## 5 Discussion  
In a large simulation study, we compared the prediction accuracy and model complexity of several popular variable selection methods. Our published protocol [12] included BSS and BE as representatives of long-established classical methods. Although FS was not part of the original protocol, we decided to include it during the analysis phase, as it is also a widely used classical method. We then compared their performances with those of penalized methods: NNG, lasso, ALASSO, and RLASSO. While RLASSO was originally proposed for high-dimensional data, we included it to assess its performance in low-dimensional data. Our focus on low-dimensional data allowed us to better understand the results. The penalized methods considered can also be applied to high-dimensional data, but among the classical methods, only FS is suitable for such settings. Due to the complexity of the simulation design, which involved seven methods, three model selection  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multicolumn{2}{c}{**Full Model**} & \multirow{2}{*}{**BE**} & \multirow{2}{*}{**BSS**} & \multirow{2}{*}{**FS**} & \multirow{2}{*}{**NNG**} & \multirow{2}{*}{**ALASSO**} & \multirow{2}{*}{**Lasso**} & \multirow{2}{*}{**RLASSO**} & \multirow{2}{*}{**VIF**} \\  & & & & & & & & & & 1.99 \\ x5 & 0.044 & ✓ (-) & ✓ (-) & & ✓ (-) & ✓ (-) & ✓ (-) & & 1.97 \\ x6 & 0.598 & & & & & & & & 1.65 \\ x7 & 0.535 & & & & & & & & 1.10 \\ x8 & 0.735 & & & & & & & & 5.56 \\ x9 & 0.471 & & & & & ✓ (-) & ✓ (-) & ✓ (-) & & 4.49 \\ x10 & 0.084 & ✓ (�) & ✓ (�) & – (�) & ✓ (�) & ✓(�) & ✓(�) & & 1.59 \\ x11 & 0.877 & & & & & & & 1.51 \\ x12 & 0.231 & & & & & ✓ (-) & ✓ (-) & ✓(-) & & 1.10 \\ x13 & 0.281 & & & & & ✓ (-) & ✓ (-) & ✓(-) & & 2.34 \\ x14 & 0.221 & & & ✓ (-) & ✓ (-) & ✓ (-) & ✓(-) & & 1.33 \\ x15 & 0.000 & ✓ (�) & ✓ (�) & ✓ (�) & ✓ (�) & ✓ (�) & ✓(�) & ✓(�) & 1.05 \\ x16 & 0.000 & ✓ (�) & ✓ (�) & ✓ (�) & ✓ (�) & ✓ (�) & ✓(�) & ✓(�) & 1.13 \\ x17 & 0.893 & & & & & & & & & 1.11 \\ x18 & 0.820 & & & & & & & & & 1.36 \\ x19 & 0.458 & & & & & & & & & 1.31 \\ x20 & 0.478 & & & & & & & & & 5.29 \\ x21 & 0.684 & & & & & & & & & 1.23 \\ x22 & 0.277 & & & & & & ✓ (-) & ✓(-) & & 7.00 \\ x23 & 0.379 & & & & & ✓ (-) & ✓(-) & ✓(-) & & 1.21 \\ x24 & 0.588 & & & & & & & & & 1.14 \\ \hline \#variables & 21 & 4 (3) & 6 (3) & 2 (3) & 8 (3) & 8 (3) & 9 (2) & 2 (2) & \\ Adjusted \(R^{2}\) & 0.18 & (0.19) & (0.19) & (0.19) & (0.20 (0.17) & 0.20 (0.17) & 0.18 (0.14) & 0.17 (0.17) & \\ RMSE (CV) & 0.356 & 0.360 & 0.358 & 0.369 & 0.361 & 0.361 & 0.367 & 0.369 & \\ RMSE (BIC) & 0.356 & 0.363 & 0.364 & 0.364 & 0.363 & 0.363 & 0.384 & 0.369 & \\ \hline \hline \end{tabular} \#variables denotes the number of covariates.  
\end{table}
Table 4: Variables selected by different approaches tuned using CV and BIC criteria under the low condition. Variables selected using BIC are enclosed in brackets. Blank spaces indicate variables excluded by a method under both criteria. A check mark (�) indicates that a variable was selected using a given criterion, while a dash (–) indicates that the corresponding variable was excluded. The last column reports the VIF, which measures the degree of multicollinearity.  
criteria, five sets of true regression coefficients, four correlation structures, four SNRs, and two sample sizes, we structured our results into five objectives and summarized the key findings for each. Below, we briefly discuss the results of each objective before drawing our final conclusions.