## 2 Materials and Methods
### BLUP, GBLUP, and ssGBLUP Models  
To estimate breeding values, we employed several methodologies utilizing different data types: BLUP for pedigree and phenotypic data [23], GBLUP for genotypic and pheno-type data [24], and ssGBLUP for a combination of pedigree, genotypic, and phenotypic data [25]. Our linear mixed model is expressed as follows:  
\[y=X\mu+Zu+e. \tag{1}\]  
In this equation, \(y\) represents the vector of observations, and \(\mu\) denotes the vector of fixed effects, which includes factors such as sex and batch. \(X\) is the design matrix corresponding to the fixed effects, \(u\) stands for the vector of additive genetic effects, \(Z\) is an incidence matrix allocating records to \(u\), and \(e\) represents random residual effects.  
For the BLUP approach, the vector of additive genetic effects \(u\) is assumed to follow a normal distribution with a mean of zero and variance \(A\sigma_{a}^{2}\), where \(A\) is the numerator relationship matrix derived from pedigree information, \(\sigma_{a}^{2}\) is the additive genetic variance, and \(\sigma_{e}^{2}\) is the residual variance. Heritability (\(h^{2}\)) is calculated as the ratio of additive genetic variance to the total phenotypic variance, given by  
\[h^{2}=\frac{\sigma_{a}^{2}}{\sigma_{a}^{2}+\sigma_{e}^{2}}. \tag{2}\]  
Heritability is classified as high (>0.4), moderate (0.2-0.4), or low (<0.2) [26]. Correlation coefficients are categorized as negligible (0.0-0.2), weak (0.2-0.4), moderate (0.4-0.7), or strong (0.7-1.0) [27].  
In the GBLUP method, the additive genetic effects vector \(u\) is modeled with a normal distribution \(N\big{(}0,~{}G\sigma_{a}^{2}\big{)}\), where \(G\) is the genomic relationship matrix calculated using [PERSON]'s method [28]. The genomic relationship matrix \(G\) is defined as follows:  
\[G=\frac{ZZ^{\prime}}{2\sum\limits_{i=1}^{m}p_{i}(1-p_{i})}, \tag{3}\]  
where \(Z\) is the centralized genotype matrix, which can be calculated using the marker genotypic matrix \(M\)[24]. The genotype matrix \(M\) is an \(n\times m\) dimensional matrix, where \(n\) is the sample size, \(m\) is the number of markers, and the elements in \(M\) are 0 (homozygote), 1 (heterozygote), or 2 (other homozygote); \(p_{i}\) is the frequency of the minor allele of the i-th marker. The matrix \(Z\) is defined as follows:  
\[Z=M-2p_{i}. \tag{4}\]  
For the ssGBLUP method, the additive genetic effects vector \(u\) follows a normal distribution \(N\big{(}0,~{}H\sigma_{a}^{2}\big{)}\), where \(H\) is a blended relationship matrix combining both pedigree and genomic information. The inverse of \(H\) is derived using the following formula:  
\[H^{-1}=A^{-1}+\begin{bmatrix}0&0\\ 0&G^{-1}-A_{22}^{-1}\end{bmatrix}, \tag{5}\]  
where \(A^{-1}\) is the inverse of the pedigree relationship matrix, \(A_{22}^{-1}\) is the inverse of the numerator relationship matrix for genotyped animals, and \(G^{-1}\) is the inverse of the genomic relationship matrix.  
Variance components and estimated breeding values (EBVs) or genomic estimated breeding values (GEBVs) were determined using restricted maximum likelihood (REML) as implemented in the ASReml-R software (v4.2) [29].