Vector space algebra for scaling and centering relationship matrices under non-Hardy-Weinberg equilibrium conditions

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.