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
)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
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.
Minimum minor allele frequency. The A matrix is not sensitive to rare alleles, so by default only monomorphic markers are removed.
Maximum proportion of missing data; default removes completely missing markers.
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).
Specifies the convergence criterion for the EM algorithm (see details).
Specifies the number of cores to use for parallel execution of the EM algorithm
set shrink=FALSE to disable shrinkage estimation. See Details for how to enable shrinkage estimation.
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 \(c = 2 \sum_k {p_k (1-p_k)}\), 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.
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 128:693-703. <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. <doi:10.1038/ng.608>