### abstract

- The most general linear phenotypic selection index (PSI) is the predetermined proportional gains phenotypic selection index (PPG-PSI) that allows imposing restrictions on the trait expected genetic gain values to make some traits change their mean values based on a predetermined level, while the rest of the traits remain without restrictions. However, due to the increasing number of restricted traits: (i) PPG-PSI accuracy decreases; (ii) the proportional constant associated with this index can be negative, in which case, its results have no meaning in practice; and (iii) the PPG-PSI can shift the population means in the opposite direction to the predetermined desired direction. Based on the eigen selection index method (ESIM), we propose a PPG-ESIM that does not require a proportional constant, and due to the properties associated with eigen analysis, it is possible to use the theory of similar matrices to change the direction of the eigenvector values without affecting PPG-ESIM accuracy, which helps to eliminate the problem indicated in the third point above, associated with the standard PPG-PSI. The PPG-ESIM uses the first eigenvector as its vector of coefficients, and the first eigenvalue in the selection response. Two simulated and one real data set, each with four traits, were used to validate PPG-ESIM efficiency vs. PPG-PSI efficiency; the simulated and real results indicated that PPG-ESIM efficiency was higher than PPG-PSI efficiency. We concluded that PPG-ESIM is an efficient selection index that can be used in any selection program as a good alternative to PPG-PSI
- The most general linear phenotypic selection index (PSI) is the predetermined proportional gains phenotypic selection index (PPG-PSI) that allows imposing restrictions on the trait expected genetic gain values to make some traits change their mean values based on a predetermined level, while the rest of the traits remain without restrictions. However, due to the increasing number of restricted traits: (i) PPGPSI accuracy decreases; (ii) the proportional constant associated with this index can be negative, in which case, its results have no meaning in practice; and (iii) the PPG-PSI can shift the population means in the opposite direction to the predetermined desired direction. Based on the eigen selection index method (ESIM), we propose a PPG-ESIM that does not require a proportional constant, and due to the properties associated with eigen analysis, it is possible to use the theory of similar matrices to change the direction of the eigenvector values without affecting PPG-ESIM accuracy, which helps to eliminate the problem indicated in the third point above, associated with the standard PPG-PSI. The PPG-ESIM uses the first eigenvector as its vector of coefficients, and the first eigenvalue in the selection response. Two simulated and one real data set, each with four traits, were used to validate PPG-ESIM efficiency vs. PPG-PSI efficiency; the simulated and real results indicated that PPG-ESIM efficiency was higher than PPGPSI efficiency. We concluded that PPG-ESIM is an efficient selection index that can be used in any selection program as a good alternative to PPG-PSI.