A linear selection index (LSI) can be a linear combination of phenotypic values, marker scores, and genomic estimated breeding values (GEBVs); phenotypic values and marker scores; or phenotypic values and GEBVs jointly. The main objective of the LSI is to predict the net genetic merit (H), which is a linear combination of unobservable individual traits’ breeding values, weighted by the trait economic values; thus, the target of LSI is not a parameter but rather the unobserved random H values. The LSI can be single-stage or multi-stage, where the latter are methods for selecting one or more individual traits available at different times or stages of development in both plants and animals. Likewise, LSIs can be either constrained or unconstrained. A constrained LSI imposes predetermined genetic gain on expected genetic gain per trait and includes the unconstrained LSI as particular cases. The main LSI parameters are the selection response, the expected genetic gain per trait, and its correlation with H. When the population mean is zero, the selection response and expected genetic gain per trait are, respectively, the conditional mean of H and the genotypic values, given the LSI values. The application of LSI theory is rapidly diversifying; however, because LSIs are based on the best linear predictor and on the canonical correlation theory, the LSI theory can be explained in a simple form. We provided a review of the statistical theory of the LSI from phenotypic to genomic selection showing their relationships, advantages, and limitations, which should allow breeders to use the LSI theory confidently in breeding programs.

中文翻译：

---

从表型到基因组选择的线性选择指标的统计理论

线性选择指数 (LSI) 可以是表型值、标记分数和基因组估计育种值 (GEBV) 的线性组合；表型值和标记分数；或表型值和 GEBV 联合。LSI 的主要目标是预测净遗传价值 ( H )，它是不可观察的个体性状育种价值的线性组合，由性状经济价值加权；因此，LSI 的目标不是参数，而是未观察到的随机H价值观。LSI 可以是单阶段或多阶段的，后者是用于选择在植物和动物的不同发育时期或发育阶段可用的一种或多种个体性状的方法。同样，LSI 可以是受约束的或不受约束的。受约束的 LSI 将预定的遗传增益强加于每个性状的预期遗传增益，并将不受约束的 LSI 包括为特定情况。LSI 的主要参数是选择响应、每个性状的预期遗传增益及其与H的相关性。当种群平均值为零时，每个性状的选择响应和预期遗传增益分别是H的条件平均值和基因型值，给定 LSI 值。LSI理论的应用正在迅速多样化；但是，由于 LSI 基于最佳线性预测器和典型相关理论，因此 LSI 理论可以用简单的形式来解释。我们对 LSI 从表型到基因组选择的统计理论进行了回顾，展示了它们之间的关系、优势和局限性，这应该允许育种者在育种计划中自信地使用 LSI 理论。