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Shirin Amiri was asking about GBLUP (genomic BLUP) and based on her example I set up the following R script to show how GBLUP works. Note that this is the so called marker model, where we estimate allele substitution effects of the markers and not individual based model, where genomic breeding values are inferred directly. The code:

library(package="MatrixModels")

dat <- data.frame( y=c(1.5, 2.35, 3.4, 2.31, 1.53),
s1=c("AA", "Aa", "Aa", "AA", "aa"),
s2=c("Bb", "BB", "bb", "BB", "bb"),
s3=c("cc", "Cc", "Cc", "cc", "CC"),
s4=c("Dd", "Dd", "DD", "dd", "Dd"))

cols <- paste("s", 1:4, sep="")
dat[, cols] <- lapply(dat[, cols], function(z) as.integer(z) - 1)

lambda <- 1

(y <- dat$y) (X <- model.Matrix(y ~ 1, data=dat, sparse=TRUE)) (Z <- model.Matrix(y ~ -1 + s1 + s2 + s3 + s4, data=dat, sparse=TRUE)) XX <- crossprod(X) XZ <- crossprod(X, Z) ZZ <- crossprod(Z) Xy <- crossprod(X, y) Zy <- crossprod(Z, y) (LHS <- rBind(cBind(XX, XZ), cBind(t(XZ), ZZ + Diagonal(n=dim(ZZ)) * lambda))) (RHS <- rBind(Xy, Zy)) (sol <- solve(LHS, RHS)) (GEBV <- Z %*% sol[-1]) And the transcript: R version 2.14.2 (2012-02-29) Copyright (C) 2012 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. Natural language support but running in an English locale R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > library(package="MatrixModels") Loading required package: Matrix Loading required package: lattice Attaching package: ‘Matrix’ The following object(s) are masked from ‘package:base’: det Attaching package: ‘MatrixModels’ The following object(s) are masked from ‘package:stats’: getCall > > dat <- data.frame( y=c(1.5, 2.35, 3.4, 2.31, 1.53), + s1=c("AA", "Aa", "Aa", "AA", "aa"), + s2=c("Bb", "BB", "bb", "BB", "bb"), + s3=c("cc", "Cc", "Cc", "cc", "CC"), + s4=c("Dd", "Dd", "DD", "dd", "Dd")) > > cols <- paste("s", 1:4, sep="") > dat[, cols] <- lapply(dat[, cols], function(z) as.integer(z) - 1) > > lambda <- 1 > > (y <- dat$y)
 1.50 2.35 3.40 2.31 1.53
> (X <- model.Matrix(y ~  1,                     data=dat, sparse=TRUE))
"dsparseModelMatrix": 5 x 1 sparse Matrix of class "dgCMatrix"
(Intercept)
1           1
2           1
3           1
4           1
5           1
@ assign:  0
@ contrasts:
named list()
> (Z <- model.Matrix(y ~ -1 + s1 + s2 + s3 + s4, data=dat, sparse=TRUE))
"dsparseModelMatrix": 5 x 4 sparse Matrix of class "dgCMatrix"
s1 s2 s3 s4
1  2  1  .  1
2  1  2  1  1
3  1  .  1  2
4  2  2  .  .
5  .  .  2  1
@ assign:  1 2 3 4
@ contrasts:
named list()
>
> XX <- crossprod(X)
> XZ <- crossprod(X, Z)
> ZZ <- crossprod(Z)
>
> Xy <- crossprod(X, y)
> Zy <- crossprod(Z, y)
>
> (LHS <- rBind(cBind(XX,    XZ),
+               cBind(t(XZ), ZZ + Diagonal(n=dim(ZZ)) * lambda)))
5 x 5 sparse Matrix of class "dgCMatrix"
(Intercept) s1 s2 s3 s4
(Intercept)           5  6  5  4  5
s1                    6 11  8  2  5
s2                    5  8 10  2  3
s3                    4  2  2  7  5
s4                    5  5  3  5  8
>
> (RHS <- rBind(Xy,
+               Zy))
5 x 1 Matrix of class "dgeMatrix"
[,1]
(Intercept) 11.09
s1          13.37
s2          10.82
s3           8.81
s4          12.18
>
> (sol <- solve(LHS, RHS))
5 x 1 Matrix of class "dgeMatrix"
[,1]
(Intercept)  1.65468864
s1           0.05223443
s2           0.08655678
s3          -0.05223443
s4           0.45586081
>
> (GEBV <- Z %*% sol[-1])
5 x 1 Matrix of class "dgeMatrix"
[,1]
1 0.6468864
2 0.6289744
3 0.9117216
4 0.2775824
5 0.3513919
>
> proc.time()
user  system elapsed
2.330   0.070   2.412