Scales are linear combinations of variables with coefficients that add up to zero and have a similar meaning to “contrast” in the analysis of variance. Scales are necessary in order to incorporate genomic information into relationship matrices for genomic selection. Statistical and biological parameterizations using scales under different assumptions have been proposed to construct alternative genomic relationship matrices. Except for the natural and orthogonal interactions approach (NOIA) method, current methods to construct relationship matrices assume Hardy–Weinberg equilibrium (HWE). The objective of this paper is to apply vector algebra to center and scale relationship matrices under non-HWE conditions, including orthogonalization by the Gram-Schmidt process. Vector space algebra provides an evaluation of current orthogonality between additive and dominance vectors of additive and dominance scales for each marker. Three alternative methods to center and scale additive and dominance relationship matrices based on the Gram-Schmidt process (GSP-A, GSP-D, and GSP-N) are proposed. GSP-A removes additive-dominance co-variation by first fitting the additive and then the dominance scales. GSP-D fits scales in the opposite order. We show that GSP-A is algebraically the same as the NOIA model. GSP-N orthonormalizes the additive and dominance scales that result from GSP-A. An example with genotype information on 32,645 single nucleotide polymorphisms from 903 Large-White × Landrace crossbred pigs is used to construct existing and newly proposed additive and dominance relationship matrices. An exact test for departures from HWE showed that a majority of loci were not in HWE in crossbred pigs. All methods, except the one that assumes HWE, performed well to attain an average of diagonal elements equal to one and an average of off diagonal elements equal to zero. Variance component estimation for a recorded quantitative phenotype showed that orthogonal methods (NOIA, GSP-A, GSP-N) can adjust for the additive-dominance co-variation when estimating the additive genetic variance, whereas GSP-D does it when estimating dominance components. However, different methods to orthogonalize relationship matrices resulted in different proportions of additive and dominance components of variance. Vector space methodology can be applied to measure orthogonality between vectors of additive and dominance scales and to construct alternative orthogonal models such as GSP-A, GSP-D and an orthonormal model such as GSP-N. Under non-HWE conditions, GSP-A is algebraically the same as the previously developed NOIA model.

中文翻译：

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非Hardy-Weinberg平衡条件下缩放和居中关系矩阵的向量空间代数

尺度是变量的线性组合，其系数加起来为零，与方差分析中的“对比”具有相似的含义。为了将基因组信息整合到基因组选择的关系矩阵中，尺度是必要的。已经提出在不同假设下使用尺度的统计和生物参数化来构建替代基因组关系矩阵。除了自然和正交交互方法 (NOIA) 方法外，当前构建关系矩阵的方法都假设哈代-温伯格平衡 (HWE)。本文的目的是将向量代数应用于非 HWE 条件下的中心和尺度关系矩阵，包括通过 Gram-Schmidt 过程进行正交化。向量空间代数提供了对每个标记的加性和优势尺度的加性和优势向量之间的当前正交性的评估。提出了三种基于 Gram-Schmidt 过程（GSP-A、GSP-D 和 GSP-N）对加性和优势关系矩阵进行中心化和缩放的替代方法。GSP-A 通过首先拟合加性然后是优势尺度来消除加性优势协变。GSP-D 以相反的顺序拟合尺度。我们证明 GSP-A 在代数上与 NOIA 模型相同。GSP-N 对由 GSP-A 产生的加性和优势尺度进行正交归一化。以来自 903 头大白 × 长白杂交猪的 32,645 个单核苷酸多态性的基因型信息为例，用于构建现有和新提出的加性和显性关系矩阵。对偏离 HWE 的精确测试表明，杂交猪的大多数基因座不在 HWE 中。除了假设 HWE 的方法外，所有方法都表现良好，可以使对角线元素的平均值等于 1，非对角线元素的平均值等于 0。记录的定量表型的方差分量估计表明，正交方法（NOIA、GSP-A、GSP-N）在估计加性遗传方差时可以调整加性优势协变，而 GSP-D 在估计优势分量时可以调整. 然而，正交化关系矩阵的不同方法导致方差的加性和优势分量的比例不同。向量空间方法可用于测量加性和优势尺度向量之间的正交性，并构建替代正交模型，例如 GSP-A、GSP-D 和正交模型，例如 GSP-N。在非 HWE 条件下，GSP-A 在代数上与之前开发的 NOIA 模型相同。