The simulated population consisted of 3,220 individuals in two generations. The first generation consisted of 20 sires and 200 dams, which were assumed to be unrelated. Each sire mated with 10 dams and each dam produced 15 progenies, leading to a total of 3,000 individuals in the second generation. Of the 15 progenies of each dam, 10 were phenotyped for a continuous trait. All of the 3,220 individuals were genotyped for 9,990 SNP markers distributed on 5 chromosomes without missing. Each chromosome had a size of 1 Morgan (M) and carried 1,998 evenly distributed SNPs.
Variance component estimation
We applied the software DMU (Version 6, release 5.0) [3
] to estimate the variance components of the simulated trait, which would be used in the subsequent association analysis, based on the following model
Where y is the vector of phenotypes of the 2,000 phenotyped individuals, μ is the overall mean, a is the vector of the residual polygenic effect with (where A is the additive genetic relationship matrix and is the additive genetic variance), Z is the incidence matrix of a, and e is the vector of residual errors with (where I is a unit matrix and is the residual error variance).
Genotype quality control
We removed the 1,000 progenies without phenotypes off the genotype data, and we calculated the minor allele frequency (MAF) for each SNP for the remained 2,220 individuals (2,000 progenies and 220 parents). We found that 2,879 SNPs were homozygous (MAF = 0) for all the tested individuals and additionally 715 SNPs had a MAF less than 0.03. These SNPs were removed and 6,396 SNPs remained for the subsequent analyses.
The mixed model based single locus analysis [2
] was performed based on the following linear mixed model:
where y is the vector of phenotypes of the 2000 phenotyped individuals, μ is the overall mean, × is the vector of the SNP genotype indicators which takes values 0, 1 or 2 corresponding to the three genotypes 11, 12 and 22 (assuming 2 is the allele with a minor frequency), b is the regression coefficient of phenotypes on SNP genotypes (i.e., the substitution effect of the SNP), a is the vector of the residual polygenic effect with , Z is the incidence matrix of a, and e is the vector of residual errors with .
For each SNP, the estimate of b and the corresponding sampling variances can be obtained via mixed model equations (MME), and a Wald chi-squared statistic with df =1 was constructed to examine whether the SNP is associated with the trait.
For the analyses above, the permutation method was adopted to adjust for multiple testing from the number of SNP loci detected. In our method, the phenotypes were permuted 10,000 times against the genotype and pedigree data and the empirical distribution of the Wald chi-squared statistic under the null hypothesis (no association existed between any SNP and the trait in genome-wide level) was obtained using the largest Wald chi-squared statistic value across all SNPs from each permuted dataset. The threshold value for declaring a significant association was determined by choosing the 95th percentile of the empirical distribution, i.e., we declared a significant SNP at a 0.05 genome-wide significance level if its raw value of the Wald chi-squared statistic was larger than the empirical threshold value.
For the significant SNPs, linkage disequilibrium (LD) in term of D' between them was quantified using Haploview  and the LD blocks were defined by the criteria of Gabriel et al.  with default parameters.