Table 1.
Results on variable selection of sequential advantage selection (SAS), S-score and LASSO methods for the scenarios with the first choice of β (three important prescriptive variables: X1, X9 and X10). Size is the number of covariates selected. For S-score method, Size S ≠ 0 is the number of covariates with nonzero S-scores. TDR stands for true discovery rate, which is the proportion correctly identified important variables among all the selected variables. TP stands for true positive, which is the number of correctly identified important variables among X1, X9 and X10. The quantity in parenthesis is standard deviation of corresponding value.
| Setting | SAS | S-score | LASSO | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Size | TDR | TP | Size S ≠ 0 | TDR | TP | Size | TDR | TP | |
| Randomized Study | |||||||||
| Model I, ρ = 0.2 | 7.67 (2.71) | 0.42 (0.09) | 2.98 (0.23) | 765.28 (163.13) | 0.23 (0.11) | 1.66 (0.68) | 17.40 (23.29) | 0.211 (0.29) | 1.60 (1.05) |
| Model I, ρ = 0.5 | 9.55 (2.91) | 0.34 (0.11) | 2.92 (0.38) | 757.18 (165.78) | 0.16 (0.08) | 1.35 (0.51) | 15.04 (18.61) | 0.24 (0.31) | 1.37 (0.79) |
| Model I, ρ = 0.8 | 14.05 (3.68) | 0.17 (0.11) | 2.05 (1.00) | 738.44 (173.40) | 0.08 (0.03) | 1.09 (0.28) | 11.44 (14.29) | 0.28 (0.30) | 1.10 (0.44) |
| Model II, ρ = 0.2 | 23.08 (3.87) | 0.12 (0.04) | 2.58 (0.76) | 762.73 (169.38) | 0.11 (0.04) | 2.43 (0.62) | 23.42 (25.22) | 0.22 (0.23) | 2.32 (0.86) |
| Model II, ρ = 0.5 | 22.16 (4.32) | 0.12 (0.06) | 2.41 (0.89) | 747.08 (176.22) | 0.09 (0.04) | 1.88 (0.57) | 20.54 (23.56) | 0.25 (0.28) | 1.91 (0.84) |
| Model II, ρ = 0.8 | 17.44 (4.56) | 0.12 (0.08) | 1.91 (0.98) | 721.28 (188.22) | 0.07 (0.04) | 1.19 (0.40) | 12.93 (15.93) | 0.27 (0.27) | 1.29 (0.58) |
| Model III, ρ = 0.2 | 22.58 (4.32) | 0.11 (0.06) | 2.35 (1.04) | 765.27 (165.52) | 0.09 (0.04) | 1.98 (0.71) | 15.84 (21.65) | 0.20 (0.28) | 1.48 (1.02) |
| Model III, ρ = 0.5 | 23.27 (4.04) | 0.09 (0.06) | 2.04 (1.02) | 758.89 (169.00) | 0.07 (0.03) | 1.46 (0.57) | 13.36 (22.14) | 0.20 (0.29) | 1.10 (0.83) |
| Model III, ρ = 0.8 | 22.75 (4.27) | 0.08 (0.05) | 1.66 (0.88) | 749.84 (170.04) | 0.05 (0.02) | 1.17 (0.40) | 11.65 (15.26) | 0.22 (0.28) | 0.98 (0.50) |
|
| |||||||||
| Observational Study | |||||||||
| Model I, ρ = 0.2 | 8.44 (2.13) | 0.38 (0.09) | 3.00 (0) | 760.58 (158.94) | 0.22 (0.11) | 1.73 (0.71) | 34.43 (24.72) | 0.14 (0.13) | 2.94 (0.25) |
| Model I, ρ = 0.5 | 9.51 (2.89) | 0.35 (0.11) | 2.99 (0.15) | 760.74 (164.11) | 0.15 (0.08) | 1.36 (0.57) | 36.43 (27.92) | 0.16 (0.20) | 2.71 (0.58) |
| Model I, ρ = 0.8 | 12.46 (3.67) | 0.21 (0.12) | 2.28 (0.96) | 746.41 (167.70) | 0.10 (0.04) | 1.11 (0.31) | 17.63 (21.87) | 0.28 (0.31) | 1.62 (0.64) |
| Model II, ρ = 0.2 | 19.58 (4.07) | 0.15 (0.05) | 2.84 (0.50) | 527.97 (207.27) | 0.13 (0.05) | 2.42 (0.64) | 28.30 (22.26) | 0.18 (0.15) | 2.96 (0.21) |
| Model II, ρ = 0.5 | 18.91 (4.00) | 0.15 (0.06) | 2.67 (0.72) | 609.91 (211.18) | 0.11 (0.04) | 1.88 (0.57) | 31.67 (24.06) | 0.16 (0.17) | 2.82 (0.45) |
| Model II, ρ = 0.8 | 16.62 (4.21) | 0.14 (0.08) | 2.08 (1.00) | 672.87 (194.65) | 0.08 (0.04) | 1.25 (0.44) | 21.04 (22.84) | 0.25 (0.28) | 1.82 (0.78) |
| Model III, ρ = 0.2 | 19.80 (3.93) | 0.16 (0.04) | 2.93 (0.34) | 680.88 (201.15) | 0.10 (0.05) | 1.91 (0.73) | 28.68 (23.58) | 0.18 (0.14) | 2.91 (0.33) |
| Model III, ρ = 0.5 | 20.55 (4.19) | 0.14 (0.06) | 2.65 (0.74) | 704.98 (199.21) | 0.08 (0.04) | 1.44 (0.63) | 27.82 (23.55) | 0.19 (0.19) | 2.52 (0.64) |
| Model III, ρ = 0.8 | 19.66 (4.14) | 0.11 (0.06) | 2.12 (0.96) | 712.65 (191.05) | 0.06 (0.03) | 1.13 (0.45) | 16.40 (19.08) | 0.264 (0.27) | 1.64 (0.59) |