Gregor Gorjanc (gg)

Nice introductory videos on DNA, Y chromosome, mitochondrial DNA and phylogeography

noreply@blogger.com (Gregor Gorjanc) — Sun, 28 Apr 2013 14:22:00 +0000

Integrating quantitative genetics and genomic in animal breeding graduate learning

noreply@blogger.com (Gregor Gorjanc) — Sun, 07 Apr 2013 09:38:00 +0000

Follow link bellow for a set of graduate courses funded by USDA:

Integrating quantitative genetics and genomic in animal breeding graduate learning

Medical and Population Genetics videos

noreply@blogger.com (Gregor Gorjanc) — Mon, 18 Mar 2013 01:05:00 +0000

Nice set of videos from Broad Institute on Medical and Population Genetics/Genomics:

Kmečka pamet

noreply@blogger.com (Gregor Gorjanc) — Mon, 11 Feb 2013 08:07:00 +0000

Zdrava kmečka pamet lahko nadomesti skoraj vsak nivo izobrazbe, Vendar noben nivo izobrazbe ne more nadomestiti zdrave kmečke pameti. Arthur Schopenha

The contribution of dominance and inbreeding depression in estimating variance components for litter size in Pannon White rabbits

noreply@blogger.com (Gregor Gorjanc) — Sun, 27 Jan 2013 02:29:00 +0000

Our paper on analysis of dominance in Pannon White rabbits has been accepted and will appear in Journal of Animal breeding and Genetics. Abstract says:

In a synthetic closed population of Pannon White rabbits, additive (VA), dominance (VD) and permanent environmental (VPe) variance components as well as doe (bFd) and litter (bFl) inbreeding depression were estimated for the number of kits born alive (NBA), number of kits born dead (NBD) and total number of kits born (TNB). The data set consisted of 18,398 kindling records of 3883 does collected from 1992 to 2009. Six models were used to estimate dominance and inbreeding effects. The most complete model estimated VA and VD to contribute 5.5 ± 1.1% and 4.8 ± 2.4%, respectively, to total phenotypic variance (VP) for NBA; the corresponding values for NBD were 1.9 ± 0.6% and 5.3 ± 2.4%, for TNB, 6.2 ± 1.0% and 8.1 ± 3.2% respectively. These results indicate the presence of considerable VD. Including dominance in the model generally reduced VA and VPe estimates, and had only a very small effect on inbreeding depression estimates. Including inbreeding covariates did not affect estimates of any variance component. A 10% increase in doe inbreeding significantly increased NBD (bFd = 0.18 ± 0.07), while a 10% increase in litter inbreeding significantly reduced NBA (bFl = −0.41 ± 0.11) and TNB (bFl = −0.34 ± 0.10). These findings argue for including dominance effects in models of litter size traits in populations that exhibit significant dominance relationships.

The contribution of dominance and inbreeding depression in estimating variance components for litter size i... by

Genomic evaluations using similarity between haplotypes

noreply@blogger.com (Gregor Gorjanc) — Sun, 27 Jan 2013 02:13:00 +0000

Our paper on using haplotypes to build relationships between individuals has been accepted and will appear in Journal of Animal breeding and Genetics. Abstract says:

Long-range phasing and haplotype library imputation methodologies are accurate and efficient methods to provide haplotype information that could be used in prediction of breeding value or phenotype. Modelling long haplotypes as independent effects in genomic prediction would be inefficient due to the many effects that need to be estimated and phasing errors, even if relatively low in frequency, exacerbate this problem. One approach to overcome this is to use similarity between haplotypes to model covariance of genomic effects by region or of animal breeding values. We developed a simple method to do this and tested impact on genomic prediction by simulation. Results show that the diagonal and off-diagonal elements of a genomic relationship matrix constructed using the haplotype similarity method had higher correlations with the true relationship between pairs of individuals than genomic relationship matrices built using unphased genotypes or assumed unrelated haplotypes. However, the prediction accuracy of such haplotype-based prediction methods was not higher than those based on unphased genotype information.

Genomic evaluations using similarity between haplotypes by

Simulation of different strategies to implement genomic selection in plant breeding

noreply@blogger.com (Gregor Gorjanc) — Fri, 25 Jan 2013 17:45:00 +0000

Our submitted abstract for the VIPCA conference.

