## 3 Results
### Results on the Case Study with Real Data
\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}{*}{hchol} & 9.63 & \multirow{2}{*}{phys\_act} & 9.10 & \multirow{2}{*}{tot\_chol} & 9.13 & \multirow{2}{*}{tot\_chol} & 9.37 \\  & (3.43) & & (2.63) & & (2.71) & & (2.62) \\ \multirow{2}{*}{tot\_chol} & 9.74 & \multirow{2}{*}{sex} & 9.27 & \multirow{2}{*}{phys\_act} & 9.42 & \multirow{2}{*}{phys\_act} & 9.52 \\  & (2.90) & & (2.85) & & (2.61) & & (2.52) \\ \multirow{2}{*}{phys\_act} & 9.85 & \multirow{2}{*}{life\_exp} & 9.52 & \multirow{2}{*}{life\_exp} & 9.58 & \multirow{2}{*}{life\_exp} & 9.82 \\  & (2.73) & & (2.63) & & (2.72) & & (2.63) \\ \multirow{2}{*}{life\_exp} & 10.17 & \multirow{2}{*}{smoke} & 10.10 & \multirow{2}{*}{smoke} & 10.33 & \multirow{2}{*}{mar\_stat} & 10.39 \\  & (2.77) & & (2.77) & & (2.69) & & (3.10) \\ \multirow{2}{*}{smoke} & 10.97 & \multirow{2}{*}{mar\_stat} & 10.24 & \multirow{2}{*}{mar\_stat} & 10.37 & \multirow{2}{*}{smoke} & 10.64 \\  & (2.79) & & (3.10) & & (2.85) & & (2.32) \\ \multirow{2}{*}{mar\_stat} & 11.06 & \multirow{2}{*}{(3.18)} & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._Table 6 shows the final models obtained with the four ranking approaches, considering the top 8 variables in Table 5 for approaches 2-4 and the top 9 variables in Table 5 for approach 1. For each approach, we report the estimate (standard error) of the coefficient in the final model trained on the entire training set. Note that, for approach 2, the top predictive variable is the pair of features describing obesity; however, for not introducing redundancy in the model, only one of these two variables was included in the final model. We tested the model with BMI (and the other 7 features) and the model with waist circumference (and the other 7 features) in a 5-fold cross-validation performed on the training set; the model with highest average C-index was the one with waist circumference. The results for this model are then reported in Table 6. We can see that, in all the models, the obesity variables have the highest, in absolute value, coefficient, although, in approach 1, the contribution of obesity is divided between BMI and waist circumference, with resulting lower coefficients for these variables compared to the other approaches. We can also observe that the standard error of obesity variables' coefficients is higher in approach 1 compared to the other approaches. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multicolumn{4}{c}{**Estimated Coefficient (Standard Error)**} \\ \cline{2-5}  & **Approach 1** & **Approach 2** & **Approach 3** & **Approach 4** \\ \hline bmi & 2.00 & - & 3.34 & - \\  & (0.62) & - & (0.38) & - \\ waist & 2.17 & 4.03 & - & \(4.14\) \\  & (0.75) & (0.48) & - & (0.48) \\ hdl & \(-1.75\) & \(-1.59\) & \(-2.11\) & \(-1.81\) \\  & (0.35) & (0.34) & (0.33) & (0.39) \\ sys\_bp & 1.69 & 1.74 & 1.77 & 1.81 \\  & (0.40) & (0.40) & (0.39) & (0.39) \\ \multirow{2}{*}{phealth} & 0.77 & 0.82 & 0.81 & 0.81 \\  & (0.23) & (0.23) & (0.23) & (0.23) \\ \multirow{2}{*}{depress} & 0.22 & 0.26 & 0.17 & 0.20 \\  & (1.25) & (0.23) & (0.23) & (0.23) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Coefficients and performance of the predictive model of T2D onset obtained in approach 1 (standard RFE-Borda method with both BMI and waist circumference), approach 2 (proposed algorithm, the model with waist is shown), approach 3 (standard RFE-Borda method with BMI only), approach 4 (standard RFE-Borda method with waist circumference only). Figure 6: Mean \(+/-\) SD of the C-index on the out-of-bag sample obtained by the proposed variable ranking algorithm.