## 3 Results
### Results on the Case Study with Real Data  
In this section, we present the results obtained by applying the two variable ranking algorithms, evaluated in-silico in Section 3.1, to the case study about T2D onset prediction, presented in Section 2.3. In this case study, the dataset includes two highly correlated features that represent the concept of obesity: BMI and waist circumference. Since obesity is the leading risk factor for T2D development [29], we expect that the variable ranking algorithms will rank these variables as the top predictors of T2D onset. The results of variable ranking are shown in Table 5; for each scenario we report the mean (SD) of the rank obtained for each feature across the B = 100 runs of the RFE algorithm. Lower rank are assigned to the most relevant features. For each scenario, the features were ordered by their mean rank value, which represents the variable importance. Figure 5: RMSE results obtained for the standard RFE-Borda count method (red boxplot, “no corr.” label) and the proposed algorithm (green boxplot, “with corr.” label) for scenarios 10–12, characterized by a fraction of censored data equal to 0.1 (**left**), 0.3 (**middle**), and 0.7 (**right**). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Approach 1**} & \multicolumn{2}{c}{**Approach 2**} & \multicolumn{2}{c}{**Approach 3**} & \multicolumn{2}{c}{**Approach 4**} \\ \hline \multirow{2}{*}{**Variable**} & **Mean (SD)** & \multirow{2}{*}{**Variable**} & **Mean (SD)** & \multirow{2}{*}{**Variable**} & **Mean (SD)** & \multirow{2}{*}{**Variable**} & **Mean (SD)** \\  & **of Rank** & & **of Rank** & & **of Rank** & **of Rank** \\ \hline \multirow{2}{*}{\(\mathrm{hdl}\)} & 2.91 & \multirow{2}{*}{bmi-waist} & 1.01 & \multirow{2}{*}{bmi} & 1.02 & \multirow{2}{*}{waist} & 1.04 \\  & (3.04) & & (0.10) & & (0.14) & & (0.24) \\ \multirow{2}{*}{\(\mathrm{sys\_bp}\)} & 3.12 & \multirow{2}{*}{hdl} & 2.83 & \multirow{2}{*}{hdl} & 2.74 & \multirow{2}{*}{sys\_bp} & 2.76 \\  & (3.42) & & (2.78) & & (3.06) & & (2.79) \\ \multirow{2}{*}{\(\mathrm{bmi}\)} & 4.47 & \multirow{2}{*}{sys\_bp} & 2.96 & \multirow{2}{*}{sys\_bp} & 2.86 & \multirow{2}{*}{\(\mathrm{hdl}\)} & 2.82 \\  & (4.66) & & (2.98) & & (3.01) & & (2.48) \\ \multirow{2}{*}{\(\mathrm{waist}\)} & 4.51 & \multirow{2}{*}{phealth} & 5.71 & \multirow{2}{*}{phealth} & 5.44 & \multirow{2}{*}{phealth} & 5.32 \\  & (4.39) & & (5.00) & & (4.69) & & (5.11) \\ \multirow{2}{*}{\(\mathrm{phealth}\)} & 6.16 & \multirow{2}{*}{depress} & 6.21 & \multirow{2}{*}{age} & 6.53 & \multirow{2}{*}{sex} & 5.48 \\  & (5.42) & & (3.29) & & (3.03) & & (3.08) \\ \multirow{2}{*}{\(\mathrm{depress}\)} & 6.72 & \multirow{2}{*}{age} & 6.70 & \multirow{2}{*}{depress} & 6.64 & \multirow{2}{*}{depress} & 6.65 \\  & (3.41) & & (3.19) & & (3.14) & & (3.05) \\ \multirow{2}{*}{\(\mathrm{age}\)} & 7.23 & \multirow{2}{*}{depriv} & 7.08 & \multirow{2}{*}{depriv} & 7.08 & \multirow{2}{*}{htm} & 7.40 \\  & (3.31) & & (3.82) & & (3.72) & & (4.21) \\ \multirow{2}{*}{\(\mathrm{depriv}\)} & 7.23 & \multirow{2}{*}{htm} & 7.32 & \multirow{2}{*}{htm} & 7.59 & \multirow{2}{*}{depriv} & 7.59 \\  & (3.31) & & (4.23) & & (4.10) & & (3.50) \\ \multirow{2}{*}{\(\mathrm{htn}\)} & 7.95 & \multirow{2}{*}{hchol} & 8.96 & \multirow{2}{*}{sex} & 8.32 & \multirow{2}{*}{age} & 7.98 \\  & (4.52) & & (3.29) & & (3.55) & & (2.91) \\ \multirow{2}{*}{\(\mathrm{sex}\)} & 8.78 & \multirow{2}{*}{tot\_chol} & 8.99 & \multirow{2}{*}{hchol} & 8.95 & \multirow{2}{*}{hchol} & 9.22 \\  & (3.74) & & (2.71) & & (3.10) & & (2.93) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of variable ranking for the predictive model of T2D onset. By applying the standard RFE-Borda count algorithm, we can note that, in the approaches 3 and 4, in which only one obesity variable is considered in the set of candidate predictors (the other is a priori excluded), the obesity variables are ranked in the top position of the list, i.e., they are identified as most important predictors.