## 4 Discussion  
In this work, a stable variable selection method with shrinkage regression was proposed and applied to the ADNI public dataset. Based on the repeated applications of a penalized regression model, we designed a procedure that measured the relative importance of each selected variable via the combination of the results of all the penalized regression models. Through its design, the procedure highlights the most commonly selected variables, giving greater emphasis to the ones selected in the models that best fit the data. Since the proposed procedure is more restrictive than the usual correspondent penalized regression approach, stability is favored during variable selection. The penalized regression models were constructed using LASSO (\(\alpha\) = 1) and ElasticNet (\(0<\alpha<1\)) methods for \(\alpha\) values in the grid {0.75, 0.50, 0.25, 0.10, 0.05, 0.01}. The alpha parameter controls the balance between the L1 and L2 regularization penalties; \(\alpha\) = 1 forces some variable' coefficients to be equal to zero, thus leading to a smaller number of selected predictors; and, in general, the number of selected variables increases as the \(\alpha\) value decreases. For each specific value of \(\alpha\), a large dispersion in the number of selected variables was observed in relation to the 100 penalized regression models. The designed procedure  
Figure 3: AUC and corresponding 95% confidence interval, accuracy and corresponding 95% confidence interval, and F1-measure (from left to right) for each final model (black) and for the corresponding logistic regression model (gray) for two scenarios: (**a1–a3**) Scenario SNPs; (**b1–b3**) Scenario SNPs + Cov. Results are organized by alpha values in grid {0.01, 0.05, 0.10, 0.25, 0.50, 0.75, 1}. allowed us to overcome this instability by defining a single set of predictor variables which were restricted to the most important variables across all the penalized models. In analyzing the final models across the \(\alpha\) values, differences were found in the set of the selected variables. A more in-depth analysis showed that the variables' rank by their weights, as defined in step 2, is the same across all \(\alpha\) values. This means that some variables were not flagged as important due to the restrictive cutoff value (defined as 0.8), and, therefore, they were not included in the final model. The lack of consistency between the \(\alpha\) models can be overcome via adjusting the threshold value for the \(\alpha\) parameter. As expected, for each fixed \(\alpha\), there were differences between scenarios with and without the inclusion of the covariates age, sex, educational level, and _APOE4_ genotype (Scenario SNPs and Scenario SNPs + Cov, respectively). In general, the results were consistent for the common SNPs of the two scenarios proposed; the significance of the coefficients and the effects of the variables (risk or protection) remained the same (Figure 2(a1,b1)). The same was verified in relation to the corresponding traditional logistic regression model. When analyzing the predictors selected by our procedure in greater detail, particularly those observed for higher \(\alpha\) values, and when comparing these results with the literature, some interesting results emerge. Both the SNP _rs2075650_ and the _APOE4_ genotype are referenced in the literature as risk factors for AD [18,19]. The first factor was selected for our procedure in both scenarios (Scenario SNPs and Scenario SNPs + Cov). The OR estimates obtained by the proposed procedure were, as expected, lower than those obtained with the traditional regression procedure both in this work and in the literature (e.g., in reference [18], OR = 4.178 and 95% CI 1.891-9.228). The second risk factor mentioned above, the _APOE4_ genotype, was also selected in the construction Scenario SNPs + Cov regression model. In addition, the SNPs _rs573399_ and _rs11906462_ were identified as important variables in the two scenarios and, in both cases, as risk factors for AD.