Gorjanc G, Crossa J, Dreisigacker S, Autrique E, Hickey J

Genomic information provides a rich resource for plant improvement. The greatest advantage of genomic information is achived when used early in a programme. The aim of this contribution was to present different ways of utilizing genomic information in cross and self pollinated species early in the bi-parental population (BP) cycle in the F2 generation by simulation. Simulation mimicked a real breeding programme with BP of different degree of relationship. Simulated trait heritability was 0.5. Predictions were always within a focal BP (BPX) avoiding the effect of inflated accuracy due to population structure. Measure of interest was the correlation between true and predicted additive genetic value (accuracy). Predictions were based on training on collected data within BPX, from BP having one parent in common with BPX (BPPC), from BP having one grand-parent in common with BPX (BPGC), and from unrelated BP (BPUR). Results show that genomic predictions can be accurate early in the cycle. Accurate predictions within the BPX can be obtained with a small number of markers and moderate phenotyping of F2 derived lines. With 200 markers and 50 phenotypes accuracy was around 0.60 and increased to 0.80 with 100 phenotypes. Use of information from relatives requires denser marker panels and more phenotypes to properly model the expanding number of linkage blocks in progressively distantly related BP. However, these requirements can be spread over many related BP reducing the cost per BP. For example accuracy of 0.6 can be achieved by using 1,000 markers and 50 phenotypes from 4 BPPC and 40 BPUR or by using 10,000 markers and 50 phenotypes from 8 BPGC and 50 BPUR. Requirement for denser marker panels increases needed investment, which can be offset by proper use of imputation method making the whole procedure cost effective for real applications.

Functions dim, nrow, and ncol for shell

noreply@blogger.com (Gregor Gorjanc) — Thu, 27 Dec 2012 23:31:00 +0000

Ever wanted to find out quickly the number of rows and/or columns in file directly from terminal. There are many ways to skin this cat. Here is what I used for number of rows for quite a while:

wc -l filename

What about number of columns? "Easy", just combine head and awk commands:

head -n 1 filename | awk '{ print NF }'

not a big problem (there are likely better ways to do this), but is long and tedious.

I got sick of typing commands above and assembled them in three easy to use function with R-like names: nrow, ncol, and dim. Functions are simply a collection of above ideas and assume that the file is of "rectangular shape", i.e., a table, a matrix, etc.

dim()
{
for FILE in $@; do
NROW=$(nrow $FILE | awk '{ print $1}')
NCOL=$(ncol $FILE | awk '{ print $1}')
echo "$NROW $NCOL $FILE"
unset NROW NCOL
done
}
export -f dim

nrow ()
{
for FILE in $@; do
wc -l $FILE
done
}
export -f nrow

ncol ()
{
for FILE in $@; do
TMP=$(head $FILE -n 1 | awk '{ print NF }')
echo "$TMP $FILE"
unset TMP
done
}
export -f ncol

Add these files to your .bashrc or .profile or something similar and you can now simply type:

nrow filename

ncol filename

dim filename

A simple test:

touch file.txt

echo "line1 with four columns" >> file.txt

echo "line2 with four columns" >> file.txt

nrow file.txt

2 file.txt

ncol file.txt

4 file.txt

dim file.txt

2 4 file.txt

Recover deleted file under linux

noreply@blogger.com (Gregor Gorjanc) — Tue, 04 Dec 2012 15:25:00 +0000

Here is a nice summary, but the bottom line is that the Ubuntu/Debian package testdisk has a nice utility called photorec that can be used to search for deleted files and recover. This is not a GUI program, though!

AWK one-liners

noreply@blogger.com (Gregor Gorjanc) — Tue, 23 Oct 2012 16:01:00 +0000

See here.

InterBull and ICAR papers

noreply@blogger.com (Gregor Gorjanc) — Thu, 11 Oct 2012 11:46:00 +0000

Reliability of breeding values in selected populations Partitioning International Genetic Trends by Origin in Holstein bulls

Whole-genome evaluation of complex traits using SNP, haplotype, or QTL information

noreply@blogger.com (Gregor Gorjanc) — Sun, 30 Sep 2012 05:07:00 +0000

This is the presentation (and abstract bellow) of my talk at local congress Genetika 2012.
Whole-genome evaluation of complex traits using SNP, haplotype, or QTL information

Abstract:

