SNP-BLUP additive genomic prediction

Model (Meuwissen, Hayes and Goddard 2001, BLUP estimation, p. 1822): \(y = 1_n \mu + Z u + e\) with \(u \sim N(0, \sigma_u^2 I)\), \(e \sim N(0, \sigma_e^2 I)\), solved from Henderson's mixed-model equations (Henderson 1975; as printed in Montesinos-Lopez et al. 2022 ch. 5.2): \([n, 1'Z; Z'1, Z'Z + \lambda I] [\mu; u] = [1'y; Z'y]\) with \(\lambda = \sigma_e^2/\sigma_u^2\). Z is the VanRaden-centred marker matrix \(Z_{ij} = M_{ij} - 2 p_j\) (VanRaden 2008 Method 1 centring), and GEBV = Z u-hat.

Usage

Snpblr(y, M, lam = NULL, h2 = NULL, freq = NULL)

Arguments

y: Phenotype vector.
M: Genotype matrix coded 0/1/2 (individuals by markers).
lam: Shrinkage ratio sigma_e^2 / sigma_u^2. Give exactly one of lam and h2.
h2: Heritability used to derive lam.
freq: Optional allele frequencies; column means over 2 by default.

Value

List with estimate (GEBV vector), u (marker effects), mu, lam, sum2pq, freq, n, m, method.

Details

If h2 is given instead of lam, the ratio is derived from the VanRaden variance split \(\sigma_g^2 = 2 \sum_j p_j (1-p_j) \sigma_u^2\): \(\lambda = ((1-h^2)/h^2) \, 2 \sum_j p_j (1-p_j)\).

References

Meuwissen, T. H. E., Hayes, B. J. and Goddard, M. E. (2001). Prediction of total genetic value using genome-wide dense marker maps. Genetics 157(4), 1819-1829, sec. BLUP estimation p. 1822. Henderson, C. R. (1975). Biometrics 31(2), 423-447. VanRaden, P. M. (2008). Journal of Dairy Science 91(11), 4414-4423, via Montesinos-Lopez et al. (2022) Multivariate Statistical Machine Learning Methods for Genomic Prediction, Springer, sec. 2.4 and ch. 5.2 (local split PDFs).

Examples

set.seed(14)
M <- matrix(rbinom(50 * 20, 2, 0.4), 50, 20)
u <- rnorm(20, sd = 0.3)
y <- as.numeric(10 + scale(M, scale = FALSE) %*% u + rnorm(50))
s <- Snpblr(y, M, h2 = 0.5)
cor(s$estimate, scale(M, scale = FALSE) %*% u)
#>           [,1]
#> [1,] 0.9083338