TABLE 4.
Simulation results for Model II with 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 | 0.36(0.011) | 89.8 | 0.29 | 1.65 (0.085) | 80.8 | 1.06 | 58.7 | 3.69 | |
| P(0.5) | 0.57 (0.017) | 86.9 | 0.46 | 0.61 (0.026) | 86.4 | 0.55 | 1.31 (0.045) | 81.7 | 0.93 | |
| P(0.25) | 0.65 (0.020) | 86.2 | 0.52 | 0.22 (0.008) | 91.7 | 0.20 | 4.67 (0.312) | 74.7 | 1.64 | |
| Huber | 0.38 (0.012) | 89.5 | 0.30 | 0.45 (0.018) | 88.3 | 0.40 | 1.70 (0.060) | 79.5 | 1.14 | |
| 200 | LS | 0.16 (0.004) | 92.9 | 0.14 | 0.74 (0.030) | 85.6 | 0.61 | 59.1 | 3.64 | |
| P(0.5) | 0.25 (0.007) | 91.2 | 0.21 | 0.26 (0.008) | 90.7 | 0.24 | 0.52 (0.017) | 87.8 | 0.41 | |
| P(0.25) | 0.30 (0.008) | 90.3 | 0.26 | 0.09 (0.003) | 94.8 | 0.08 | 1.69 (0.074) | 81.3 | 0.92 | |
| Huber | 0.17 (0.005) | 92.8 | 0.14 | 0.19 (0.006) | 92.2 | 0.17 | 0.70 (0.022) | 86.2 | 0.53 | |
| 400 | LS | 0.08 (0.002) | 95.1 | 0.06 | 0.36 (0.013) | 89.7 | 0.30 | 58.0 | 3.79 | |
| P(0.5) | 0.12 (0.003) | 93.8 | 0.10 | 0.12 (0.003) | 93.8 | 0.10 | 0.22 (0.006) | 91.6 | 0.19 | |
| P(0.25) | 0.14 (0.004) | 93.3 | 0.12 | 0.04 (0.001) | 96.5 | 0.03 | 0.63 (0.021) | 86.5 | 0.49 | |
| Huber | 0.08 (0.002) | 95.0 | 0.07 | 0.09 (0.002) | 94.8 | 0.07 | 0.30 (0.009) | 90.3 | 0.26 | |
| 800 | LS | 0.04 (0.001) | 96.5 | 0.03 | 0.18 (0.006) | 92.3 | 0.16 | 58.2 | 3.76 | |
| P(0.5) | 0.06 (0.002) | 95.6 | 0.05 | 0.06 (0.002) | 95.6 | 0.05 | 0.10 (0.003) | 94.4 | 0.09 | |
| P(0.25) | 0.07 (0.002) | 95.3 | 0.06 | 0.02 (0.001) | 97.5 | 0.02 | 0.29 (0.009) | 90.6 | 0.23 | |
| Huber | 0.04 (0.001) | 96.4 | 0.03 | 0.04 (0.001) | 96.3 | 0.04 | 0.14 (0.004) | 93.2 | 0.12 | |
| Heterogeneous Error | ||||||||||
|
Normal |
Log-Normal |
Cauchy |
||||||||
| n | method | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 | MSE | PCD | δ0.5 |
| 100 | LS | 1.45 (0.059) | 82.9 | 0.85 | 8.53 (0.784) | 72.4 | 2.01 | 54.9 | 4.22 | |
| P(0.5) | 0.94 (0.034) | 85.6 | 0.61 | 1.29 (0.058) | 83.3 | 0.86 | 2.27 (0.132) | 78.9 | 1.24 | |
| P(0.25) | 1.46 (0.051) | 81.5 | 0.96 | 1.78 (0.071) | 78.2 | 1.30 | 7.88 (0.422) | 68.1 | 2.46 | |
| Huber | 0.89 (0.034) | 86.1 | 0.57 | 1.46 (0.067) | 81.7 | 0.99 | 3.28 (0.157) | 75.1 | 1.65 | |
| 200 | LS | 0.84 (0.035) | 86.6 | 0.53 | 3.85 (0.358) | 77.6 | 1.43 | 55.9 | 4.09 | |
| P(0.5) | 0.44 (0.016) | 90.0 | 0.29 | 0.60 (0.024) | 89.0 | 0.39 | 0.87 (0.034) | 86.3 | 0.56 | |
| P(0.25) | 0.69 (0.025) | 87.0 | 0.49 | 0.75 (0.024) | 85.5 | 0.59 | 3.08 (0.179) | 75.3 | 1.58 | |
| Huber | 0.43 (0.016) | 90.3 | 0.28 | 0.66 (0.025) | 87.7 | 0.47 | 1.32 (0.050) | 82.4 | 0.87 | |
| 400 | LS | 0.44 (0.020) | 90.3 | 0.28 | 2.34 (0.393) | 82.4 | 0.95 | 55.9 | 4.09 | |
| P(0.5) | 0.23 (0.009) | 92.9 | 0.16 | 0.28 (0.011) | 92.5 | 0.19 | 0.39 (0.015) | 90.8 | 0.26 | |
| P(0.25) | 0.33 (0.011) | 91.0 | 0.23 | 0.36 (0.012) | 90.1 | 0.27 | 1.25 (0.048) | 82.8 | 0.82 | |
| Huber | 0.22 (0.008) | 93.1 | 0.15 | 0.31 (0.012) | 91.7 | 0.21 | 0.60 (0.022) | 88.0 | 0.43 | |
| 800 | LS | 0.23 (0.009) | 93.0 | 0.15 | 0.90 (0.057) | 86.2 | 0.60 | 56.3 | 4.03 | |
| P(0.5) | 0.11 (0.004) | 95.0 | 0.07 | 0.14 (0.005) | 94.8 | 0.09 | 0.18 (0.006) | 93.6 | 0.12 | |
| P(0.25) | 0.17 (0.006) | 93.7 | 0.12 | 0.18 (0.006) | 93.0 | 0.14 | 0.59 (0.017) | 87.3 | 0.44 | |
| Huber | 0.10 (0.004) | 95.1 | 0.07 | 0.15 (0.006) | 94.2 | 0.11 | 0.29 (0.010) | 91.4 | 0.21 | |
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.