Expectation and variance of the estimator of the maximized selection response of linear selection indices with normal distribution.

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.

Efficiency of a Constrained Linear Genomic Selection Index To Predict the Net Genetic Merit in Plants.

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.

Optimum and Decorrelated Constrained Multistage Linear Phenotypic Selection Indices Theory.

Some authors have evaluated the unconstrained optimum and decorrelated multistage linear phenotypic selection indices (OMLPSI and DMLPSI, respectively) theory. We extended this index theory to the constrained multistage linear phenotypic selection index context, where we denoted OMLPSI and DMLPSI as OCMLPSI and DCMLPSI, respectively. The OCMLPSI (DCMLPSI) is the most general multistage index and includes the OMLPSI (DMLPSI) as a particular case. The OCMLPSI (DCMLPSI) predicts the individual net genetic merit at different individual ages and allows imposing constraints on the genetic gains to make some traits change their mean values based on a predetermined level, while the rest of them remain without restrictions. The OCMLPSI takes into consideration the index correlation values among stages, whereas the DCMLPSI imposes the restriction that the index correlation values among stages be null. The criteria to evaluate OCMLPSI efficiency vs. DCMLPSI efficiency were that the total response of each index must be lower than or equal to the single-stage constrained linear phenotypic selection index response and that the expected genetic gain per trait values should be similar to the constraints imposed by the breeder. We used one real and one simulated dataset to validate the efficiency of the indices. The results indicated that OCMLPSI accuracy when predicting the selection response and expected genetic gain per trait was higher than DCMLPSI accuracy when predicting them. Thus, breeders should use the OCMLPSI when making a phenotypic selection.