### Introduction

*G*) and e~MVN(0,R), where MVN indicates a multivariate normal distribution and E(y) = Xb, cov(u, e') = 0. The general solution using the maximum likelihood (ML) method, which shows the effects of b and u, is in the form of:

_{u}*G*

_{u}^{-1}is the inverse matrix of the variance-covariance matrix of random effects, which indicates the SNP-SNP variance-covariance matrix. The GRM, G, can be calculated using R package "rrBLUP." For the feasible and precise prediction of the breeding value of organisms, genome-wide studies are quickly becoming mainstream. SNP information can show us the precise and broad application of BLUP, because it is widely applicable, and hundreds of thousands of genetic markers (SNPs) can be tested for association with a phenotype [7].

*G*matrix, developed by Fernando and Grossman [12], uses the IBD concept, which is complicated and vulnerable to incorrect calculations in organisms. There is no R package available in

_{u}*G*matrix calculation (http://www.r-project.org). However, the G matrix can be calculated from R codes [12, 13]. We proposed a simple method for calculating the

_{u}*G*matrix from the G matrix. The SNP-SNP relationship matrix means the variance-covariance matrix between SNPs.

_{u}### Methods

### Data preparation

### The method of statistical SNP-SNP variance-covariance matrix calculation

*Zu*as the independent variable instead of

*u*, which denotes the breeding value or genetic value

*Zu*for the derivation of BLUP. We assumed the normality condition of the genetic value

*Zu*and random effect u. Based on these assumptions, the variance-covariance matrix of genetic values,

*Zu*, is the GRM, G.

*G*

_{u}^{-1}=

*Z*

^{T}G^{-1}

*Z*. This important relationship links the SNP-SNP variance covariance matrix

*G*to the GRM G.

_{u}### The generalized least squares for the BLUP

### The Sherman-Morrison-Woodbury lemma

*G*

_{u}^{-1}=

*Z*

^{T}

*G*

^{-1}

*Z*, we should calculate the inverse matrix of the

*G*

_{u}^{-1}matrix. Because a lot of time was needed for calculating the inverse matrix directly, we used the Sherman-Morrison-Woodbury (SMW) lemma [20, 21, 22, 23]. The formula was as follows: where A and G are both invertible, and A + YGZ

^{*}are invertible if and only if

*G*

^{-1}+

*Z*

^{*}

*A*

^{-1}

*Y*are invertible. Practically, to reduce the error, we used A as the identity matrix (I matrix), and the practically used formula was as follows:

*G*using the lemma was faster than the direct calculation, because we used larger SNPs than the sample size.

_{u}### Results

### Genetic value prediction and SNP-GBLUP

### SNP-SNP relationship matrix

*G*square matrix obtained using the SMW lemma. The variance components (diagonal parts) were the values close to 1. It suggests that the

_{u}*G*matrix can be interpreted as the SNP-SNP relationship matrix. The covariance components can be interpreted as the SNP-SNP relationships or, in other words, SNP-SNP interaction terms, because the variance components represent the SNPs' relationships themselves.

_{u}### Estimated heritability

^{2}) using simple regression between the genetic values and the height phenotypic values. The heritability was 0.24 in G-BLUP and SNP-GBLUP and 0.20 in SNP-BLUP. According to Yang et al.'s article [2], only 45% of variance can be explained by total SNPs. Therefore, our estimated heritability was not poor, and of the generally accepted narrow-sense heritability of 0.8, we explained 33% in G-BLUP and SNP-GBLUP. We explained 53% of the heritability when using total SNPs. Also, we found the method that is widely applicable, SNP-GBLUP, because it can predict the SNP effects. It is impossible in G-BLUP, and

*G*matrix components can be used widely, such as in genome-wide association studies (GWASs), because they can be interpreted as representing the interaction terms between SNPs.

_{u}### Discussion

### SNP-GBLUP and its applicability

### Heritability and the GLS approach in BLUP

^{2}) reflects the additive effects of QTL. The estimated heritability from our data was smaller than the generally accepted heritability. This is because causal variants were not in complete LD with the SNPs that were genotyped. Incomplete LD might occur if causal variants have a lower MAF than genotyped. The effects of the SNPs are treated statistically as random, and the SNPs have a small effect on the trait [2]. However, we achieved better heritability than the predicted value in common GWASs-i.e., only ~5% of the phenotypic variance in human height [27].