## 4 Example: Respiratory Health Data  
The ozone data originates from a study investigating the effects of ozone on school children's lung growth. The study was conducted from February 1996 to October 1999 in Germany, involving 1101 school children who were initially in the first and second primary school classes (ages 6-8) [57]. Over the four years, lung function measurements were collected three times per year (spring, summer, and autumn), except for spring 1998 [57]. A subset of 496 children with complete data has been used in previous studies to investigate medical issues [57], and as an example in methodological papers to assess the stability of model-building strategies [57] and bootstrap model averaging [58]. The same subset is analyzed here to illustrate various issues relevant to our simulation study. The outcome variable is forced vital capacity (FVC), which measures the amount of air that can be forcibly exhaled from the lungs after taking the deepest breath possible, with 24 covariates as shown in Table 3. For further details, see [PERSON] et al. [57] and [PERSON] et al. [99]. To assess the predictive accuracy of the models, we split the dataset into a training set (70%) and a test set (30%). This approach was chosen because it allows for the comparison of the number of variables selected by each method and their corresponding RMSE. The full OLS model with 24 covariates fitted to the training data yielded an \(R^{2}\) of 65%, which we refer to as the adequate information setting. To illustrate how the amount of information in the data affects variable selection and prediction accuracy, we conducted two analyses. The first analysis used the full set of candidate variables (Section 4.1). The second analysis \begin{table}
\begin{tabular}{l 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**} \\  & \multicolumn{1}{c}{_p-Value_} & & & & & & & & \\ \hline FLGROSS (x1) & 0.000 & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & 2.52 \\ SEX (z2) & 0.000 & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & 1.10 \\ FLGEM (x3) & 0.000 & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & ✓(✓) & 2.23 \\ HOCHOZON (x4) & 0.023 & & & & ✓(-) & ✓(-) & ✓(-) & 2.00 \\ FNOH24 (x5) & 0.026 & & & & ✓(-) & ✓(-) & ✓(-) & 1.97 \\ SFPET (x6) & 0.052 & & & & ✓(-) & ✓(-) & ✓(✓) & 1.68 \\ FLOTM/MED (x7) & 0.053 & & & & ✓(-) & ✓(-) & ✓(-) & 1.10 \\ FO3H24 (x8) & 0.070 & & & & & & & 5.62 \\ FTEH24 (x9) & 0.144 & & & & & & & 4.50 \\ PSATATM (x10) & 0.194 & & & ✓(-) & ✓(-) & ✓(-) & 1.60 \\ FTHE (x11) & 0.241 & & & ✓(-) & ✓(-) & ✓(-) & 1.54 \\ AVATOP (x12) & 0.264 & & & ✓(-) & ✓(-) & ✓(-) & 1.11 \\ FMHL (x13) & 0.293 & & & ✓(-) & ✓(-) & ✓(-) & 2.36 \\ PSALQE (x14) & 0.321 & & & & & & & 1.34 \\ AGEBGEW (x15) & 0.332 & & & ✓(-) & ✓(-) & ✓(-) & 1.13 \\ ALTER (x16) & 0.519 & & & ✓(-) & ✓(-) & ✓(-) & 1.47 \\ ARAUCH (x17) & 0.521 & & & & & & & 1.17 \\ ADHCH (x18) & 0.741 & & & & & & & 1.37 \\ FSILLAUF (x19) & 0.766 & & & & & & & 1.31 \\ FPOLL (x20) & 0.771 & & & & & & & 5.30 \\ AMATOP (x21) & 0.917 & & & & & & & 1.23 \\ PSPT (x22) & 0.938 & & & & & & & & 7.06 \\ FSNIGHT (x23) & 0.941 & & & & & & & & 1.23 \\ ADEKZ (x24) & 0.986 & & & & & & & & 1.14 \\ \hline \hline \#variables & 24 & 3 (3) & 3 (3) & 3 (3) & 13 (3) & 13 (3) & 13 (4) & 3 (3) & \\ Adjusted \(R^{2}\) & 0.63 & 0.62 & 0.62 & 0.62 & 0.63 (0.57) & 0.63 (0.57) & 0.63 (0.61) & 0.62 (0.62) & \\ RMSE (CV) & 0.238 & 0.242 & 0.242 & 0.242 & 0.237 & 0.237 & 0.241 & 0.242 & \\ RMSE (BIC) & 0.238 & 0.242 & 0.242 & 0.242 & 0.259 & 0.256 & 0.249 & 0.242 & \\ \hline \hline \end{tabular} \#variables denotes the number of covariates.  
\end{table} 
Table 3: Variables selected by different approaches tuned using CV and BIC criteria under the high condition. Variables selected using BIC are in brackets. Blank spaces indicate variables excluded by a method under both criteria.