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Population Structure and Quantitative Characters Copyright 0 1983 by the Genetics Society of America POPULATION STRUCTURE AND QUANTITATIVE CHARACTERS ALAN R. ROGERS’ AND HENRY C. HARPENDING Department of Anthropology, University of New Mexico, Albuquerque, New Mexico 87131 Manuscript received March 3, 1983 Revised copy accepted July 8, 1983 ABSTRACT A migration matrix model is used to investigate the behavior of neutral polygenic characters in subdivided populations. It is shown that gametic dise- quilibrium has a large effect on the variance among groups but none at all on its expectation. The variance of among-group variance is substantial and does not depend on the number of loci contributing to variance in the character. It is just as large for polygenic characters as for single loci with the same additive variance. This implies that one polygenic character contains exactly as much information about population relationships as one single-locus marker. The theory is compared with observed differentiation of dermatoglyphic and anthropometric characters among Bougainville islanders. N studies of population structure it is common practice to compare variation I of quantitative characters with that of genetic markers. Such comparisons are potentially interesting for several reasons. They may, for example, shed light on the factors maintaining variation in quantitative traits. HIERNAUX (1977) argued that, since the concordance between patterns of variation in stature and in (presumably) neutral gene frequencies is low in equatorial Africa, something other than drift must be affecting variation in stature. Several other authors have suggested using models of neutral differentiation to test the hypothesis that observed variation is due to drift (LANDE19’76; CHAKRABORTY and NEI 1982; K. MORGAN and M. HOLMES,unpublished data). In practice, the value of this method has been limited by our meager understanding of the dynamics of quantitative variation. There has been no basis for deciding when departures from the predictions of a neutral model are large enough to war- rant its rejection. We need to know not only the variance expected under a neutral model but also the standard error of this expectation. Other studies have tried to infer patterns of mate exchange and local pop- ulation history from patterns of variation in metric characters. FROEHLICHand GILES(1981) argue that, since quantitative characters are affected by many loci, they are somehow buffered against random forces such as drift and should give a better picture of pcjpulation history than can be obtained from a handful of genetic markers (see also BIRDSELL1950; RELETHFORDand LEES 1982). ’ Present address: Department of Anthropology, SUNY-Albany, Albany, New York 12222. Genetics 105 985-1002 December, 1983. 986 A. R. ROGERS AND H. C. HARPENDING This implies that drift and migration have more predictable effects on the variation of polygenic characters than on that of single loci. The truth of this assertion is not at all obvious. As before, the unresolved issue is the standard error of the variance predicted under a neutral model. In this paper we wish to investigate both the expectation and standard error of variance of group means about the population mean. Our model assumes additive gene effects and two alleles per locus. We will apply results from the migration matrix model of a structured population to study the effects of migration and drift on neutral characters. We show that gametic disequili- brium, which has been neglected in earlier theory, has a substantial effect on the variance among groups, but that WRIGHT’S (195 1) formula continues to hold in expectation. We derive a formula for the variance of the variance, show that it does not depend on the number of loci contributing to genetic variance and compare the mathematical theory to data on anthropometric characters and dermatoglyphics in Bougainville Island populations. GAMETIC DISEQUILIBRIUM AND QUANTITATIVE CHARACTERS We are primarily interested in the dynamics of local variation over relatively short periods of time. Recent investigations have shown that gametic disequi- librium has a large effect on the short-run behavior of variation within groups (BULMER1980), and we wish to investigate its effect on variation among groups. To simplify the analysis, we use a model of additive effects and two alleles per locus. Complications arising from multiple alleles are ignored in order to deal with those arising from gametic disequilibrium. Let a] denote the average effect of a gene substitution (FISHER 1958) at locus j. We refer to the allele whose average effect is largest as the “plus” allele and denote its frequency in the ith group by p, (= 1 - qI,). The differ- ence between the mean group of i and the