Characteristics of restricted selection indices and geometrical interpretation of restricted breeding values

Restricted selection is used to control genetic changes in one or more characters. Three main selection indices are adopted for this purpose. First, Kempthorne's index is used to maximize aggregate breeding value (BV) with changes in some traits restricted to zero; second, Harville's index is used to maximize aggregate BV with proportional changes for some traits; and third, Yamada's index is mathematically used to achieve the relative desired changes for all traits. Kempthorne's index is equivalent to Harville's index. However, the relationship between Kempthorne's and Yamada's indices has not been clarified. In addition, the characteristics of restricted selection indices and the relationship between BV and restricted BV (RBV) are also unknown. The aim of this study was to clarify the characteristics of restricted selection indices and describe the relationship between BV and RBV by using linear algebra and geometric techniques, respectively. First, I proved that Yamada's index is part of Kempthorne's index. Second, I investigated the relationship between BVs that were estimated using an ordinary selection index (EBVs) and RBVs estimated using a restricted selection index (ERBVs) and proved that the ERBVs of the restricted traits are proportional to the relative desired changes. Third, I proved that RBV is represented by a linear function of BV and geometrically represented the relationship between BV and RBV. In this study, new findings on restricted selection indices and RBV were obtained. This useful clarification of the relationship between BV and RBV will make it possible to evaluate the response to selection using not only a restricted selection index, but also a restricted BLUP in computer simulation studies.