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Table 3 Simulation example 3: effect of multicollinearity among mandatory covariates

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=0.75 Ridge 6.353 (0.022)     
  Lasso 4.649 (0.167) 0.802 (0.011) 0.800 (0.000) 0.700 (0.048) 0.908 (0.004)
  Elastic net 4.410 (0.128) 0.804 (0.005) 1.000 (0.009) 0.700 (0.006) 0.858 (0.006)
  \(\mathcal {M}\)-unpenalized lasso 4.776 (0.260) 0.829 (0.005) 1.000 (0.000) 0.700 (0.031) 0.902 (0.007)
  \(\mathcal {M}\)-unpenalized elastic net 5.402 (0.190) 0.823 (0.006) 1.000 (0.000) 0.700 (0.013) 0.871 (0.009)
  Ridle 2.699 (0.152) 0.893 (0.007) 1.000 (0.000) 0.900 (0.048) 0.904 (0.004)
ρ=0.9 Ridge 6.270 (0.026)     
  Lasso 4.914 (0.148) 0.784 (0.010) 0.600 (0.089) 0.700 (0.036) 0.908 (0.004)
  Elastic net 4.336 (0.135) 0.816 (0.005) 0.800 (0.092) 0.700 (0.018) 0.867 (0.008)
  \(\mathcal {M}\)-unpenalized lasso 6.992 (0.337) 0.828 (0.008) 1.000 (0.000) 0.700 (0.031) 0.902 (0.006)
  \(\mathcal {M}\)-unpenalized elastic net 7.245 (0.237) 0.827 (0.005) 1.000 (0.000) 0.700 (0.045) 0.860 (0.011)
  Ridle 3.000 (0.214) 0.890 (0.006) 1.000 (0.000) 0.800 (0.045) 0.900 (0.004)
ρ=0.99 Ridge 6.231 (0.031)     
  Lasso 7.322 (0.200) 0.745 (0.005) 0.400 (0.000) 0.700 (0.000) 0.913 (0.003)
  Elastic net 5.003 (0.155) 0.804 (0.006) 0.800 (0.049) 0.700 (0.019) 0.883 (0.006)
  \(\mathcal {M}\)-unpenalized lasso 36.214 (2.064) 0.824 (0.006) 1.000 (0.000) 0.700 (0.046) 0.904 (0.005)
  \(\mathcal {M}\)-unpenalized elastic net 33.583 (2.197) 0.830 (0.004) 1.000 (0.010) 0.700 (0.045) 0.867 (0.010)
  Ridle 4.193 (0.343) 0.890 (0.005) 1.000 (0.000) 0.800 (0.029) 0.904 (0.004)
  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, \(|\mathcal {M}|=5\). The smallest rpe and largest two g-measures are boldfaced