Dominance relationship matrix

C++ implementation of the dominance matrix. Calculates the realized dominance relationship matrix. Can help to increase the prediction accuracy when 2 conditions are met; 1) The trait has intermediate to high heritability, 2) The population contains a big number of individuals that are half or full sibs (HS & FS).

Usage

D.mat(X,nishio=TRUE,min.MAF=0,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.
nishio: If TRUE, Nishio and Satoh (2014), which uses the orthogonal parameterization of Vitezica et al. (2013). Otherwise Su et al. (2012). See Details and references.
min.MAF: Minimum minor allele frequency. The D matrix is not sensitive to rare alleles, so by default only monomorphic markers are removed.
return.imputed: When TRUE, the imputed marker matrix is returned.

Details

Missing values are replaced by the column (marker) mean before the matrix is computed. Markers with minor allele frequency less than or equal to min.MAF (and therefore all monomorphic markers) are removed. For each marker $k$, $p_k$ is the frequency of the allele counted by the code 1, estimated as the column mean of $(X+1)/2$, and $q_k = 1-p_k$.

For the nishio method (Nishio and Satoh, 2014; Vitezica et al., 2013): genotypes are coded as $w_{ik} = -2q_k^2$ for $x_{ik}=1$, $2p_kq_k$ for $x_{ik}=0$ and $-2p_k^2$ for $x_{ik}=-1$, computed as $w_{ik} = -x_{ik}^2 + (p_k-q_k)x_{ik} + 2p_kq_k$. Fractional values are passed through the same formula; note that a mean-imputed value $x_{ik} = p_k-q_k$ receives the heterozygote code $2p_kq_k$. Then $D = W W'/c$, where $c = \sum_k (2p_kq_k)^2$. Under Hardy-Weinberg equilibrium this coding is orthogonal to the additive coding.

For the Su method (Su et al., 2012): heterozygosity is coded as $h_{ik} = 1-|x_{ik}|$ and centered as $m_{ik} = h_{ik} - 2p_kq_k$. Then $D = M M'/c$, where $c = \sum_k 2p_kq_k(1-2p_kq_k)$.

Versions of sommer earlier than 4.4.9 used a different formula for nishio=TRUE: $h_{ik}$ centered by its column mean and divided by $\sum_k (2p_kq_k)^2$. That was not the Nishio and Satoh (2014) method, so results from those versions will differ.

Value

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

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

$D: the D matrix
$X: the imputed marker matrix

References

Covarrubias-Pazaran G (2016) Genome assisted prediction of quantitative traits using the R package sommer. PLoS ONE 11(6): doi:10.1371/journal.pone.0156744

Nishio M and Satoh M. 2014. Including Dominance Effects in the Genomic BLUP Method for Genomic Evaluation. Plos One 9(1), doi:10.1371/journal.pone.0085792

Vitezica ZG, Varona L, Legarra A. 2013. On the Additive and Dominant Variance and Covariance of Individuals Within the Genomic Selection Scope. Genetics 195(4): 1223-1230. doi:10.1534/genetics.113.155176

Su G, Christensen OF, Ostersen T, Henryon M, Lund MS. 2012. Estimating Additive and Non-Additive Genetic Variances and Predicting Genetic Merits Using Genome-Wide Dense Single Nucleotide Polymorphism Markers. PLoS ONE 7(9): e45293. doi:10.1371/journal.pone.0045293

Examples

####=========================================####
#### EXAMPLE 1
####=========================================####
####random population of 200 lines with 1000 markers
X <- matrix(rep(0,200*1000),200,1000)
for (i in 1:200) {
  X[i,] <- sample(c(-1,0,0,1), size=1000, replace=TRUE)
}

D <- D.mat(X)