## 3 Results  
### Statistical Properties  
We compared the statistical properties of the new HRePML method with those of the multi-locus mixed-model (MLMM) [26], fixed and random model circulating probability unification (FarmCPU) [27], and genome-wide efficient mixed model association (GEMMA) [9] methods. Here, the statistical properties mainly included statistical power and mean squared error (MSE). In the first simulation study, the dataset consisted of 500 individuals and 10,000 SNP markers with 1000 replicates. Eight true QTNs were set in each replicate. Then, this dataset was regarded as having 10,000,000 SNPs and 8000 true QTNs in total. The average power of the four methods were 60.39%, 53.15%, 53.21%, and 36.34%, respectively. HRePML obtained the highest statistical power, which was at least about 7% higher than the other three methods. In particular, HRePML performed well on QTNs with lower heritability, such as QTN 1, 3, 4, and 6 (Tables 1 and 2 and Figure 1A). The mean squared error was used to measure the accuracy of the QTN effect estimates, and smaller MSE represented better accuracy. The average MSE of the above four methods were 0.0772, 0.1068, 0.1165, and 0.1933, respectively, demonstrating that the average MSE of HRePML was the minimum (Tables 1 and 2 and Figure 2A).  
To further validate the performance of HRePML, an additive polygenic effect was involved in the second simulation study. The dataset was the same with that used in the first simulation study, except that polygenic effect was added to the phenotype. The same trend in statistical power was observed, and the average powers of HRePML, MLMM, FarmCPU, and GEMMA were 62.45%, 57.08%, 55.18%, and 40.46%, respectively, which showed that HRePML was still powerful and robust under polygenic interference (Tables S1 and S2 and Figure 1B). As far as the mean squared error was concerned, the average MSE of the above four methods were 0.0926, 0.1343, 0.1184, and 0.2125, respectively. HRePML had the most accuracy of the QTN effect estimates, followed by FarmCPU, MLMM, and GEMMA (Tables S1 and S2 and Figure 2B).  
In the third simulation study, we investigated the effect of the sample size on the statistical power of HRePML. There were four datasets consisting of 500, 1000, 2000, and 4000 individuals, respectively, and 10,000 SNP markers, with 100 replicates. Eight true QTNs were set in each replicate. Then, each dataset could be regarded as having 1,000,000 SNPs and 800 true QTNs in total. The average powers of sample sizes 500, 1000, 2000, and 4000 were 59.88%, 75.75%, 82.00%, and 91.13%, respectively. Clearly, the statistical power improved as the sample size increased. The results demonstrated that the statistical power could be more than 80% for QTN, with heritability equal or greater than 0.03 when the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Statistical Properties** & **HRePML** & **MLMM** & **FarmCPU** & **GEMMA** \\ \hline Average power (\%) & 60.39 & 53.15 & 53.21 & 36.34 \\ Average MSE & 0.0772 & 0.1068 & 0.1165 & 0.1933 \\ Running time (Hour) & 3.1419 & 22.7274 & 4.6653 & 2.4186 \\ \hline \hline \end{tabular}  
* The dataset used in Table 2 is the same as that used in Table 1.  
\end{table}
Table 2: Comparison of average statistical power, average mean squared errors (MSE), and running time among the HRePML, MLMM, FarmCPU, and GEMMA methods in the first simulation study *.