Table 1.
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 continuous 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) | 2.53(1.74) | 9 | 3.00(0.00) | 6.17(3.50) | 0 | 3.00(0.00) | 8.59(4.47) | 0 |
| Lasso2 | 3.00(0.00) | 0.72(1.26) | 62 | 3.00(0.00) | 1.50(1.92) | 37 | 3.00(0.00) | 2.25(2.40) | 22 |
| Lasso3 | 3.00(0.00) | 0.71(0.94) | 54 | 3.00(0.00) | 1.42(1.52) | 33 | 3.00(0.00) | 1.98(1.84) | 22 |
| A-Lasso1 | 3.00(0.00) | 0.48(0.79) | 66 | 2.99(0.08) | 1.91(2.06) | 26 | 2.85(0.37) | 11.07(5.72) | 0 |
| A-Lasso2 | 3.00(0.00) | 0.21(0.55) | 84 | 2.99(0.09) | 0.77(1.31) | 56 | 2.82(0.40) | 7.38(4.31) | 0 |
| A-Lasso3 | 3.00(0.00) | 0.17(0.47) | 87 | 2.99(0.08) | 0.65(1.11) | 61 | 2.84(0.37) | 6.63(4.27) | 1 |
| LR1 | 3.00(0.03) | 3.59(1.73) | 2 | 3.00(0.03) | 15.69(4.93) | 0 | 3.00(0.05) | 32.05(8.90) | 0 |
| LR2 | 2.99(0.08) | 2.03(1.44) | 13 | 2.99(0.11) | 5.01(3.21) | 3 | 2.98(0.13) | 8.01(5.21) | 2 |
| (β0, β1, β6, β12) = (−0.5, −1.0, 1.5, 1.0) | |||||||||
| Lasso1 | 2.99(0.11) | 2.95(1.64) | 4 | 2.94(0.31) | 6.96(3.73) | 0 | 2.88(0.41) | 8.66(4.59) | 0 |
| Lasso2 | 2.95(0.25) | 1.21(1.43) | 39 | 2.83(0.47) | 2.61(2.45) | 14 | 2.71(0.60) | 3.27(2.90) | 8 |
| Lasso3 | 2.99(0.11) | 1.18(1.17) | 33 | 2.94(0.26) | 2.51(2.07) | 12 | 2.85(0.41) | 3.11(2.59) | 10 |
| A-Lasso1 | 2.97(0.16) | 0.82(0.94) | 44 | 2.61(0.63) | 2.75(2.34) | 9 | 2.22(0.82) | 10.68(5.42) | 0 |
| A-Lasso2 | 2.94(0.24) | 0.35(0.71) | 71 | 2.50(0.67) | 1.41(1.58) | 23 | 2.13(0.85) | 7.37(4.21) | 0 |
| A-Lasso3 | 2.98(0.15) | 0.42(0.73) | 68 | 2.72(0.54) | 2.42(2.64) | 19 | 2.19(0.82) | 6.49(4.03) | 1 |
| LR1 | 2.68(0.52) | 3.35(1.62) | 2 | 2.70(0.50) | 9.95(3.96) | 0 | 2.70(0.49) | 18.29(6.41) | 0 |
| LR2 | 2.28(0.69) | 1.57(1.29) | 8 | 1.98(0.70) | 1.77(1.68) | 4 | 1.85(0.73) | 1.90(1.89) | 3 |