## 3 Results
### An Example for the Use of HRePML  
To run HRePML requires four input files. The first input file is a genotype file, where each row represents the SNP marker, and each column represents an individual. The first two columns of the genotype file provide the chromosome and physical position information about the SNP marker. The genotype is coded as 0, 1, and 2, representing aa, Aa, and AA, where \"A\" is a dominant allele and \"a\" is a recessive allele. The second input file is the phenotype file, which is a column vector. The third input file is the genetic relatedness matrix or kinship file, which is a \(N\times N\) matrix, and \(N\) is the number of individuals. The fourth input file is the covariates file, where the first column is the unit column vector, followed by the population structure or principal component matrix. The example data can be found at [[https://github.com/wenlongren/HRePML/tree/master/Example%20](https://github.com/wenlongren/HRePML/tree/master/Example%20) Data]([https://github.com/wenlongren/HRePML/tree/master/Example%20](https://github.com/wenlongren/HRePML/tree/master/Example%20) Data).  
In a Linux system (Ubuntu), the compiling command is: g++ -l/path/liblbfgs-1.10/include HRePML-Linux.cpp -llapack -lblas -lbfgs -o output, where it needs to include the C++ library of limited-memory BFGS. If Math Kernel Library (MKL) has been installed for Intel CPU users, the following compiling command is recommended: g++ -l/path/liblbfgs-1.10/include HRePML-Linux.cpp -lml_gf_lp64 -lml_sequential -lml_core -lbfgs -o output. In order to save the results, the HRePML program requires two output files, which are the results file and the computational time file. After compilation, the execution command is:./output Genotype.csv Phenotype.csv Kinship.csv Fixed.csv Results.csv Time.csv. Then, the results and running time are output into Results.csv and Time.csv files, respectively.  
Figure 5: **(A)**. Effect of the sample size on running time in the third simulation. (B) Effect of markers on the running time in the fourth simulation.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Detected Genes** & **Associated Trait** & **Chr.** & **Position** & **Effect Estimate** & **LOD/p-Value** & **Methods** \\ \hline _AT2G16440_ & LFS GH & 2 & 7140030 & \(-\)7.461, \(-\)9.107, \(-\)5.16 & 3.90\(\times 10^{-11}\), \(1.28\times 10^{-17}\), \(9.56\times 10^{-8}\) & FarmCPU, MLMM, GEMMA \\ \hline _AT3G07160_ & LFS GH & 3 & 2280271 & \(-\)5.934, \(-\)8.845 & \(1.16\times 10^{-7}\), \(9.37\times 10^{-15}\) & FarmCPU, MLMM \\ \hline _AT3C54280_ & MT GH & 3 & 20090780 & 1.002, 1.762 & \(9.90\times 10^{-13}\), \(5.65\times 10^{-8}\) & FarmCPU, MLMM \\ \hline _AT4G09960_ & FT Duration GH & 4 & 6228754 & 0.822, 1.136 & 3.74, \(3.69\times 10^{-8}\) & HRePML, FarmCPU \\ \hline _AT4G33620_ & LC Duration GH & 4 & 16140068 & 2.996, 2.540 & 4.78, \(4.29\times 10^{-29}\) & HRePML, MLMM \\ \hline _AT5G45900_ & LC Duration GH & 5 & 18625634, & \(-\)3.707, \(-\)6.051 & 4.78, \(2.51\times 10^{-28}\) & HRePML, FarmCPU \\ \hline _AT5G45900_ & LFS GH & 5 & 18625634, & \(-\)4.318, \(-\)5.147, \(-\)5.616 & 5.23, \(1.83\times 10^{-8}\), \(1.05\times 10^{-7}\) & HRePML, FarmCPU, GEMMA \\ \hline _AT5G53360_ & MT GH & 5 & 21646741 & 0.236, 0.267 & \(3.05\times 10^{-14}\), \(1.55\times 10^{-7}\) & FarmCPU, GEMMA \\ \hline \end{tabular}
\end{table}
Table 4: Previously reported genes that were identified at least by two methods simultaneously with HRePML, MLMM, FarmCPU, and GEMMA.