observed data. Hence a n appropriate
transformation of both matrices is not problematic for this testing strategy. In addition,
Hubert and Golledge (1981:218) note that
“randomization can be interpreted as a conservative paradigm with respect to the usual
population model.”
A second related point is that the Mantel
inference procedure does not address the
usual statistical issues of degrees of freedom,
power, and the like. At present, there is no
way of incorporating information on the differential degrees of freedom involved in constructing the two matrices being compared
to a third matrix. However, this appears to
be a more important concern when the comparison involves matrices generated from a
data matrix using different mathematical
models, since differences in the complexity of
the models will not be taken into account in
assessing relative superiority of fit. Clearly
in this latter situation some caution should
be exercised in interpreting minimally significant differences.
RESULTS
Two separate analyses of genetic, geographic, and cultural distances matrices are
reported. Correlations between pairs of such
matrices and their associated significance
levels are estimated using Mantel’s approach. In addition, estimates of the significance of differences between several dependent pairs of correlations are assessed using the recent extension of the Mantel strategy to three matrices.
19 Yanomamo villages
Spielman (1973) reports four distance matrices for 19 Yanomamo viIIages. DetaiIs on
the sources of these data and locations of
villages are provided by Spielman (1973:465).
Genetic distances were derived from allele
frequencies by the method of Cavalli-Sforza
and Edwards (1967). Geographic distances
were estimated by the straight-line method,
and are reported in units of approximately
100 km. Anthropometric distances were calculated using the Mahalanobis distance measure. A fourth genetic distance matrix is
reported for a subsample of the population,
but is not included in the following analysis.
Table 1 reports the pairwise Pearson product-moment correlations, corresponding Zscores, and type I11 probability levels estimated for the marker gene (genetic), geographic, and anthropometric distance ma-
371
COMPARISON OF DISTANCE MATRICES
TABLE 1. Correlations, 2-scores, and estimated type III
significance levels (P11.d for geographical (D), blood
marker gene (G), and anthropometric (A) distance
matrix comparisons among 19 Yanomamo villages
Comparison
D
D
G
D
xG
xA
xA
x G-A
Correlation
Z-score
PI11
0.327
0.843
0.113
-0.557
2.559
5.127
0.830
-3.322
0.012
0.000
0.194
0.004
trices. These statistics were calculated using
Costanzo et al.’s (1983)program.
The correlations between geographic distances and both anthropometric and genetic
distances are substantial and highly statistically significant. This confirms Spielman’s
(1973:470) finding that the latter two distance matrices are “both excellent representations of the geographic relationships” based
on his “best-fitting” path lengths procedure.
However, the small and statistically nonsignificant correlation (rG,A = .113) found between the genetic and anthropometric distance matrices appears to conflict with Spielman’s (1973:470)claim that “the best anthropometric net is a good representation of the
marker gene distance relationships,” and
Dietz’ (1983)finding that these two matrices
were statistically significantly related using
an unnamed measure of association related
to Kendall’s Tau.
Although the results in Table 1 and those
reported by Spielman indicate very significant fits of both genetic and anthropometric
distance to the geographic distances, the relative magnitudes of the bivariate correlations reported here suggest that perhaps the
anthropometric distances provide the better
fit of the two. To test this latter conjecture,
both genetic marker distances and anthropometric distances matrices were standardized to Z-scores by treating the elements in
each matrix as a sample (ignoring the main
diagonal elements), after which the standardized anthropometric matrix was subtracted from the standardized genetic marker
matrix. Table 1 also reports the correlation,
Z-score and type I11 probability level for this
difference matrix and the original geographic distances matrix. The resulting significant correlation (rD,G-A = - .557) clearly
suggests anthropometric distances yield a
statistically better fit to the geographic distances. This significantly higher correlation
may be taken as supporting the reliability of
anthropometric measurements relative to
gene markers in microevolutionary studies.
18 Bougainuille Island villages
Friedlaender et al. (1971) and Friedlaender
(1975) report a number of findings on the
relationships among a variety of biological,
linguistic, geographic, and migration distances estimated for 18 villages on Bougainville Island. The analyses reported here are
based on distance matrices kindly supplied
to us by Professor Friedlaender. Details on
the measurement of each kind of distance
matrix are provided in the above cited publications. Only a subset of the distance matrices are analyzed here.
Friedlaender et al. (1971) report a table of
Pearson product-moment correlations between six distance matrices: serological (A
and G), anthropometrics, geographic, linguistic, and migrational. However, they state (p.
268) that “it is not clear how to determine
the correct number of degrees of freedom.
