Table 1 Prediction & Variable Selection Performance for LR with Shrinkage Priors

Prediction

Priors

BS1

BS2

BS3

BS4

BS5

BS6

BS7

BS8

BS9

N,P, ρ

1000,10,0.5

200,10,0.5

400,20,0.5

500,50,0.3

300,10,0.5

100,10,0.5

100,130,0.5

1000,10,0

1000,10,0

β

(10,10,10,10,5,5,0.1,0.1,0.1,0.1)’

(10,10,10,10,5,5,0.1,0.1,0.1,0.1)’

\(\left (\begin {array}{l}{\underbrace {10,\ldots,10}_{5}},{\underbrace {5,\ldots,5}_{5}}, {\underbrace {0.1,\ldots,0.1}_{10}}\end {array}\right)\)

40% non-zero β

(1,1.5,-2,2.5,0,0,0,0,0,0)’

(5,5,3,0.74,-0.9,0,0,0,0,0)’

\(\left (\begin {array}{l}{\underbrace {5,\ldots,5}_{30}},{\underbrace {0,\ldots,5}_{100}}\end {array}\right)\)

β=(3.5,0.2,0.1,1,2,3,5,5,5,5)

β=(3.5,3.5,3.5,3.5,5,5,0.1,0.1,0.1,0.1)

Accuracy

Horseshoe

0.968(0.012)

0.958(0.031)

0.965(0.021)

0.856(0.032)

0.843(0.048)

0.894(0.074)

0.954(0.053)

0.909(0.018)

0.899(0.024)

Dirichlet Laplace

0.968(0.012)

0.958(0.030)

0.965(0.020)

0.839(0.039)

0.842(0.050)

0.894(0.075)

0.938(0.062)

0.911(0.018)

0.898(0.024)

Double Pareto

0.968(0.012)

0.958(0.031)

0.964(0.021)

0.834(0.040)

0.839(0.050)

0.894(0.073)

0.940(0.050)

0.912(0.018)

0.898(0.024)

Sensitivity

Horseshoe

0.967(0.016)

0.965(0.043)

0.964(0.031)

0.853(0.047)

0.840(0.067)

0.897(0.107)

0.940(0.076)

0.909(0.027)

0.914(0.030)

Dirichlet Laplace

0.967(0.016)

0.964(0.043)

0.964(0.031)

0.840(0.053)

0.837(0.068)

0.898(0.106)

0.944(0.073)

0.910(0.028)

0.914(0.030)

Double Pareto

0.967(0.016)

0.964(0.044)

0.964(0.030)

0.835(0.055)

0.835(0.067)

0.898(0.105)

0.946(0.073)

0.912(0.028)

0.922(0.029)

Specificity

Horseshoe

0.969(0.017)

0.953(0.046)

0.966(0.031)

0.859(0.052)

0.847(0.070)

0.893(0.094)

0.943(0.084)

0.910(0.027)

0.877(0.038)

Dirichlet Laplace

0.969(0.017)

0.955(0.046)

0.967(0.028)

0.838(0.058)

0.847(0.070)

0.890(0.097)

0.935(0.079)

0.911(0.026)

0.875(0.039)

Double Pareto

0.969(0.017)

0.954(0.045)

0.964(0.030)

0.833(0.058)

0.847(0.070)

0.893(0.094)

0.934(0.079)

0.912(0.027)

0.863(0.041)

Area Under Curve

Horseshoe

0.968(0.012)

0.958(0.031)

0.965(0.021)

0.856(0.032)

0.844(0.049)

0.898(0.074)

0.944(0.052)

0.910(0.018)

0.896(0.025)

Dirichlet Laplace

0.968(0.012)

0.958(0.029)

0.966(0.020)

0.839(0.039)

0.842(0.050)

0.897(0.075)

0.939(0.063)

0.911(0.018)

0.895(0.025)

Double Pareto

0.968(0.012)

0.958(0.030)

0.964(0.021)

0.834(0.040)

0.849(0.050)

0.897(0.073)

0.941(0.051)

0.912(0.018)

0.897(0.024)

Brier Score

Horseshoe

0.023(0.007)

