## 2 Materials and Method
### Data Generating Mechanisms  
We considered 128 simulation scenarios (see Table 1). For each scenario, we generated 2000 training datasets, each consisting of a continuous response variable (y) and a matrix X of 15 continuous covariates. The number of simulation repetitions was set to 2000 to ensure that the Monte Carlo standard error (MCSE) of the model error was smaller than 0.005 for better precision (see [PERSON] and [PERSON] (1999) for details). The X matrix was sampled from a multivariate normal distribution with a mean vector of zero and a variance-covariance matrix \(\Sigma\), where \(\Sigma_{ij}\) is the pairwise correlation coefficient between covariate \(x_{i}\) and \(x_{j}\) (Table 1). After generating X, we standardized each covariate to have a mean of zero and unit variance to ensure they were on a comparable scale. Using the standardized \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{2}{l}{**Training/development dataset**} \\ \(\bullet\) & \(X\sim N_{p}(0,\Sigma)\) where \(p=15\) and \(\Sigma\in\mathbb{R}^{p\times p}\); \(\Sigma_{ij}\) is equal to the correlation coefficient between covariate \(\mathbf{x_{i}}\) and \(\mathbf{x_{j}}\) \\ \(\bullet\) & \(y=X\beta+e\) where \(\beta\in(\beta_{A},\ \beta_{B},\ \beta_{C},\ \beta_{D})\) and \(e\sim N\big{(}0,\ \sigma^{2}I_{n}\big{)}\) \\ \multicolumn{2}{l}{**True regression coefficients (8) for 15 covariates**} \\ \(\beta_{A}\): 1.5, 0, 1, 0, 1, 0, 0.5, 0, 0.5, 0, 0.5, 0, 0.5, 0, \(-0.5, 0, 0\) \\ \(\beta_{B}\): 1.50, 0, 0.50, 0, 0.50, 0, 0.25, 0, 0.25, 0, 0.25, 0, \(-0.25, 0, 0\) \\ \(\beta_{C}\): 1.0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0 \\ \(\beta_{D}\): 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 \\ \multicolumn{2}{l}{**Correlation structure (C)**} \\ \(\mathrm{C1}\): Correlation structure used by [PERSON] and [PERSON] (2000)-low correlation \\ \(\mathrm{C2}\): Autoregressive structure with \(\Sigma_{ij}=0.3^{[1-i]}\)—moderate correlation \\ \(\mathrm{C3}\) Autoregressive structure with \(\Sigma_{ij}=0.8^{[1-i]}\)—high correlation \\ Data-generating \\ \(\mathrm{mechanism}\) & \(R^{2}\) **and sample size (\(n\))** \\ \(R^{2}\) = [0.20, 0.30, 0.50, 0.71], corresponding to SNR = [0.25, 0.43, 1.00, 2.45]. \\ \(n\) = [100, 400] \\ \multicolumn{2}{l}{**Number of scenarios (full factorial design) and simulation runs**} \\ \(\beta\times\mathrm{C}\times R^{2}\times n=4\times 4\times 4\times 2=128\)  scenarios \\ \(\mathrm{N}\) = 2000 simulation repetitions per scenario \\ \multicolumn{2}{l}{**Test dataset**} \\ \(\bullet\) & New simulations with the same design as training dataset (\(n_{test}=100,000\) \\ \multicolumn{2}{l}{**Additional analyses (specified in the published protocol but \(n\) = 100 added)**} \\ \(n\) = (**100**, 400), \(\beta_{A}\), \(\mathrm{C1}\), and \(R^{2}\) with 30 covariates (i.e., the 15 original covariates and 15 uncorrelated noise variables) \\ \hline \multirow{4}{*}{Target of analysis} & \(\bullet\)  Model complexity, which is measured by the average number of variables selected.  We do not distinguish between signal and noise variables, as this distinction is more relevant for descriptive modelling. \\ \(\bullet\) & Model prediction errors on new data. \\ \hline \multirow{3}{*}{Methods} & Variable selection methods: Lasso, NNG, ALASSO, RLASSO, BSS, BE, and \(FS\) \\  & Tuning criteria: AIC, BIC, and 10-fold CV \\  & Initial estimates for ALASSO and NNG: OLS, ridge, and Lasso \\ \hline \multirow{2}{*}{Performance measures} & \(\bullet\)  Model complexity: average number of selected variables (nonzero coefficients) \\  & Prediction accuracy: **Relative risk (RR) and relative test error (RTE) (see Section 2.3)** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._covariates, we calculated the response variable \(y\) with the formula \(y=X\beta+\varepsilon\), where \(\beta\) is a vector of true regression coefficients (see Table 1). The error term \(\varepsilon_{i}\) was sampled from a normal distribution with a mean of zero and variance of \(\sigma^{2}\).