Whole-genome technologies provide rich data for dissection of complex traits. While gene discovery is still largely limited, the data at hand can be successfully used for evaluation of genetic merit. The aim of this work was to demonstrate the value of different sources of information (pedigrees, Single Nucleotide Polymorphisms – SNP, haplotypes, or Quantitative Trait Loci – QTL) for genetic evaluation of non-phenotyped individuals in a typical animal breeding scenario via simulation. In the first step a coalescent simulation was used to create a base population with structured chromosomes that were in the second step dropped and recombined through the pedigree of 10 generations with 50 sires per generation, 10 dams per sire, and 2 offspring per dam. Phenotypic values were simulated with different genetic architectures (QTL effects were sampled from Gaussian or gamma distribution and minor allele frequency less than 0.3) and heritability of 0.25. Genotypic data was available for all individuals from generation 4 onwards, while phenotypic data was available for individuals in generations 4 and 5. Genetic evaluation was based on linear mixed models with relationship matrix between individuals. This matrix was built using pedigree, SNP, haplotype, or QTL data. Haplotypes of different length were considered (from 5 to all the way up to 2000 SNP) with an option to account for similarities between haplotypes while building relationship matrices. The accuracy of different methods was assessed by correlation between true and evaluated additive genetic values for individuals in generations 6, 8 and 10. Average accuracy over ten replications for Gaussian trait over generations was between 0.45 to 0.10 for pedigree data, 0.50 to 0.35 for SNP and haplotype data and 0.6 to 0.4 for QTL data. In the case of long haplotypes accuracies dropped considerably, but accounting for similarities between haplotypes prevented this drop. In the case of gamma trait accuracies were slightly higher in generation 6 and dropped faster in the later generations in the case of pedigree, SNP, and haplotype data due to recombinations. On the other hand accuracies were substantially higher with QTL data and quite stable over generations (from 0.75 to 0.65) though still far from perfect (even though QTL genotypes are known), due to estimation errors. Results demonstrate the value and limitations of genotypic information for the evaluation of additive genetic merit in animal populations.

Software Tools for Animal Gene Mapping

noreply@blogger.com (Gregor Gorjanc) — Sat, 29 Sep 2012 00:00:00 +0000

Here are some cool tools for animal genetics:

AGDP for the analysis of genome differences between opulations
SNPEVG graphical tool for SNP effect viewing and graphing
...

Google drive drives in

noreply@blogger.com (Gregor Gorjanc) — Sun, 23 Sep 2012 15:02:00 +0000

I just realized that Google offers me "Google drive". The pricing options for larger disk space than 5GB are very competitive with Dropbox! I am using Dropbox for all my files - yes, it costs me quite some money when the bill comes, but if we consider this is "few" dollars per month for backup and synchronization I am more than willing to pay this amount for this!!! Google drives is now yet another option, but the desktop folder does not work on Linux so for now I will stick with Dropbox.

AIPL presentations on animal breeding and genetics/genomics

noreply@blogger.com (Gregor Gorjanc) — Mon, 20 Aug 2012 14:34:00 +0000

You can find here some top-notch presentations on animal breeding and genetics/genomics.

“Livestock Conservation Genomics: Data, Tools and Trends“ Summer School, Croatia, Oct 1-7, 2012

noreply@blogger.com (Gregor Gorjanc) — Sat, 11 Aug 2012 11:01:00 +0000

There will be an ESF GENOMIC-RESOURCES (link1, link2) summer school “Livestock Conservation Genomics: Data, Tools and Trends“ Summer School, Croatia, Oct 1-7, 2012. You can find more details here.

Nice SNP tools from the Pevsner laboratory

noreply@blogger.com (Gregor Gorjanc) — Thu, 09 Aug 2012 14:32:00 +0000

SNPduo - compare SNPs from two individuals
SNPtrio - compare SNPs from a family trio - very cool plots and nice diagnostics
kcoeff - estimate K0, K1, and K2 coefficients from SNP data

EU FP7 Livestock Genomics projects

noreply@blogger.com (Gregor Gorjanc) — Wed, 08 Aug 2012 21:58:00 +0000

The Quantomics group had a meeting in September 2011 and published was a very nice set of presentations on EU FP7 Livestock Genomics projects.

InterBull and ICAR talks from Cork

noreply@blogger.com (Gregor Gorjanc) — Sat, 30 Jun 2012 15:07:00 +0000

Follow links bellow to see slides and videos!

Other talks as are available from here.

2 min HOWTO in R

noreply@blogger.com (Gregor Gorjanc) — Wed, 06 Jun 2012 11:38:00 +0000

Lots of short videos on how to do several things in R.

Animal breeding and genetics progress video

noreply@blogger.com (Gregor Gorjanc) — Tue, 22 May 2012 11:18:00 +0000

See here.

