*Theor Appl Genet ; 2020 Jun 20.*

##### RESUMO

KEY MESSAGE: The expectation and variance of the estimator of the maximized index selection response allow the breeders to construct confidence intervals and to complete the analysis of a selection process. The maximized selection response and the correlation of the linear selection index (LSI) with the net genetic merit are the main criterion to compare the efficiency of any LSI. The estimator of the maximized selection response is the square root of the variance of the estimated LSI values multiplied by the selection intensity. The expectation and variance of this estimator allow the breeder to construct confidence intervals and determine the appropriate sample size to complete the analysis of a selection process. Assuming that the estimated LSI values have normal distribution, we obtained those two parameters as follows. First, with the Fourier transform, we found the distribution of the variance of the estimated LSI values, which was a Gamma distribution; therefore, the expectation and variance of this distribution were the expectation and variance of the variance of the estimated LSI values. Second, with these results, we obtained the expectation and the variance of the estimator of the selection response using the Delta method. We validated the theoretical results in the phenotypic selection context using real and simulated dataset. With the simulated dataset, we compared the LSI efficiency when the genotypic covariance matrix is known versus when this matrix is estimated; the differences were not significant. We concluded that our results are valid for any LSI with normal distribution and that the method described in this work is useful for finding the expectation and variance of the estimator of any LSI response in the phenotypic or genomic selection context.

*G3 (Bethesda) ; 10(6): 2087-2101, 2020 Jun 01.*

##### RESUMO

A combined multistage linear genomic selection index (CMLGSI) is a linear combination of phenotypic and genomic estimated breeding values useful for predicting the individual net genetic merit, which in turn is a linear combination of the true unobservable breeding values of the traits weighted by their respective economic values. The CMLGSI is a cost-saving strategy for improving multiple traits because the breeder does not need to measure all traits at each stage. The optimum (OCMLGSI) and decorrelated (DCMLGSI) indices are the main CMLGSIs. Whereas the OCMLGSI takes into consideration the index correlation values among stages, the DCMLGSI imposes the restriction that the index correlation values among stages be zero. Using real and simulated datasets, we compared the efficiency of both indices in a two-stage context. The criteria we applied to compare the efficiency of both indices were that the total selection response of each index must be lower than or equal to the single-stage combined linear genomic selection index (CLGSI) response and that the correlation of each index with the net genetic merit should be maximum. Using four different total proportions for the real dataset, the estimated total OCMLGSI and DCMLGSI responses explained 97.5% and 90%, respectively, of the estimated single-stage CLGSI selection response. In addition, at stage two, the estimated correlations of the OCMLGSI and the DCMLGSI with the net genetic merit were 0.84 and 0.63, respectively. We found similar results for the simulated datasets. Thus, we recommend using the OCMLGSI when performing multistage selection.

*G3 (Bethesda) ; 9(12): 3981-3994, 2019 12 03.*

##### RESUMO

The constrained linear genomic selection index (CLGSI) is a linear combination of genomic estimated breeding values useful for predicting the net genetic merit, which in turn is a linear combination of true unobservable breeding values of the traits weighted by their respective economic values. The CLGSI is the most general genomic index and allows imposing constraints on the expected genetic gain per trait to make some traits change their mean values based on a predetermined level, while the rest of them remain without restrictions. In addition, it includes the unconstrained linear genomic index as a particular case. Using two real datasets and simulated data for seven selection cycles, we compared the theoretical results of the CLGSI with the theoretical results of the constrained linear phenotypic selection index (CLPSI). The criteria used to compare CLGSI vs. CLPSI efficiency were the estimated expected genetic gain per trait values, the selection response, and the interval between selection cycles. The results indicated that because the interval between selection cycles is shorter for the CLGSI than for the CLPSI, CLGSI is more efficient than CLPSI per unit of time, but its efficiency could be lower per selection cycle. Thus, CLGSI is a good option for performing genomic selection when there are genotyped candidates for selection.

##### Assuntos

Genômica , Seleção Genética , Zea mays/genética , Simulação por Computador , Cruzamentos Genéticos , Bases de Dados Genéticas , Genoma de Planta , Fenótipo , Melhoramento Vegetal , Característica Quantitativa Herdável*G3 (Bethesda) ; 5(10): 2155-64, 2015 Aug 18.*

##### RESUMO

A genomic selection index (GSI) is a linear combination of genomic estimated breeding values that uses genomic markers to predict the net genetic merit and select parents from a nonphenotyped testing population. Some authors have proposed a GSI; however, they have not used simulated or real data to validate the GSI theory and have not explained how to estimate the GSI selection response and the GSI expected genetic gain per selection cycle for the unobserved traits after the first selection cycle to obtain information about the genetic gains in each subsequent selection cycle. In this paper, we develop the theory of a GSI and apply it to two simulated and four real data sets with four traits. Also, we numerically compare its efficiency with that of the phenotypic selection index (PSI) by using the ratio of the GSI response over the PSI response, and the PSI and GSI expected genetic gain per selection cycle for observed and unobserved traits, respectively. In addition, we used the Technow inequality to compare GSI vs. PSI efficiency. Results from the simulated data were confirmed by the real data, indicating that GSI was more efficient than PSI per unit of time.

##### Assuntos

Simulação por Computador , Modelos Genéticos , Seleção Genética , Algoritmos , Conjuntos de Dados como Assunto*Genetics ; 180(1): 547-57, 2008 Sep.*

##### RESUMO

The traditional molecular selection index (MSI) employed in marker-assisted selection maximizes the selection response by combining information on molecular markers linked to quantitative trait loci (QTL) and phenotypic values of the traits of the individuals of interest. This study proposes an MSI based on an eigenanalysis method (molecular eigen selection index method, MESIM), where the first eigenvector is used as a selection index criterion, and its elements determine the proportion of the trait's contribution to the selection index. This article develops the theoretical framework of MESIM. Simulation results show that the genotypic means and the expected selection response from MESIM for each trait are equal to or greater than those from the traditional MSI. When several traits are simultaneously selected, MESIM performs well for traits with relatively low heritability. The main advantages of MESIM over the traditional molecular selection index are that its statistical sampling properties are known and that it does not require economic weights and thus can be used in practical applications when all or some of the traits need to be improved simultaneously.