## 3 Results  
As expected, multiple runs of the same \(\alpha\)-penalized regression procedure led to different results, confirming the instability of such procedures in contexts where the number of variables (i.e., SNPs) is much higher than the number of individuals. Indeed, there was a large variability in the number of selected variables across the 100 penalized regression models, both in cases where models were fitted to covariates (Scenario SNPs + Cov) and in which they were not (Scenario SNPs). As shown in Table 1, for all values of \(\alpha\), there was a large dispersion in the number of selected variables. For example, in Scenario SNPs and \(\alpha\) = 0.01, the number of selected variables ranged from 0 to 6886. In both scenarios, the range and the maximum number of selected variables generally increased as the \(\alpha\) value decreased. In relation to the median number of selected variables, a tendency to increase as the value of \(\alpha\) decreased was also observed in Scenario SNPs + Cov. Although this trend was not observed in the SNP scenario for alpha values ranging between 0.05 and 1, the median number of the selected variables increased substantially for the model where \(\alpha\) = 0.01 (Figure 1). \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Alpha Parameter**} & \multicolumn{2}{c}{**Number of Selected Variables**} \\ \cline{2-3}  & **Scenario SNPs** & **Scenario SNPs + Cov** \\ \hline
0.01 & 4958 (0–6886) & 4843 (9–6399) \\
0.05 & 9 (1–779) & 156 (9–1427) \\
0.10 & 41 (0–225) & 107 (5–419) \\
0.25 & 48 (0–153) & 61 (4–207) \\
0.50 & 33 (0–70) & 49 (4–103) \\
0.75 & 22 (1–52) & 6 (1–14) \\
1 & 27 (0–45) & 16 (1–66) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Minimum, maximum, and median number of selected variables in 100 penalized regression models, organized by alpha value, for each scenario: models considering only SNPs (left); and models adjusted to covariates age, sex, educational level, and _APOE4_ genotype (right). Results are shown as the median (minimum–maximum). Regarding the final models, which were built based on the proposed procedure, the largest number of selected variables occurred when \(\alpha=1\) (with 11 variables) in Scenario SNPs, and when \(\alpha=0.01\) and \(\alpha=0.05\) (with 9 variables) in Scenario SNPs + Cov. In the latter scenario, the _APOE4_ genotype covariate was always selected. It should be noted that, in the SNP scenario, there were several alpha values for which no variables were selected (Table 2). The magnitude of the coefficient estimates on the proposed models increased as the value of \(\alpha\) increased. In general, as the alpha value increased, the odds ratio deviated further from the value 1 (Figure 2(a1,b1)). A high similarity existed between the selected variables in both scenarios. The main difference was that the model fitted in Scenario SNPs + Cov contained the covariate _APOE4_ genotype, instead of the SNPs _rs6090754_, _rs1052242_, and _rs4982401_. Naturally, the magnitude of the coefficient estimates was higher in the \(\alpha=1\) model and, consequently, the risk and protection effects were increased. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Alpha Parameter**} & \multicolumn{3}{c}{**Number of Selected Variables**} \\ \cline{2-3}  & **Scenario SNPs** & **Scenario SNPs + Cov** \\ \hline
0.01 & 0 & \(8+1\) \\
0.05 & 1 & \(8+1\) \\
0.10 & 0 & \(4+1\) \\
0.25 & 0 & \(3+1\) \\
0.50 & 0 & \(3+1\) \\
0.75 & 1 & \(0+1\) \\
1 & 11 & \(2+1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of selected variables for each final model, organized by alpha parameter, for the two scenarios: model considering only SNPs (left); and model adjusted to covariates age, sex, educational level, and _APOE4_ genotype (right). Scenario SNPs + Cov results are shows as number of SNPs + number of covariates.