## 2 Materials and Methods
Following the same reasoning as the selection of the relevant variables, the coefficients associated with better quality models will greatly contribute to the overall value of the coefficient of the predictor variable. This final coefficient indicates whether the variable contributes positively or negatively to the outcome of the study. In the practical application being analyzed, this coefficient allows us to conclude whether the variable is a risk factor or a protection factor for Alzheimer's disease. In short, from several sets of selected variables, a single set of (the most relevant) variables is obtained, with a regression coefficient being assigned to each one. To enhance the clarity of the process, the description of the proposed procedure was divided into three steps. First, several runs of the same penalized regression method are performed, and the result of each model is associated with a relative goodness-of-fit weight (step 1). Then, a global weight is assigned to each variable, defined as the sum of the weights of all the penalized models in which the variable was selected, representing its relative importance across the penalized models. If a variable's weight exceeds a defined threshold, it is then selected for the final model (step 2). The coefficient estimate for each selected variable is determined as the average of its estimated coefficients in each penalized model, weighted by the relative goodness-of-fit weight of the model (step 3). The procedure is detailed below, and the graphical summary of the algorithm is available in the Supplementary Materials. **Penalized regression models (Step 1). ** Repeated applications of a penalized regression model are conducted on the training dataset. Each repetition of the penalized regression model generates a set of predictor variables, which may not be the same. The Akaike's information criterion (AIC) was calculated for each model, and the difference between the AIC and the AIC of the best model was calculated (a higher AIC means a lower fitness):  
\[\Delta_{i}\ =\ AIC_{i}-AIC_{min} \tag{1}\]  
The smaller this difference, the better the model's fit for the data. These differences allowed us to obtain the Akaike's weight for each model using the following equation:  
\[w_{i}\ =\ \frac{exp\left(-\frac{1}{2}\Delta_{i}\right)}{\sum_{r\ =\ 1}^{R}exp\left(-\frac{1}{2}\Delta_{r}\right)}. \tag{2}\]  
Now, the higher the value of this measure, the better the model's quality of fit. The model Akaike's weight is used in the next steps to define the variables across all penalized models and the regression coefficient in the final model. **Stable variable selection (Step 2). ** At this point, for each \(\alpha\)-penalized regression technique, we obtained R models, and each one was assigned an Akaike's weight. The next challenge was how to combine these weights for the estimation of the relative importance of each predictor. We defined the variable weight as the sum of the Akaike's weight of all repeated penalized models in which the variable appears. \[w_{j}\ =\ \sum_{i\ =\ 1}^{R}w_{i}a_{ij},\ \ \ j\ =\ 1,\ldots,p. \tag{3}\]  
where \(\left[a_{1j},\ a_{2j},\ \ldots,a_{Rj}\right]\) is an indicator vector of the non-null \(j^{\text{th}}\) coefficient, and \(p\) is the number of variables. The criterion for classifying a variable as important was its weight, \(w_{j}\), being at least 0.8. This importance threshold, in general, demonstrated good properties, allowing the minimization of the occurrence of type I and II errors [17]. **Model coefficients estimation (Step 3). ** In the final model, only the important predictors were considered; that is, the predictors with a weight of at least 0.8. For these, the estimated coefficient was defined by an overall weighted coefficient as follows:  
\[\beta\ =\ \frac{\sum_{i\ =\ 1}^{R}w_{i}\hat{\beta}_{i}}{\sum_{r\ =\ 1}^{R}w_{i}}, \tag{4}\]  
where \(\hat{\beta}_{i}\) is the coefficient estimate of the predictor in the model obtained in run \(i\), and \(w_{i}\) is the Akaike's weight of that model.