# A.mat: Additive relationship matrix In rrBLUP: Ridge Regression and Other Kernels for Genomic Selection

## Description

Calculates the realized additive relationship matrix.

## Usage

 1 2 A.mat(X,min.MAF=NULL,max.missing=NULL,impute.method="mean",tol=0.02, n.core=1,shrink=FALSE,return.imputed=FALSE) 

## Arguments

 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.

## Details

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.rep=5), where the parameters n.qtl and n.rep can be varied to adjust the number of simulated QTL and number of replicates, 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.

## Value

If return.imputed = FALSE, the n \times n additive relationship matrix is returned.

If return.imputed = TRUE, the function returns a list containing

$A the A matrix$imputed

the imputed marker matrix

## References

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.

## Examples

 1 2 3 4 5 6 7 #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) 

rrBLUP documentation built on June 20, 2017, 9:06 a.m.