Some more study video material

noreply@blogger.com (Gregor Gorjanc) — Sat, 21 Apr 2012 09:14:00 +0000

Mathematical Monk channel featuring Information Theory, Machine Learning, and Probability Primer
Coursera featuring loads of material from Princeton, Stanford, Michigan, and Penn
Udacity

Simulated data for genomic selection and GWAS using a combination of coalescent and gene drop methods

noreply@blogger.com (Gregor Gorjanc) — Tue, 17 Apr 2012 14:27:00 +0000

Our (Hickey and Gorjanc) paper describing simulation using coalescent and gene drop methods has been published in the new open G3 journal. This contribution is a part of the Genomic selection collection of the Genetic Society of America that is nicely described here.

Chalk letter generator

noreply@blogger.com (Gregor Gorjanc) — Sat, 14 Apr 2012 23:07:00 +0000

Here is a cool online application that takes chalkboard with Bart Simpsons and adds your text on it.

Small pedigree based mixed model example

noreply@blogger.com (Gregor Gorjanc) — Sun, 08 Apr 2012 17:14:00 +0000

Pedigree based mixed models (often called animal models, due to modelling animal performance) are the cornerstone of animal breeding and quantitative genetics. There are many programs that can be used for analyzing your data with these models, e.g., ASREML, BLUPf90, MATVEC, MiXBLUP & MiX99, SurvivalKit, PEST/VCE, WOMBAT, ...). There are also R packages you can use: pedigreemm and MCMCglmm. If you want to run your own program you can take the example code bellow and start from it. The code shows the essence of building the system of equations that needs to be solved on a simple example. Note that this is mean only for demonstration purposes and small scale analyses. In addition, variance components are assumed known here. In order to understand the model for this simple example a bit better the graphical model view is shown first.

Graphical model view of simple pedigree based mixed model example

The code:

options(width=200)
 
### --- Required packages ---
 
## install.packages(pkg=c("pedigreemm", "MatrixModels"))
 
library(package="pedigreemm")   ## pedigree functions
library(package="MatrixModels") ## sparse matrices
 
### --- Data ---
 
## NOTE:
## - some individuals have one or both parents (un)known
## - some individuals have phenotype (un)known
## - some indididuals have repeated phenotype observations 
example <- data.frame(
  individual=c( 1,   2,   2,   3,   4,   5,   6,   7,   8,   9,  10),
      father=c(NA,  NA,  NA,   2,   2,   4,   2,   5,   5,  NA,   8),
      mother=c(NA,  NA,  NA,   1,  NA,   3,   3,   6,   6,  NA,   9),
   phenotype=c(NA, 103, 106,  98, 101, 106,  93,  NA,  NA,  NA, 109),
       group=c(NA,   1,   1,   1,   2,   2,   2,  NA,  NA,  NA,   1))
 
## Variance components
sigma2e <- 1
sigma2a <- 3
(h2 <- sigma2a / (sigma2a + sigma2e))
 
### --- Setup data ---
 
## Make sure each individual has only one record in pedigree
ped <- example[!duplicated(example$individual), 1:3]
 
## Factors (this eases buliding the design matrix considerably)
example$individual <- factor(example$individual)
example$group      <- factor(example$group)
 
## Phenotype data
dat <- example[!is.na(example$phenotype), ]
 
### --- Setup MME ---
 
## Phenotype vector
(y <- dat$phenotype)
 
## Design matrix for the "fixed" effects (group)
(X <- model.Matrix( ~ group,          data=dat, sparse=TRUE))
 
## Design matrix for the "random" effects (individual)
(Z <- model.Matrix(~ individual - 1, data=dat, sparse=TRUE))
 
## Inverse additive relationship matrix
ped2 <- pedigree(sire=ped$father, dam=ped$mother, label=ped$individual)
TInv <- as(ped2, "sparseMatrix")
DInv <- Diagonal(x=1/Dmat(ped2))
AInv <- crossprod(sqrt(DInv) %*% TInv)
 
## Variance ratio
alpha <- sigma2e / sigma2a
 
## Mixed Model Equations (MME)
 
## ... Left-Hand Side (LHS) without pedigree prior
(LHS0 <- rBind(cBind(crossprod(X),    crossprod(X, Z)),
               cBind(crossprod(Z, X), crossprod(Z, Z))))
 
## ... Left-Hand Side (LHS) with    pedigree prior
round(
 LHS <- rBind(cBind(crossprod(X),    crossprod(X, Z)),
              cBind(crossprod(Z, X), crossprod(Z, Z) + AInv * alpha)), digits=1)
 
## ... Right-Hand Side (RHS)
(RHS <- rBind(crossprod(X, y),
              crossprod(Z, y)))
 
### --- Solutions ---
 
## Solve
LHSInv <- solve(LHS)
sol <- LHSInv %*% RHS
 
## Standard errors
se <- diag(LHSInv) * sigma2e
 
## Reliabilities
r2 <- 1 - diag(LHSInv) * alpha
 
## Accuracies
r <- sqrt(r2)
 
## Printout
cBind(sol, se, r, r2)

Created by Pretty R at inside-R.org

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.
 
