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Table 2 Simulation example 2: effect of correlation between mandatory and irrelevant predictors

From: Ridle for sparse regression with mandatory covariates with application to the genetic assessment of histologic grades of breast cancer

  Method rpe g-measure Sensitivity (\(\mathcal {M}\)) Sensitivity (\(\mathcal {O}\)) Specificity (\(\mathcal {O}\))
ρ 0=0.25 Ridge 1.671 (0.012)     
  Lasso 1.911 (0.022) 0.383 (0.034) 0.100 (0.032) 0.200 (0.028) 0.975 (0.008)
  Elastic net 1.744 (0.019) 0.585 (0.015) 0.400 (0.054) 0.600 (0.050) 0.835 (0.036)
  \(\mathcal {M}\)-unpenalized lasso 1.741 (0.028) 0.742 (0.012) 1.000 (0.000) 0.200 (0.037) 0.938 (0.003)
  \(\mathcal {M}\)-unpenalized elastic net 1.657 (0.017) 0.757 (0.008) 1.000 (0.000) 0.500 (0.064) 0.833 (0.022)
  Ridle 1.492 (0.031) 0.773 (0.006) 1.000 (0.000) 0.200 (0.048) 0.931 (0.006)
ρ 0=0.5 Ridge 1.807 (0.014)     
  Lasso 2.045 (0.035) 0.571 (0.013) 0.300 (0.046) 0.400 (0.039) 0.925 (0.007)
  Elastic net 1.773 (0.034) 0.667 (0.008) 0.600 (0.014) 0.800 (0.048) 0.756 (0.020)
  \(\mathcal {M}\)-unpenalized lasso 1.922 (0.044) 0.794 (0.003) 1.000 (0.000) 0.400 (0.047) 0.929 (0.004)
  \(\mathcal {M}\)-unpenalized elastic net 1.729 (0.040) 0.796 (0.007) 1.000 (0.000) 0.700 (0.048) 0.785 (0.022)
  Ridle 1.438 (0.057) 0.852 (0.006) 1.000 (0.000) 0.600 (0.049) 0.900 (0.004)
ρ 0=0.75 Ridge 1.564 (0.022)     
  Lasso 1.365 (0.029) 0.684 (0.008) 0.400 (0.032) 0.600 (0.012) 0.900 (0.003)
  Elastic net 1.237 (0.030) 0.745 (0.005) 0.700 (0.048) 0.900 (0.011) 0.775 (0.014)
  \(\mathcal {M}\)-unpenalized lasso 1.423 (0.037) 0.839 (0.005) 1.000 (0.000) 0.700 (0.026) 0.904 (0.006)
  \(\mathcal {M}\)-unpenalized elastic net 1.310 (0.041) 0.847 (0.005) 1.000 (0.000) 0.800 (0.012) 0.840 (0.008)
  Ridle 0.886 (0.029) 0.875 (0.003) 1.000 (0.000) 0.700 (0.038) 0.908 (0.003)
  1. The \(\mathcal {M}\)-unpenalized lasso and \(\mathcal {M}\)-unpenalized elastic net were performed without penalization on the mandatory covariates. g-measure is estimated from all predictors. Sensitivity (\(\mathcal {M}\)) is computed in terms of the mandatory variables only, whereas sensitivity (\(\mathcal {O}\)) and specificity (\(\mathcal {O}\)) are computed in terms of the optional variables only
  2. n=50, p=250, \(|\mathcal {M}|=10\). The smallest rpe and largest two g-measures are boldfaced