Additive relationship matrix
Calculates the realized additive relationship matrix.
A.mat(X,min.MAF=NULL,max.missing=NULL,impute.method="mean",tol=0.02,
n.core=1,shrink=FALSE,return.imputed=FALSE)X |
Matrix (n \times m) of unphased genotypes for n lines and m biallelic markers, coded as {-1,0,1}. Fractional (imputed) and missing values (NA) are allowed. |
min.MAF |
Minimum minor allele frequency. The A matrix is not sensitive to rare alleles, so by default only monomorphic markers are removed. |
max.missing |
Maximum proportion of missing data; default removes completely missing markers. |
impute.method |
There are two options. The default is "mean", which imputes with the mean for each marker. The "EM" option imputes with an EM algorithm (see details). |
tol |
Specifies the convergence criterion for the EM algorithm (see details). |
n.core |
Specifies the number of cores to use for parallel execution of the EM algorithm (use only at UNIX command line). |
shrink |
Set shrink=FALSE to disable shrinkage estimation. See Details for how to enable shrinkage estimation. |
return.imputed |
When TRUE, the imputed marker matrix is returned. |
At high marker density, the relationship matrix is estimated as A=W W'/c, where W_{ik} = X_{ik} + 1 - 2 p_k and p_k is the frequency of the 1 allele at marker k. By using a normalization constant of c = 2 ∑_k {p_k (1-p_k)}, the mean of the diagonal elements is 1 + f (Endelman and Jannink 2012).
The EM imputation algorithm is based on the multivariate normal distribution and was designed for use with GBS (genotyping-by-sequencing) markers, which tend to be high density but with lots of missing data. Details are given in Poland et al. (2012). The EM algorithm stops at iteration t when the RMS error = n^{-1} \|A_{t} - A_{t-1}\|_2 < tol.
Shrinkage estimation can improve the accuracy of genome-wide marker-assisted selection, particularly at low marker density (Endelman and Jannink 2012). The shrinkage intensity ranges from 0 (no shrinkage) to 1 (A=(1+f)I). Two algorithms for estimating the shrinkage intensity are available. The first is the method described in Endelman and Jannink (2012) and is specified by shrink=list(method="EJ"). The second involves designating a random sample of the markers as simulated QTL and then regressing the A matrix based on the QTL against the A matrix based on the remaining markers (Yang et al. 2010; Mueller et al. 2015). The regression method is specified by shrink=list(method="REG",n.qtl=100,n.iter=5), where the parameters n.qtl and n.iter can be varied to adjust the number of simulated QTL and number of iterations, respectively.
The shrinkage and EM-imputation options are designed for opposite scenarios (low vs. high density) and cannot be used simultaneously. When the EM algorithm is used, the imputed alleles can lie outside the interval [-1,1]. Polymorphic markers that do not meet the min.MAF and max.missing criteria are not imputed.
If return.imputed = FALSE, the n \times n additive relationship matrix is returned.
If return.imputed = TRUE, the function returns a list containing
the A matrix
the imputed marker matrix
Endelman, J.B., and J.-L. Jannink. 2012. Shrinkage estimation of the realized relationship matrix. G3:Genes, Genomes, Genetics. 2:1405-1413. doi: 10.1534/g3.112.004259
Mueller et al. 2015. Shrinkage estimation of the genomic relationship matrix can improve genomic estimated breeding values in the training set. Theor Appl Genet doi: 10.1007/s00122-015-2464-6
Poland, J., J. Endelman et al. 2012. Genomic selection in wheat breeding using genotyping-by-sequencing. Plant Genome 5:103-113. doi: 10.3835/plantgenome2012.06.0006
Yang et al. 2010. Common SNPs explain a large proportion of the heritability for human height. Nat. Genetics 42:565-569.
#random population of 200 lines with 1000 markers
X <- matrix(rep(0,200*1000),200,1000)
for (i in 1:200) {
X[i,] <- ifelse(runif(1000)<0.5,-1,1)
}
A <- A.mat(X)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.