[Previously saved workspace restored]
 
> 
> options(width=200)
> 
> ### --- Required packages ---
> 
> ## install.packages(pkg=c("pedigreemm", "MatrixModels"))
> 
> library(package="pedigreemm")   ## pedigree functions
Loading required package: lme4
Loading required package: Matrix
Loading required package: lattice
 
Attaching package: ‘Matrix’
 
The following object(s) are masked from ‘package:base’:
 
    det
 
 
Attaching package: ‘lme4’
 
The following object(s) are masked from ‘package:stats’:
 
    AIC, BIC
 
> library(package="MatrixModels") ## sparse matrices
> 
> ### --- Data ---
> 
> ## NOTE:
> ## - some individuals have one or both parents (un)known
> ## - some individuals have phenotype (un)known
> ## - some indididuals have repeated phenotype observations 
> example <- data.frame(
+   individual=c( 1,   2,   2,   3,   4,   5,   6,   7,   8,   9,  10),
+       father=c(NA,  NA,  NA,   2,   2,   4,   2,   5,   5,  NA,   8),
+       mother=c(NA,  NA,  NA,   1,  NA,   3,   3,   6,   6,  NA,   9),
+    phenotype=c(NA, 103, 106,  98, 101, 106,  93,  NA,  NA,  NA, 109),
+        group=c(NA,   1,   1,   1,   2,   2,   2,  NA,  NA,  NA,   1))
> 
> ## Variance components
> sigma2e <- 1
> sigma2a <- 3
> (h2 <- sigma2a / (sigma2a + sigma2e))
[1] 0.75
> 
> ### --- Setup data ---
> 
> ## Make sure each individual has only one record in pedigree
> ped <- example[!duplicated(example$individual), 1:3]
> 
> ## Factors (this eases buliding the design matrix considerably)
> example$individual <- factor(example$individual)
> example$group      <- factor(example$group)
> 
> ## Phenotype data
> dat <- example[!is.na(example$phenotype), ]
> 
> ### --- Setup MME ---
> 
> ## Phenotype vector
> (y <- dat$phenotype)
[1] 103 106  98 101 106  93 109
> 
> ## Design matrix for the "fixed" effects (group)
> (X <- model.Matrix( ~ group,          data=dat, sparse=TRUE))
"dsparseModelMatrix": 7 x 2 sparse Matrix of class "dgCMatrix"
   (Intercept) group2
2            1      .
3            1      .
4            1      .
5            1      1
6            1      1
7            1      1
11           1      .
@ assign:  0 1 
@ contrasts:
$group
[1] "contr.treatment"
 
> 
> ## Design matrix for the "random" effects (individual)
> (Z <- model.Matrix(~ individual - 1, data=dat, sparse=TRUE))
"dsparseModelMatrix": 7 x 10 sparse Matrix of class "dgCMatrix"
   [[ suppressing 10 column names ‘individual1’, ‘individual2’, ‘individual3’ ... ]]
 
2  . 1 . . . . . . . .
3  . 1 . . . . . . . .
4  . . 1 . . . . . . .
5  . . . 1 . . . . . .
6  . . . . 1 . . . . .
7  . . . . . 1 . . . .
11 . . . . . . . . . 1
@ assign:  1 1 1 1 1 1 1 1 1 1 
@ contrasts:
$individual
[1] "contr.treatment"
 
> 
> ## Inverse additive relationship matrix
> ped2 <- pedigree(sire=ped$father, dam=ped$mother, label=ped$individual)
> TInv <- as(ped2, "sparseMatrix")
> DInv <- Diagonal(x=1/Dmat(ped2))
> AInv <- crossprod(sqrt(DInv) %*% TInv)
> 
> ## Variance ratio
> alpha <- sigma2e / sigma2a
> 
> ## Mixed Model Equations (MME)
> 
> ## ... Left-Hand Side (LHS) without pedigree prior
> (LHS0 <- rBind(cBind(crossprod(X),    crossprod(X, Z)),
+                cBind(crossprod(Z, X), crossprod(Z, Z))))
12 x 12 sparse Matrix of class "dgCMatrix"
   [[ suppressing 12 column names ‘(Intercept)’, ‘group2’, ‘individual1’ ... ]]
 
