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We describe the main characteristics of two approaches to the linear selection indices theory. The first approach is called whereas the second of them is called In the first approach, the economic weights are fixed and known, whereas in the second approach the economic weights are fixed but unknown. This is the main difference between both approaches and implies that the eigen selection index methods include to the standard linear selection indices because they do not require that the economic weights be known. Both types of indices predict the net genetic merit and maximize the selection response, and they give the breeder an objective criterion to select individuals as parents for the next selection cycle. In addition, in the prediction they can use phenotypic, markers, and genomic information. In both approaches, the indices can be unrestricted, null restricted or predetermined proportional gains and can be used in the context of single-stage or multistage breeding selection schemes. We describe the main characteristics of the two approaches to the linear selection indices theory and we finish this chapter describing the Lagrange multiplier method, which is the main tool to maximize the selection index responses.

The main distinction in the linear phenotypic selection index (LPSI) theory is between the net genetic merit and the LPSI. The net genetic merit is a linear combination of the true unobservable breeding values of the traits weighted by their respective economic values, whereas the LPSI is a linear combination of several observable and optimally weighted phenotypic trait values. It is assumed that the net genetic merit and the LPSI have bivariate normal distribution; thus, the regression of the net genetic merit on the LPSI is linear. The aims of the LPSI theory are to predict the net genetic merit, maximize the selection response and the expected genetic gains per trait (or multi-trait selection response), and provide the breeder with an objective rule for evaluating and selecting parents for the next selection cycle based on several traits. The selection response is the mean of the progeny of the selected parents, whereas the expected genetic gain per trait, or multi-trait selection response, is the population means of each trait under selection of the progeny of the selected parents. The LPSI allows extra merit in one trait to offset slight defects in another; thus, with its use, individuals with very high merit in one trait are saved for breeding even when they are slightly inferior in other traits. This chapter describes the LPSI theory and practice. We illustrate the theoretical results of the LPSI using real and simulated data. We end this chapter with a brief description of the quadratic selection index and its relationship with the LPSI.

The linear phenotypic selection index (LPSI), the null restricted LPSI (RLPSI), and the predetermined proportional gains LPSI (PPG-LPSI) are the main phenotypic selection indices used to predict the net genetic merit and select parents for the next selection cycle. The LPSI is an unrestricted index, whereas the RLPSI and the PPG-LPSI allow restrictions equal to zero and predetermined proportional gain restrictions respectively to be imposed on the expected genetic gain values of the trait to make some traits change their mean values based on a predetermined level while the rest of the trait means remain without restrictions. One additional restricted index is the desired gains LPSI (DG-LPSI), which does not require economic weights and, in a similar manner to the PPG-LPSI, allows restrictions to be imposed on the expected genetic gain values of the trait to make some traits change their mean values based on a predetermined level. The aims of RLPSI and PPG-LPSI are to maximize the selection response, the expected genetic gains per trait, and provide the breeder with an objective rule for evaluating and selecting parents for the next selection cycle based on several traits. This chapter describes the theory and practice of the RLPSI, PPG-LPSI, and DG-LPSI. We show that the PPG-LPSI is the most general index and includes the LPSI and the RLPSI as particular cases. Finally, we describe the DG-LPSI as a modification of the PPG-LPSI. We illustrate the theoretical results of all the indices using real and simulated data.

There are two main linear marker selection indices employed in marker-assisted selection (MAS) to predict the net genetic merit and to select individual candidates as parents for the next generation: the linear marker selection index (LMSI) and the genome-wide LMSI (GW-LMSI). Both indices maximize the selection response, the expected genetic gain per trait, and the correlation with the net genetic merit; however, applying the LMSI in plant or animal breeding requires genotyping the candidates for selection; performing a linear regression of phenotypic values on the coded values of the markers such that the selected markers are statistically linked to quantitative trait loci that explain most of the variability in the regression model; constructing the marker score, and combining the marker score with phenotypic information to predict and rank the net genetic merit of the candidates for selection. On the other hand, the GW-LMSI is a single-stage procedure that treats information at each individual marker as a separate trait. Thus, all marker information can be entered together with phenotypic information into the GW-LMSI, which is then used to predict the net genetic merit and select candidates. We describe the LMSI and GW-LMSI theory and show that both indices are direct applications of the linear phenotypic selection index theory to MAS. Using real and simulated data we validated the theory of both indices.

