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 ∑_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

- $A
the A matrix

- $imputed
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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