(Intercept)  7 3 . 2 1 1 1 1 . . . 1
group2       3 3 . . . 1 1 1 . . . .
individual1  . . . . . . . . . . . .
individual2  2 . . 2 . . . . . . . .
individual3  1 . . . 1 . . . . . . .
individual4  1 1 . . . 1 . . . . . .
individual5  1 1 . . . . 1 . . . . .
individual6  1 1 . . . . . 1 . . . .
individual7  . . . . . . . . . . . .
individual8  . . . . . . . . . . . .
individual9  . . . . . . . . . . . .
individual10 1 . . . . . . . . . . 1
> 
> ## ... Left-Hand Side (LHS) with    pedigree prior
> round(
+  LHS <- rBind(cBind(crossprod(X),    crossprod(X, Z)),
+               cBind(crossprod(Z, X), crossprod(Z, Z) + AInv * alpha)), digits=1)
12 x 12 sparse Matrix of class "dgCMatrix"
   [[ suppressing 12 column names ‘(Intercept)’, ‘group2’, ‘individual1’ ... ]]
 
(Intercept)  7 3  .    2.0  1.0  1.0  1.0  1.0  .    .    .    1.0
group2       3 3  .    .    .    1.0  1.0  1.0  .    .    .    .  
individual1  . .  0.5  0.2 -0.3  .    .    .    .    .    .    .  
individual2  2 .  0.2  2.8 -0.2 -0.2  .   -0.3  .    .    .    .  
individual3  1 . -0.3 -0.2  2.0  0.2 -0.3 -0.3  .    .    .    .  
individual4  1 1  .   -0.2  0.2  1.6 -0.3  .    .    .    .    .  
individual5  1 1  .    .   -0.3 -0.3  2.1  0.4 -0.4 -0.4  .    .  
individual6  1 1  .   -0.3 -0.3  .    0.4  2.1 -0.4 -0.4  .    .  
individual7  . .  .    .    .    .   -0.4 -0.4  0.8  .    .    .  
individual8  . .  .    .    .    .   -0.4 -0.4  .    1.0  0.2 -0.4
individual9  . .  .    .    .    .    .    .    .    0.2  0.5 -0.4
individual10 1 .  .    .    .    .    .    .    .   -0.4 -0.4  1.8
> 
> ## ... Right-Hand Side (RHS)
> (RHS <- rBind(crossprod(X, y),
+               crossprod(Z, y)))
12 x 1 Matrix of class "dgeMatrix"
             [,1]
(Intercept)   716
group2        300
individual1     0
individual2   209
individual3    98
individual4   101
individual5   106
individual6    93
individual7     0
individual8     0
individual9     0
individual10  109
> 
> ### --- Solutions ---
> 
> ## Solve
> LHSInv <- solve(LHS)
> sol <- LHSInv %*% RHS
> 
> ## Standard errors
> se <- diag(LHSInv) * sigma2e
> 
> ## Reliabilities
> r2 <- 1 - diag(LHSInv) * alpha
> 
> ## Accuracies
> r <- sqrt(r2)
Warning message:
In sqrt(r2) : NaNs produced
> 
> ## Printout
> cBind(sol, se, r, r2)
12 x 4 Matrix of class "dgeMatrix"
                         se         r         r2
 [1,] 104.76444809 1.901479 0.6051228  0.3661736
 [2,]  -4.65061921 1.376348 0.7356748  0.5412175
 [3,]  -2.62685846 2.432448 0.4349528  0.1891839
 [4,]  -0.77797260 1.983301 0.5821509  0.3388997
 [5,]  -4.32927399 1.979138 0.5833415  0.3402874
 [6,]   1.54997883 2.316720 0.4772421  0.2277600
 [7,]   3.18706240 2.604540 0.3630701  0.1318199
 [8,]  -5.07852788 2.508673 0.4046922  0.1637758
 [9,]  -0.94573274 3.459904       NaN -0.1533013
[10,]  -0.08765651 3.534636       NaN -0.1782121
[11,]   2.11218764 2.511203 0.4036487  0.1629323
[12,]   2.82742681 2.009539 0.5745901  0.3301538
> 
> proc.time()
   user  system elapsed 
  3.350   0.070   3.425

Created by Pretty R at inside-R.org