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

Abstract

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.

Fig. 1

Histograms and quantile–quantile plots of the estimated LPSI (Fig. 1a, d, respectively) and CLPSI (Fig. 1b, c, respectively) values for a real dataset with four traits and 247 genotypes