Genomic BLUP

Henderson's mixed model equations give the fixed and genomic effects jointly. With G the identity the random block is ridge regression with penalty sigma_e^2/sigma_u^2, which is the closed form the tests check; as the penalty tends to zero the system tends to ordinary least squares on the concatenated design. G is ridged before inversion because VanRaden's G is singular whenever markers outnumber individuals.

Usage

Gblupr(y, X, Z, G, var_u = 1, var_e = 1, ridge = 1e-08)

Arguments

y

Response vector.

X

Fixed-effect design matrix.

Z

Random-effect incidence matrix.

G

Genomic relationship matrix.

var_u, var_e

Variance components, positive.

ridge

Ridge added to G before inversion.

Value

List with estimate, beta, u, fitted, residual_ss, lambda, n, method.

Details

Formula: [X'X, X'Z; Z'X, Z'Z + G^{-1} k] [beta; u] = [X'y; Z'y], k = sigma_e^2 / sigma_u^2.

References

VanRaden (2008), Efficient methods to compute genomic predictions, Journal of Dairy Science 91(11):4414-4423. doi:10.3168/jds.2007-0980

Examples

set.seed(1)
n <- 30; q <- 5
X <- cbind(1, rnorm(n))
Z <- matrix(rbinom(n * q, 1, 0.5), n, q)
G <- diag(q)
u <- rnorm(q)
y <- X %*% c(1, 0.5) + Z %*% u + rnorm(n)
Gblupr(y, X, Z, G)
#> $estimate
#> [1] 1.702658
#> 
#> $beta
#> [1] 1.0592993 0.6084397
#> 
#> $u
#> [1] 1.7026579 0.7036307 0.7868724 0.1840810 1.5995863
#> 
#> $fitted
#>  [1]  3.3517511  1.3551161  2.8540867  4.3331485  6.2366131  1.2637244
#>  [7]  3.9264101  4.8107695  3.7128446  5.6662361  5.0692164  3.1832342
#> [13]  0.6813118  2.9859583  3.4183360  5.1210764  3.5389787  4.0200254
#> [19]  4.9362048  6.2133997  3.2930257  4.9124240  2.8883349 -0.1511013
#> [25]  3.9259561  2.8088156  5.1544630  1.1353884  0.9524547  4.5877619
#> 
#> $residual_ss
#> [1] 12.21944
#> 
#> $lambda
#> [1] 1
#> 
#> $n
#> [1] 30
#> 
#> $method
#> [1] "Henderson mixed model equations with G from VanRaden (2008)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"