The Estimation of Admixture in Racial Hybrids

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The Estimation of Admixture in Racial Hybrids

Annals of Human Genetics
Volume 35, Issue 1 (July 1971)
pages 9–17
DOI: 10.1111/j.1469-1809.1956.tb01373.x

Robert C. Elston, Professor & Chair, Distinguished University Professor
Case Western Reserve University

When a racial hybrid population has arisen from the intermarriage of two or more parental populations, a problem of interest is to determine what the relative contributions are from each parental population to the hybrid. Various distance measures have been proposed whereby, on the basis of several traits, the distance between the hybrid and each of the parental populations can be estimated: these distances are then sometimes interpreted, as a first approximation, as being inversely proportional to the parental contributions (Pollitzer, 1964). In the particular case that all the traits considered are discrete in nature and each is determined by alleles at a single locus (or system of tightly linked loci), it is possible to estimate the parental contributions more directly. It in the purpose of this paper to reconsider two main methods of doing this when the traits involved are determined by a random set of independently assorting loci.

Robers &. Hiorns (1962, 1965) proposed a least-squares solution to the problem, and Krieger et al. (1965) gave a maximum-likelihood solution. Both methods, as given by these authors, can be improved. We shall here restate both methods, using a common notation, and point out the improvements possible; furthermore, some resumes of using these method will also be presented, so that the methods may be compared empirically.

Least-Squares Method

Suppose ther are p (> 1) parental populations and for each we have gene frequency estimates of the same k genes.  Let X = (xij) be a k x p matrix, xij being the estimate of the ith gene frequency in the jth parental population. Let the k x 1 vector y have as its elements the corresponding gene frequency estimates in the hybrid population; and let the proportion of the hybrid population’s genes that come from the jth parental populatio be µij the jth element of th p x 1 vector µ.  Then if the estimates are all exactly equal to the gene frequencies; and if the k chosen genes represent perfectly all the genes for which there has been no selection or drift, y-Xµ = 0, where 0 is the null vector. The least squares estimate of µ is that value of µ, m say, which minimizes the sum of squares of the diserepancies given by y-Xµ, i.e. which minimizes (y — Xµ)′(y — Xµ), where the prime denotes transposition. The least squares estimate is accordingly

m = (X′X)-1 X′y provided X′X is non-singular.

Now it should be noted that the k genes fall into allelic systems, the sum of the gene frequencies for each syatem being unity in each population. Thus, for example, the gene frequency for M and N add to unity, and it is impossible to estimate a gene  frequency for M without at the same time implicitly estimating a gene frequency for N. When Roberts & Hiorns (1963) use (1) to obtain least squares estimates they eliminate one allele from each system, so that tho rows of X and y can no longer be grouped by system with the column totals for each group adding…

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