Since tests of significance cannot be appropriately applied, the coefficients presented
. . . give only crude measures of association.”
In a later publication, however, Friedlaender
(1975) reports tables of pairwise Kendall’s
Taus and their associated normal deviates
for these six (and some additional) distance
matrices as an attempt to provide corroborating statistical tests of the corresponding
Gower’s (1971) residual distances statistics.
Here, we focus on the Pearson product-moment correlations reported in the earlier paper. Table 2 reports these latter pairwise
correlations and their associated Z-scores and
type I11 probability levels for two of the biological distances (serological A and anthropometrics) and three historical/cultural
distances (geographic, linguistic, and migration). Since the two blood marker distance
matrices are highly correlated (r = .96), the
TABLE 2. Correlations, 2-scores, and estimated type I l l
significance levels (PIId for geographical (D),
serological
(G), anthropometric (A), migrational (M), and language
(L) distance matrices for 18 Bougainuille Island villages
Comoarison
D
D
G
M
M
L
L
xG
x A
xA
xG
x A
x G
x A
Correlation
Z-score
PTll
0.406
0.170
0.416
0.374
0.509
0.565
0.547
3.249
1.691
4.501
4.545
6.244
6.109
6.377
0.003
0.069
0.000
0.000
0.000
0.000
0.000
M.M. DOW AND J.M. CHEVERUD
372
one with the highest correlations to each of
the other four matrices was employed here
(serological A). Table 2 shows very high significance levels for each of the pairwise comparisons, with the exception of the geographic-anthropometric comparison. This
finding appears to conflict with the previous
analysis of the Yanomamo data, where this
latter correlation was the highest found. This
indicates that the relation of population processes to geography may be quite different in
Venezuela and Bougainville.
Friedlaender et aI. (1971:268) conjecture on
the basis of a simple comparison of bivariate
correlation coefficients that “neither blood
polymorphism gene variation nor anthropometric variation has correlated with spatial
distance at significantly stronger level than
the other.” We test this conjecture using the
three-matrix comparison procedure and find
that it receives statistical support. The appropriate correlation is shown in Table 3
(rDA.G). We also test whether either of the
biological distances yields a significantly better fit to the language and migration matrices. Table 3 also shows the results of these
comparisons. The only significant difference
is found with respect to the migration matrix, to which the anthropometric distances
give a statistically better fit than the blood
gene distances,
In both examples, then, anthropometric
measurements show a significantly better fit
to population processes, as represented by
geography in the Yanomamo and migration
distance at Bougainville, than single locus
markers, and thus may be the preferred metric in microevolutionary reconstructions. The
superior performance of anthropometric
measurements may be due to the averaging
of effects from a larger number of segregating loci and the possibility that antigenic,
serum, and red cell protein genes may be a
biased sample with respect to evolutionary
processes.
TABLE 3. Correlations, Z-scores, and estimated type 111
sigrrificance levels iP,,S for anthropometric (A)and
serological fGJ difference matrix with geographical (D),
migrational (M), and language (L) distance matrices for
18 Bouaaircuille Island uillages
Comparison
D x A-G
M x A-G
L x A-G
Correlation
Z-score
PIJI
-0.182
0.178
-1.280
2.151
0.050
0.514
0.157
0.041
0.309
CONCLUSIONS
In genetic microdifferentiation and population structure studies questions concerning
the relative effects of the various evo1utior.ary processes have frequently been investigated by comparing a variety of genetic and
culturalhistorical distance matrices. Evidence in favor of a particular a priori theoretical conjecture is usually based on some
assessment of fit between relevant distance
matrices. A major methodological difficulty
with this line of research, however, has been
the lack of formal testing procedures with
which to assess the degree of confirmation or
disconfirmation of a n estimated measure of
fit. In particular, it has not previously been
possible to assess relative degree of fit of two
distance matrices to a third, although statements asserting such relationships are quite
common in the literature. We have introduced a very flexible matrix combinatorial
approach to hypothesis testing and inference
which overcomes previous difficulties. Applications of the new significance testing procedures to two distinct data sets illustrate the
simplicity and generality of these matrix
combinatorial methods and their potential
for finally solving a long-standing problem in
anthropological genetics.
ACKNOWLEDGMENTS
We wish to thank Professor Jonathan
Friedlaender for providing us with the various Bougainville Island distances matrices.
Costanzo et al.’s (1984) Quadratic Assignment Program used here was implemented
on the Northwestern University CDC Cyber
170 by Richard Kerber. We also thank John
Terrell for helpful comments on this paper.
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population, structure, matrices, quadratic, assignments, genetics, studies, comparison, microdifferentiation, distance
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