{"id":10207,"date":"2010-11-23T03:05:50","date_gmt":"2010-11-23T03:05:50","guid":{"rendered":"http:\/\/www.mixedracestudies.org\/wordpress\/?p=10207"},"modified":"2010-11-23T03:12:19","modified_gmt":"2010-11-23T03:12:19","slug":"the-estimation-of-admixture-in-racial-hybrids","status":"publish","type":"post","link":"https:\/\/mixedracestudies.org\/wp\/?p=10207","title":{"rendered":"The Estimation of Admixture in Racial Hybrids"},"content":{"rendered":"

The Estimation of Admixture in Racial Hybrids<\/a><\/em><\/strong><\/p>\n

Annals of Human Genetics<\/a>
\nVolume 35, Issue 1<\/a> (July 1971)
\npages 9\u201317
\nDOI: 10.1111\/j.1469-1809.1956.tb01373.x<\/a><\/p>\n

Robert C. Elston<\/a>,<\/strong> Professor & Chair, Distinguished University Professor
\nCase Western Reserve University<\/em><\/p>\n

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<\/a> 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.<\/p>\n

Robers &. Hiorns (1962, 1965) proposed a least-squares<\/a> solution to the problem, and Krieger et al<\/em>. (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.<\/p>\n

Least-Squares Method<\/strong><\/p>\n

Suppose ther are p (> 1)<\/em> parental populations and for each we have gene frequency estimates of the same k<\/em> genes.\u00a0 Let X = (xij<\/sub>)<\/em> be a k x p<\/em> matrix, xij<\/sub> being the estimate of the i<\/em>th gene frequency in the j<\/em>th parental population. Let the k x 1<\/em> vector y<\/em> 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 j<\/em>th parental populatio be \u00b5<\/em>ij<\/sub> the jth element of th p x 1<\/em> vector \u00b5. \u00a0Then if the estimates are all exactly equal to the gene frequencies; and if the k<\/em> chosen genes represent perfectly all the genes for which there has been no selection or drift, y-X\u00b5<\/em> = 0, where 0 is the null vector. The least squares estimate of\u00a0\u00b5 is that value of \u00b5, m say, which minimizes the sum of squares of the diserepancies given by y-X\u00b5, i.e.\u00a0which minimizes (y \u2014 X\u00b5)\u2032(y \u2014 X\u00b5), where the prime denotes transposition. The least squares estimate is accordingly<\/p>\n

m = (X\u2032X)-1<\/sup> X\u2032y provided X\u2032X\u00a0is non-singular.<\/p>\n

Now it should be noted that the\u00a0k<\/em> 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<\/em> and N<\/em> add to unity, and it is impossible to estimate a gene\u00a0 frequency for M<\/em> without at the same time implicitly estimating a gene frequency for N<\/em>. 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…<\/p>\n

Read or purchase the article here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

The Estimation of Admixture in Racial Hybrids Annals of Human Genetics Volume 35, Issue 1 (July 1971) pages 9\u201317 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,2039,8],"tags":[4484,4482,4483,4485,4486],"class_list":["post-10207","post","type-post","status-publish","format-standard","hentry","category-articles","category-health-medicine","category-media-archive","tag-annals-of-human-genetics","tag-r-c-elston","tag-r-elston","tag-robert-c-elston","tag-robert-elston"],"_links":{"self":[{"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/10207","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10207"}],"version-history":[{"count":0,"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/10207\/revisions"}],"wp:attachment":[{"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mixedracestudies.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}