TABLE 3.
Simulation results for Model I with the non-constant propensity scores.
| Homogeneous Error | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Normal |
Log-Normal |
Cauchy |
||||||||
| n | method | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 |
| 100 | LS | 1.70 (0.061) | 81.9 | 0.91 | 2.90 (0.114) | 77.6 | 1.34 | 59.3 | 3.61 | |
| P(0.5) | 1.90 (0.069) | 80.1 | 1.09 | 2.13 (0.073) | 78.3 | 1.25 | 3.54 (0.128) | 75.7 | 1.57 | |
| P(0.25) | 2.35 (0.080) | 78.2 | 1.33 | 1.95 (0.076) | 80.4 | 1.08 | 8.45 (0.431) | 69.8 | 2.28 | |
| Huber | 1.51 (0.053) | 82.1 | 0.89 | 1.77 (0.065) | 80.6 | 1.02 | 3.67 (0.127) | 75.4 | 1.60 | |
| 200 | LS | 0.77 (0.026) | 86.8 | 0.50 | 1.35(0.045) | 82.2 | 0.91 | 59.2 | 3.63 | |
| P(0.5) | 0.88 (0.028) | 85.5 | 0.60 | 1.00 (0.029) | 83.0 | 0.79 | 1.54 (0.050) | 81.1 | 1.00 | |
| P(0.25) | 1.06 (0.035) | 84.5 | 0.68 | 0.83 (0.027) | 85.9 | 0.59 | 3.61 (0.143) | 74.7 | 1.70 | |
| Huber | 0.68 (0.022) | 87.3 | 0.46 | 0.81 (0.025) | 85.2 | 0.62 | 1.58 (0.052) | 80.7 | 1.03 | |
| 400 | LS | 0.39 (0.012) | 90.2 | 0.28 | 0.65 (0.020) | 86.9 | 0.48 | 58.0 | 3.79 | |
| P(0.5) | 0.43 (0.013) | 89.3 | 0.32 | 0.47 (0.014) | 88.4 | 0.38 | 0.73 (0.022) | 86.5 | 0.51 | |
| P(0.25) | 0.53 (0.016) | 88.5 | 0.38 | 0.41 (0.013) | 90.5 | 0.27 | 1.50 (0.049) | 81.7 | 0.96 | |
| Huber | 0.34 (0.010) | 90.6 | 0.25 | 0.39(0.012) | 89.6 | 0.30 | 0.72 (0.022) | 86.3 | 0.53 | |
| 800 | LS | 0.18 (0.006) | 93.3 | 0.13 | 0.32 (0.010) | 90.2 | 0.27 | 58.3 | 3.75 | |
| P(0.5) | 0.21 (0.007) | 92.7 | 0.15 | 0.24 (0.007) | 91.5 | 0.20 | 0.36(0.011) | 90.3 | 0.27 | |
| P(0.25) | 0.28 (0.009) | 92.4 | 0.17 | 0.21 (0.007) | 93.4 | 0.13 | 0.78 (0.026) | 86.9 | 0.50 | |
| Huber | 0.16 (0.005) | 93.7 | 0.11 | 0.19(0.006) | 92.6 | 0.15 | 0.37 (0.010) | 89.9 | 0.28 | |
| Heterogeneous Error | ||||||||||
|
Normal |
Log-Normal |
Cauchy |
||||||||
| n | method | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 |
| 100 | LS | 2.84 (0.111) | 78.2 | 1.33 | 9.96 (0.773) | 72.0 | 2.06 | 55.2 | 4.18 | |
| P(0.5) | 2.01 (0.082) | 80.6 | 1.09 | 2.18 (0.080) | 79.2 | 1.21 | 4.18(0.189) | 74.1 | 1.81 | |
| P(0.25) | 2.91 (0.110) | 76.7 | 1.52 | 3.22 (0.105) | 74.2 | 1.76 | 10.62 (0.475) | 65.3 | 2.87 | |
| Huber | 1.90 (0.074) | 80.9 | 1.06 | 2.38 (0.090) | 78.1 | 1.32 | 5.06 (0.230) | 71.9 | 2.04 | |
| 200 | LS | 1.46 (0.053) | 83.1 | 0.83 | 4.47 (0.371) | 76.8 | 1.51 | 56.3 | 4.04 | |
| P(0.5) | 0.92 (0.033) | 86.4 | 0.55 | 0.98 (0.035) | 85.3 | 0.64 | 1.69 (0.065) | 81.5 | 0.98 | |
| P(0.25) | 1.35 (0.049) | 83.3 | 0.81 | 1.47 (0.049) | 81.6 | 0.97 | 4.73 (0.241) | 71.9 | 2.05 | |
| Huber | 0.86 (0.030) | 86.6 | 0.52 | 1.02 (0.036) | 84.7 | 0.68 | 2.11 (0.079) | 79.3 | 1.18 | |
| 400 | LS | 0.74 (0.029) | 87.4 | 0.47 | 2.65 (0.402) | 81.4 | 1.04 | 56.2 | 4.06 | |
| P(0.5) | 0.45 (0.016) | 90.2 | 0.29 | 0.44 (0.017) | 89.5 | 0.34 | 0.79 (0.029) | 87.2 | 0.49 | |
| P(0.25) | 0.66 (0.025) | 88.3 | 0.41 | 0.70 (0.023) | 86.9 | 0.50 | 2.12(0.091) | 79.5 | 1.19 | |
| Huber | 0.43 (0.016) | 90.2 | 0.28 | 0.48 (0.018) | 89.0 | 0.36 | 1.01 (0.036) | 85.0 | 0.65 | |
| 800 | LS | 0.36 (0.013) | 90.8 | 0.25 | 1.09 (0.066) | 85.0 | 0.69 | 56.3 | 4.02 | |
| P(0.5) | 0.21 (0.008) | 93.2 | 0.14 | 0.24 (0.009) | 92.3 | 0.19 | 0.39 (0.014) | 90.5 | 0.27 | |
| P(0.25) | 0.33 (0.013) | 91.7 | 0.21 | 0.36(0.012) | 90.8 | 0.25 | 1.01 (0.034) | 84.9 | 0.65 | |
| Huber | 0.20 (0.008) | 93.2 | 0.14 | 0.25 (0.009) | 92.1 | 0.19 | 0.49 (0.016) | 89.1 | 0.34 | |
LS stands for lsA-learning. P(0.5) stands for robust regression with pinball loss and parameter τ = 0.5. P(0.25) stands for robust regression with pinball loss and parameter τ = 0.25. Huber stands for robust regression with Huber loss, where parameter α is tuned automatically with R function rlm. Column δ0.5 is multiplied by 10.