population mean is k Y? = E 24%- PI) ]=I where k is the number of loci, and PI is frequency of the “plus” allele in the population as a whole. This can be written economically in matrix notation. Let yT = (4’1 4‘2 4’3 . yg) (1) ar = (a1 a2 a3 . ak), U] = 2a,(p,q,)’/2 Here, y is a vector of deviations of group means from the population mean, g is the number of groups and k is the number of loci. The elements of the vector a measure the contribution of each locus to variance in the quantitative character y. Z is a g by k matrix of standardized deviations of group gene POPULATION STRUCTURE 987 frequencies from population mean gene frequencies. This yields a familiar formula (FALCONER1981, p. 116). aTa = 2 I] 2aj2pJqJ= 20:. j U: is the additive genetic variance expected under panmixia with the same gene frequencies. Using these definitions, the equation for deviations of group means from the population mean can be rewritten as y = Za. The variance of the y,, weighted by relative group sizes, is U,’ = yTwy = aT[ZTwz]a (3) = aTSa W is a diagonal matrix whose jth diagonal entry is wj = nj/N, the relative size of the jth group. S is a k by k matrix whose 9th entry is the normalized covariance of the ith and jth loci. The diagonal entries of S are which is simply WRIGHT’S (1951) Fst for the jth locus. Var(p,) denotes the sample variance of p, about the population mean. F,, is sometimes defined differently, in terms of variance about a theoretical expectation, and this am- biguity has led to a good deal of confusion. We will use a different symbol, Ro, when the former definition is intended. The estimate of Ro generally used is an average of the diagonal entries of S. The off-diagonal terms in S measure the between-groups component of gametic disequilibrium (NEI and LI 1973). If we assume gametic equilibrium and also that the diagonal elements of S are all equal, then S = RoI (where I is the identity matrix) and (3) reduces to U,’ = 2o:Ro (4) This equation was first derived by WRIGHT(1943, 1951), and equivalent for- mulas have been obtained for different models by LANDE (1976, 1979) and CHAKRABORTYand NEI (1982). But there is little reason to assume that finite subdivided populations will be in gametic equilibrium. Disequilibrium is gen- erated by a variety of forces including drift and migration, and its effect on variance within groups can be quike large (BULMER1980). However, it can be shown that (4) holds in expectation in spite of gametic disequilibrium (see APPENDIX). Gametic disequilibrium has no effect on the expectation of variance among groups. Its effect is to produce variation about this expectation. To deal with this effect we must drop the assumption that S is a diagonal matrix. It will be convenient to reexpress (3) in terms of the eigenvalues and eigenvectors of S. Since S is a covariance matrix it is nonnegative definite and can be written as S = XXT, where X is a k by k matrix containing the column 988 A. R. ROGERS AND H. C. HARPENDING eigenvectors of S scaled so that the norm of the ith vector is equal to the square root of its eigenvalue, +!. Then a; = aTXXTaand aTX = (a'x, aTxz aTx3 . a'xk). Let 81 represent the angle subtended by the vectors a and xl. The cosine of 8, is defined by With these definitions the variance of group means can be written as k c,' = 2ag 1 +iCOS%i (5) i= 1 In the sections to follow (5) is used to investigate the variance of c,'. SOME RESULTS FROM SINGLE-LOCUS THEORY Equation (5) expresses among-group variance in terms of the eigenvalues and eigenvectors of the matrix s = [W'/'Z]'[W'/'Z] (6) The nonzero eigenvalues of S, +i, are equal to +i = Mi (7) where PI is an eigenvalue of the related matrix Here, z., represents the ith column of Z. The matrix R, has been studied by HARPENDINGand WARD(1982) and A. R. ROGERS(unpublished results). It is an average of k terms, each associated with a single locus, and each of these terms has the same expectation. Hence, the expectation of the average is the same as that of a single term. This section will summarize some properties of R,, derived from single-locus models. We will then use these properties to investigate the dual matrix S. Our analysis is based on a migration matrix model of population structure (BODMERand CAVALLI-SFORZA1968; SMITH 1969). We assume that drift oc- curs at reproduction and is followed by migration. Migration patterns are characterized by a discrete transition matrix, M. The zjth entry of M is the proportion of individuals in group j (after migration) who originated in group i. We assume that migration does not change relative group sizes, that endo- gamy (the proportion of individuals who do not migrate) is greater than zero in each group and that a gene starting in any group may eventually reach any of the other groups.
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