The linear genomic selection index (LGSI) is a linear combination of genomic estimated breeding values (GEBVs) used to predict the individual net genetic merit and select individual candidates from a nonphenotyped testing population as parents of the next selection cycle. In the LGSI, phenotypic and marker data from the training population are fitted into a statistical model to estimate all individual available genome marker effects; these estimates can then be used in subsequent selection cycles to obtain GEBVs that are predictors of breeding values in a testing population for which there is only marker information. The GEBVs are obtained by multiplying the estimated marker effects in the training population by the coded marker values obtained in the testing population in each selection cycle. Applying the LGSI in plant or animal breeding requires the candidates to be genotyped for selection to obtain the GEBV, and predicting and ranking the net genetic merit of the candidates for selection using the LGSI. We describe the LGSI and show that it is a direct application of the linear phenotypic selection index theory in the genomic selection context; next, we present the combined LGSI (CLGSI), which uses phenotypic and GEBV information jointly to predict the net genetic merit. The CLGSI can be used only in training populations when there are phenotypic and maker information, whereas the LGSI is used in testing populations where there is only marker information. We validate the theoretical results of the LGSI and CLGSI using real and simulated data.

The constrained linear genomic selection indices are null restricted and predetermined proportional gain linear genomic selection indices (RLGSI and PPG-LGSI respectively), which are a linear combination of genomic estimated breeding values (GEBVs) to predict the net genetic merit. They are the results of a direct application of the restricted and the predetermined proportional gain linear phenotypic selection index theory to the genomic selection context. The RLGSI can be extended to a combined RLGSI (CRLGSI) and the PPG-LGSI can be extended to a combined PPG-LGSI (CPPG-LGSI); the latter indices use phenotypic and GEBV information jointly in the prediction of net genetic merit. The main difference between the RLGSI and PPG-LGSI with respect to the CRLGSI and the CPPG-LGSI is that although the RLGSI and PPG-LGSI are useful in a testing population where there is only marker information, the CRLGSI and CPPG-LGSI can be used only in training populations when there are joint phenotypic and marker information. The RLGSI and CRLGSI allow restrictions equal to zero to be imposed on the expected genetic advance of some traits, whereas the PPG-LGSI and CPPG-LGSI allow predetermined proportional restriction values to be imposed on the expected trait genetic gains to make some traits change their mean values based on a predetermined level. We describe the foregoing four indices and we validated their theoretical results using real and simulated data.

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.

The three main linear phenotypic eigen selection index methods are the eigen selection index method (ESIM), the restricted ESIM (RESIM) and the predetermined proportional gain ESIM (PPG-ESIM). The ESIM is an unrestricted index, but the RESIM and PPG-ESIM allow null and predetermined restrictions respectively to be imposed on the expected genetic gains of some traits, whereas the rest remain without any restrictions. These indices are based on the canonical correlation, on the singular value decomposition, and on the linear phenotypic selection indices theory. We extended the ESIM theory to the molecular-assisted and genomic selection context to develop a molecular ESIM (MESIM), a genomic ESIM (GESIM), and a genome-wide ESIM (GW-ESIM). Also, we extend the RESIM and PPG-ESIM theory to the restricted genomic ESIM (RGESIM), and to the predetermined proportional gain genomic ESIM (PPG-GESIM) respectively. The latter five indices use marker and phenotypic information jointly to predict the net genetic merit of the candidates for selection, but although MESIM uses only statistically significant markers linked to quantitative trait loci, the GW-ESIM uses all genome markers and phenotypic information and the GESIM, RGESIM, and PPG-GESIM use the genomic estimated breeding values and the phenotypic values to predict the net genetic merit. Using real and simulated data, we validated the theoretical results of all five indices.

Multistage linear selection indices select individual traits available at different times or stages and are applied mainly in animals and tree breeding, where the traits under consideration become evident at different ages. The main indices are: the unrestricted, the restricted, and the predetermined proportional gain selection index. The restricted and predetermined proportional gain indices allow null and predetermined restrictions to be imposed on the trait expected genetic gain (or multi-trait selection response) values, whereas the rest of the traits remain changed without any restriction. The three indices can use phenotypic, genomic, or both sets of information to predict the unobservable net genetic merit values of the candidates for selection and all of them maximize the selection response, the expected genetic gain for each trait, have maximum accuracy, are the best predictor of the net genetic merit, and provide the breeder with an objective rule for evaluating and selecting several traits simultaneously. The theory of the foregoing indices is based on the independent culling method and on the linear phenotypic selection index, and is described in this chapter in the phenotypic and genomic selection context. Their theoretical results are validated in a two-stage breeding selection scheme using real and simulated data.

Stochastic simulation can contribute to a better understanding of the problem, and has already been successfully applied to evaluate other breeding scenarios. Despite all the theories developed in this book concerning different types of indices, including phenotypic data and/or data on molecular markers, no examples have been presented showing the long-term behavior of different indices. The objective of this chapter is to present some results and insights into the in silico (computer simulation) performance comparison of over 50 selection cycles of a recurrent and generic population breeding program with different selection indices, restricted and unrestricted. The selection indices included in this stochastic simulation were the linear phenotypic selection index (LPSI), the eigen selection index method (ESIM), the restrictive LPSI, and the restrictive ESIM.