Abstract
Based on the canonical correlation, on the singular value decomposition (SVD), and on the linear phenotypic selection indices theory, we describe the eigen selection index method (ESIM), the restricted ESIM (RESIM), and the predetermined proportional gain ESIM (PPG-ESIM), which use only phenotypic information to predict the net genetic merit. The ESIM is an unrestricted linear selection index, but the RESIM and PPG-ESIM are linear selection indices that allow null and predetermined restrictions respectively to be imposed on the expected genetic gains of some traits, whereas the rest remain without any restrictions. The aims of the three indices are to predict the unobservable net genetic merit values of the candidates for selection, maximize the selection response, and the accuracy, and provide the breeder with an objective rule for evaluating and selecting several traits simultaneously. Their main characteristics are: they do not require the economic weights to be known, the first multi-trait heritability eigenvector is used as its vector of coefficients; and because of the properties associated with eigen analysis, it is possible to use the theory of similar matrices to change the direction and proportion of the expected genetic gain values without affecting the accuracy. We describe the foregoing three indices and validate their theoretical results using real and simulated data.
Highlights
Based on the canonical correlation, on the singular value decomposition (SVD), and on the linear phenotypic selection indices theory, we describe the eigen selection index method (ESIM), the restricted ESIM (RESIM), and the predetermined proportional gain ESIM (PPG-ESIM), which use only phenotypic information to predict the net genetic merit
The ESIM index can be written as I 1⁄4 b0y, where b0 1⁄4 [b1 b2 Á Á Á bt] is the unknown index vector of coefficients, t is the number of traits, and y0 1⁄4 1⁄2 y1 y2 Á Á Á yt is a known vector of trait phenotypic values
In the ESIM, the first eigenvector of matrix PÀ1C should be used on IE 1⁄4 b0E1 y; the first eigenvalue (λ21 ) and bE1 of PÀ1C should be used on the ESIM selection response and on the ESIM expected genetic gain per trait, because, in this case, the ESIM has maximum accuracy compared with other indices, such as the linear phenotypic selection index (LPSI)
Summary
Gt is the unknown vector of true breeding values for an individual and w0 1⁄4 1⁄2 w1 w2 . Wt is a vector of unknown economic weights. In the context of the LPSI w is a known and fixed vector of economic weights, in the ESIM w is fixed, but unknown and its values must be estimated in each selection cycle. This latter assumption is the fundamental difference between the ESIM and the LPSI and implies that the ESIM is more general than the LPSI. When w is known, the LPSI and ESIM give the same results
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