Table 4 Simulation example 4: mandatory covariates are irrelevant

Method rpe g-measure Specificity ($$\mathcal {M}$$) Sensitivity ($$\mathcal {O}$$) Specificity ($$\mathcal {O}$$)
ρ 0=0.25 Ridge 1.671 (0.012)
Lasso 1.911 (0.022) 0.383 (0.034) 1.000 (0.000) 0.200 (0.028) 0.975 (0.008)
Elastic net 1.744 (0.019) 0.585 (0.015) 0.600 (0.053) 0.600 (0.050) 0.835 (0.036)
$$\mathcal {M}$$-unpenalized lasso 2.357 (0.032) 0.215 (0.103) 0.000 (0.000) 0.050 (0.024) 0.995 (0.003)
$$\mathcal {M}$$-unpenalized elastic net 2.210 (0.034) 0.308 (0.054) 0.000 (0.000) 0.525 (0.065) 0.732 (0.057)
Ridle 1.854 (0.012) 0.309 (0.029) 0.000 (0.000) 0.100 (0.024) 0.982 (0.005)
ρ 0=0.5 Ridge 1.807 (0.014)
Lasso 2.045 (0.035) 0.571 (0.013) 0.800 (0.006) 0.400 (0.039) 0.925 (0.007)
Elastic net 1.773 (0.034) 0.667 (0.008) 0.500 (0.048) 0.800 (0.048) 0.756 (0.020)
$$\mathcal {M}$$-unpenalized lasso 2.242 (0.023) 0.299 (0.035) 0.000 (0.000) 0.100 (0.021) 0.982 (0.004)
$$\mathcal {M}$$-unpenalized elastic net 2.080 (0.028) 0.305 (0.094) 0.000 (0.000) 0.550 (0.072) 0.700 (0.079)
Ridle 1.801 (0.039) 0.528 (0.032) 0.000 (0.000) 0.300 (0.038) 0.943 (0.005)
ρ 0=0.75 Ridge 1.564 (0.022)
Lasso 1.365 (0.029) 0.684 (0.008) 0.700 (0.041) 0.600 (0.012) 0.900 (0.003)
Elastic net 1.237 (0.030) 0.745 (0.005) 0.300 (0.046) 0.900 (0.011) 0.775 (0.014)
$$\mathcal {M}$$-unpenalized lasso 1.747 (0.043) 0.428 (0.003) 0.000 (0.000) 0.200 (0.000) 0.964 (0.003)
$$\mathcal {M}$$-unpenalized elastic net 1.662 (0.043) 0.514 (0.016) 0.000 (0.000) 0.350 (0.023) 0.900 (0.015)
Ridle 1.253 (0.042) 0.596 (0.017) 0.000 (0.000) 0.400 (0.026) 0.945 (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. specificity ($$\mathcal {M}$$) 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