Table 2.
Simulation results to evaluate the proposed penalized likelihood approaches for selecting the variables that mediate the co-expression patterns between two genes, where the candidate regulators were simulated as discrete variables. Both the Lasso penalty (Lasso) and the adaptive Lasso penalty (A-Lasso) are considered, cross-validation (Lasso1 and A-Lasso1), two-stage cross-validation (Lasso2 and A-Lasso2) and the BIC (Lasso3 and A-Lasso3) are used to select the tuning parameter λ. As a comparison, results from single variable analysis based on the likelihood ratio test using p-value of 0.05 (LR1) and p-value of 0.05/p (LR2) are also presented. For each procedure, the column labeled with C (or IC) represents the average number of correctly (or incorrectly) identified variables and their SEs, and E represents the percentage of the simulations that identify the three relevant variables exactly
| p = 12 | p= 50 | p= 100 | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Method | C | IC | E | C | IC | E | C | IC | E |
| (β0, β1, β2, β3) = (−3.2, 3.0, 1.5, 2.0) | |||||||||
| Lasso1 | 3.00(0.00) | 2.65(1.88) | 11 | 3.00(0.00) | 6.56(3.78) | 0 | 3.00(0.00) | 8.96(5.04) | 0 |
| Lasso2 | 3.00(0.00) | 0.65(1.31) | 69 | 3.00(0.00) | 1.16(1.70) | 47 | 3.00(0.00) | 1.80(2.27) | 33 |
| Lasso3 | 3.00(0.00) | 0.67(0.98) | 57 | 3.00(0.00) | 1.19(1.44) | 39 | 3.00(0.00) | 1.68(1.81) | 29 |
| A-Lasso1 | 3.00(0.00) | 0.39(0.73) | 72 | 3.00(0.05) | 1.45(1.80) | 39 | 2.90(0.30) | 11.48(6.37) | 0 |
| A-Lasso2 | 3.00(0.00) | 0.16(0.48) | 88 | 3.00(0.05) | 0.55(1.01) | 67 | 2.87(0.33) | 7.15(4.51) | 1 |
| A-Lasso3 | 3.00(0.00) | 0.13(0.40) | 88 | 3.00(0.03) | 0.54(1.16) | 69 | 2.89(0.31) | 6.57(4.91) | 2 |
| LR11 | 3.00(0.07) | 3.19(1.67) | 4 | 3.00(0.07) | 15.03(5.11) | 0 | 3.00(0.06) | 30.80(10.36) | 0 |
| LR2 | 2.97(0.18) | 1.62(1.30) | 19 | 2.95(0.22) | 4.42(3.20) | 4 | 2.95(0.23) | 7.57(5.90) | 2 |
| (β0, β1, β6, β12) = (−0.5, −1.0, 1.5, 1.0) | |||||||||
| Lasso1 | 3.00(0.07) | 2.94(1.71) | 4 | 2.97(0.19) | 6.89(3.46) | 0 | 2.93(0.31) | 8.54(4.67) | 0 |
| Lasso2 | 2.98(0.16) | 0.92(1.31) | 50 | 2.92(0.32) | 2.16(2.27) | 24 | 2.83(0.47) | 2.68(2.77) | 15 |
| Lasso3 | 2.99(0.09) | 1.05(1.14) | 38 | 2.97(0.21) | 2.10(1.89) | 21 | 2.94(0.28) | 2.59(2.41) | 16 |
| A-Lasso1 | 2.99(0.10) | 0.72(0.91) | 51 | 2.78(0.48) | 2.50(2.29) | 12 | 2.36(0.79) | 11.48(5.84) | 0 |
| A-Lasso2 | 2.98(0.14) | 0.26(0.63) | 80 | 2.68(0.57) | 1.07(1.42) | 34 | 2.26(0.82) | 7.58(4.39) | 0 |
| A-Lasso3 | 2.99(0.09) | 0.28(0.61) | 78 | 2.85(0.39) | 1.91(2.27) | 27 | 2.31(0.78) | 6.47(4.38) | 1 |
| LR1 | 2.60(0.54) | 2.38(1.50) | 5 | 2.56(0.56) | 9.06(3.82) | 0 | 2.62(0.53) | 17.41(6.45) | 0 |
| LR2 | 2.12(0.69) | 0.89(1.02) | 12 | 1.82(0.69) | 1.34(1.42) | 3 | 1.73(0.69) | 1.69(1.77) | 2 |