Given the pedigree numerator relationship matrix A over all individuals, a genomic relationship matrix G over the genotyped subset (block 1) and the non-genotyped remainder (block 2), the joint matrix of Christensen and Lund (2010), eq. (4) (the H of Legarra, Aguilar and Misztal 2009 and Aguilar et al. 2010), is \(H_{11} = G_w\), \(H_{12} = G_w A_{11}^{-1} A_{12}\), \(H_{21} = A_{21} A_{11}^{-1} G_w\), \(H_{22} = A_{22} + A_{21} A_{11}^{-1} (G_w - A_{11}) A_{11}^{-1} A_{12}\), with \(G_w = (1-w) G + w A_{11}\) blending a polygenic fraction w (their sec. The combined genetic effect; w = 0 gives raw eq. 4). The inverse is the sparse update of their eq. (8) (eq. 6 at w = 0): \(H^{-1} = A^{-1} + [G_w^{-1} - A_{11}^{-1}, 0; 0, 0]\).
Arguments
- A
Pedigree numerator relationship matrix (n x n).
- G
Genomic relationship matrix over the genotyped subset, in the order of
genotyped.- genotyped
1-based indices of genotyped individuals within A.
- w
Polygenic blending weight in [0, 1); eq. (8) requires w greater than 0 for guaranteed invertibility when G is singular.
Details
Limiting cases exercised in the tests: G = A11 gives H = A exactly; with every individual genotyped, H = Gw.
References
Christensen, O. F. and Lund, M. S. (2010). Genomic prediction when some animals are not genotyped. Genetics Selection Evolution 42, 2; eqs. (4), (6), (8), p. 3. Legarra, A., Aguilar, I. and Misztal, I. (2009). Journal of Dairy Science 92(9), 4656-4663. Aguilar, I., et al. (2010). Journal of Dairy Science 93(2), 743-752.