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

код для вставкиСкачатьAMERICAN 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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