0.030(0.018)

0.025(0.013)

0.103(0.019)

0.110(0.026)

0.073(0.042)

0.046(0.026)

0.067(0.009)

0.072(0.012)

Dirichlet Laplace

0.023(0.007)

0.029(0.016)

0.025(0.011)

0.114(0.022)

0.111(0.026)

0.073(0.041)

0.044(0.026)

0.066(0.009)

0.072(0.012)

Double Pareto

0.023(0.007)

0.030(0.017)

0.025(0.012)

0.117(0.023)

0.112(0.026)

0.073(0.043)

0.045(0.024)

0.063(0.010)

0.072(0.012)

Variable Selection

Accuracy

Horseshoe

0.989(0.035)

0.923(0.072)

0.922(0.053)

0.999(0.004)

0.980(0.043)

0.827(0.066)

0.422(0.262)

0.747(0.056)

0.868(0.085)

Dirichlet Laplace

0.994(0.024)

0.920(0.070)

0.914(0.049)

0.972(0.024)

0.981(0.044)

0.829(0.064)

0.504(0.275)

0.758(0.062)

0.856(0.107)

Double Pareto

0.985(0.039)

0.927(0.071)

0.926(0.052)

0.947(0.034)

0.977(0.047)

0.832(.071)

0.527(0.274)

0.820(0.065)

0.940(0.078)

Sensitivity

Horseshoe

1.000(0.000)

0.885(0.113)

0.860(0.096)

0.998(0.025)

0.962(0.090)

0.662(0.123)

0.163(0.094)

0.689(0.072)

0.820(0.100)

Dirichlet Laplace

1.000(0.000)

0.868(0.114)

0.838(0.093)

0.998(0.025)

0.978(0.072)

0.668(0.114)

0.136(0.096)

0.704(0.079)

0.848(0.106)

Double Pareto

1.000(0.000)

0.882(0.112)

0.864(0.096)

1.000(0.000)

0.978(0.072)

0.682(0.121)

0.129(0.096)

0.785(0.076)

0.978(0.056)

Specificity

Horseshoe

0.972(0.086)

0.980(0.068)

0.985(0.039)

0.999(0.004)

0.992(0.037)

0.992(0.039)

0.999(0.004)

0.980(0.098)

0.940(0.113)

Dirichlet Laplace

0.985(0.060)

0.998(0.025)

0.989(0.031)

0.969(0.026)

0.983(0.058)

0.990(0.044)

0.999(0.003)

0.975(0.110)

0.868(0.165)

Double Pareto

0.962(0.096)

0.995(0.035)

0.988(0.033)

0.942(0.037)

0.977(0.050)

0.982(0.058)

0.999(0.002)

0.960(0.136)

0.882(0.165)

L1 error

Horseshoe

2.358(0.417)

3.444(3.895)

2.152(1.194)

0.057(0.017)

0.197(0.064)

1.311(1.776)

1.977(0.229)

1.998(0.109)

1.696(0.053)

Dirichlet Laplace

2.474(0.326)

2.953(0.388)

2.148(0.329)

0.187(0.045)

0.213(0.073)

0.644(0.289)

1.955(0.220)

1.970(0.068)

1.689(0.055)

Double Pareto

2.421(0.387)

2.669(0.601)

1.997(0.438)

0.230(0.053)

0.231(0.076)

0.938(0.720)

1.960(0.237)

1.669(0.077)

1.479(0.089)

L2 error

Horseshoe

3.063(0.572)

4.435(4.931)

2.971(1.722)

0.102(0.036)

0.259(0.091)

1.942(2.797)

2.562(0.299)

2.384(0.128)

2.128(0.060)

Dirichlet Laplace

3.231(0.446)

3.849(0.507)

3.015(0.585)

0.277(0.081)

0.276(0.103)

0.896(0.456)

2.582(0.293)

2.349(0.075)

2.118(0.063)

Double Pareto

3.146(0.529)

3.469(0.772)

2.781(0.459)

0.330(0.092)

0.293(0.106)

1.359(1.176)

2.594(0.333)

1.980(0.088)

1.855(0.102)