R-matrix in Constructing a Selection Index
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概要
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A modified alternative procedure in constructing a selection index by introducing a R-matrix is described. The technique is essentially the same in principle to that of HENDERSON (1963) but is more straightforward. The advantage of this methos is that the normal equation is derived routinely. The necessary modification of the formulae in computing a restricted index, the correlation of index and the aggregate breeding value, and the correlated response in each component trait are also presented with a numerical example.The essential part of the procesure is outlined briefly as:The aggregate breeding value and the selection index are defined as H=∑mj=1ajGj, I=∑ni=1biXi, then the normal equation is Pb=H. Suppose Xi is the average phenotypic value of qi relatives in which qik (k=1, 2, …, pi) are the same group, say full-sib, σXiH=Cov[∑qiα=1X*iα/qi, Σmj=1ajGj]=Σpik=1qikγGik/qiΣmj=1ajσGijwhere γGik is Wright's coefficient of relationship, and σGij is the genetic covariance between the ith record in the index and the jth genotype in the aggregate breeding value on the individual basis.The normal equation is derived as follows:[σ2<SUB>X1</SUB> σ<SUB>X12</SUB>…σ<SUB>X1n</SUB>σ<SUB>X21</SUB> σ2<SUB>X2</SUB>…σ<SUB>X2n</SUB>σ<SUB>Xn1</SUB> σ<SUB>Xn2</SUB>…σ2<SUB>Xn</SUB>][b1 b2… bn]=[γ1 0 γ2 0 γn][σ<SUB>G11</SUB> σ<SUB>G12</SUB>… σ<SUB>G1m</SUB>σ<SUB>G21</SUB> σ<SUB>G22</SUB>… σ<SUB>G2m</SUB>σ<SUB>Gn1</SUB> σ<SUB>Gn2</SUB>… σ<SUB>Gnm</SUB>][a1 a2… am]Pn×n bn×1 Rn×n Gn×m am×1The first matrix in the right hand side may be called the diagonal matrix of relationship coefficient which has a linear combination of wright's coefficients of relationship (γi=Σpik=1qikγGik/qi). If all variates in the index are taken from candidate's own records of several traits, R-matrix is equal to unit matrix, thus the above equation comes to Pb=Ga.To have a restricted selection index using candidate's own and its relative's records of several traits, the normal equation must be modified as:
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