A.mat | R Documentation |
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 ( |
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 |
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 \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.
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 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>
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