Optimum and Decorrelated Constrained Multistage Linear Phenotypic Selection Indices Theory

Abstract

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.

Fig. 1

Theoretical relationship between the truncation points (u), the proportion retained (p), and the density values [z(u)] of the truncation points (after Ceron-Rojas et al., 2019).

Fig. 2

Distribution of the total estimated optimum and decorrelated constrained multistage linear phenotypic selection index (OCMLPSI and DCMLPSI, respectively) selection response values for a real dataset, and the fixed total proportion retained (p) = 0.05 and 0.10.

Fig. 3

Histograms of the estimated optimum and decorrelated constrained multistage linear phenotypic selection index (OCMLPSI and DCMLPSI, respectively) values at Stage 2 for simulated dataset in Cycle 2, and the fixed total proportion retained (p) = q₁q₂ = 0.05 and 0.30when the number of genotypes was 110 (Fig. 3a) and 270 (Fig. 3b) for OCMLPSI, and 30 (Fig. 3c) and 155 (Fig. 3d) for DCMLPSI.