Table 1 Summary of analyses from simulation setups 1-4
Data Simulated: | Gaussian True Surrogate Effects | Non-Gaussian True Surrogate Effects | |||
|---|---|---|---|---|---|
FP-RE Summary | NP-RE Summary | PP-RE Summary | PP-RE Summary | PP-FE Summary | |
Setup 1 (Truth) | |||||
\(\alpha _1\) (0) | 0.00(-0.09,0.11) | 0.02(-0.46,0.57) | -0.01(-0.14,0.13) | 0.00(-0.12,0.13) | -0.05(-0.15,0.06) |
\(\alpha _2\) (0) | 0.00(-0.45,0.49) | -0.01(-0.13,0.14) | 0.00(-0.12,0.14) | -0.05(-0.15,0.06) | |
\(\alpha _3\) (0) | 0.01(-0.46,0.57) | -0.01(-0.131,0.13) | 0.01(-0.12,0.15) | -0.05(-0.16,0.07) | |
\(\beta _1\) (-0.45) | -0.45(-0.63,-0.31) | -0.49(-1.44,0.33) | -0.44(-0.67,-0.24) | -0.46(-0.68,-0.27) | -0.35(-0.49,-0.22) |
\(\beta _2\) (-0.45) | -0.46(-1.52,0.51) | -0.45(-0.68,-0.25) | -0.46(-0.69,-0.27) | -0.35(-0.50,-0.22) | |
\(\beta _3\) (-0.45) | -0.47(-1.53,0.49) | -0.44(-0.65,-0.25) | -0.45(-0.67,-0.26) | -0.35(-0.49,-0.21) | |
\(\sigma _{e1}\) (0.05) | 0.06(0.02,0.14) | 0.08(0.02,0.22) | 0.06(0.01,0.17) | 0.06(0.01,0.17) | 0.07(0.01,0.18) |
\(\sigma _{e2}\) (0.05) | 0.08(0.02,0.22) | 0.06(0.01,0.17) | 0.07(0.01,0.17) | 0.07(0.01,0.18) | |
\(\sigma _{e3}\) (0.05) | 0.08(0.02,0.21) | 0.06(0.01,0.16) | 0.06(0.01,0.17) | 0.07(0.01,0.18) | |
\(R^2\) 1 (0.95) | 0.90(0.60,0.99) | 0.80(0.20,0.99) | 0.89(0.41,1.00) | 0.91(0.49,1.00) | 0.92(0.57,1.00) |
\(R^2\) 2 (0.95) | 0.78(0.23,0.98) | 0.88(0.40,1.00) | 0.90(0.48,1.00) | 0.91(0.55,1.00) | |
\(R^2\) 3 (0.95) | 0.80(0.23,0.98) | 0.90(0.46,1.00) | 0.90(0.48,1.00) | 0.92(0.56,1.00) | |
Setup 2 (Truth) | |||||
\(\alpha _1\) (0) | 0.00(-0.11,0.13) | -0.01(-0.43,0.39) | 0.01(-0.12,0.15) | 0.01(-0.11,0.14) | -0.02(-0.12,0.09) |
\(\alpha _2\) (0) | 0.01(-0.57,0.62) | 0.01(-0.14,0.18) | 0.01(-0.12,0.16) | -0.03(-0.14,0.09) | |
\(\alpha _3\) (0) | 0.03(-0.54,0.77) | -0.02(-0.20,0.18) | -0.02(-0.20,0.17) | -0.09(-0.25,0.05) | |
\(\beta _1\) (-0.25) | 0.47(-0.70,-0.27) | -0.25(-1.93,1.54) | -0.32(-0.73,0.06) | -0.33(-0.66,0.00) | -0.24(-0.42,-0.05) |
\(\beta _2\) (-0.35) | -0.40(-1.85,0.97) | -0.40(-0.75,-0.09) | -0.40(-0.72,-0.11) | -0.28(-0.46,-0.11) | |
\(\beta _3\) (-0.6) | -0.65(-1.88,0.29) | -0.57(-0.88,-0.30) | -0.55(-0.85,-0.30) | -0.42(-0.61,-0.24) | |
\(\sigma _{e1}\) (0.15) | 0.13(0.06,0.22) | 0.11(0.03,0.27) | 0.10(0.02,0.23) | 0.11(0.03,0.23) | 0.11(0.03,0.24) |
\(\sigma _{e2}\) (0.115) | 0.09(0.03,0.24) | 0.09(0.02,0.20) | 0.10(0.02,0.22) | 0.10(0.03,0.22) | |
\(\sigma _{e3}\) (0.06) | 0.09(0.03,0.26) | 0.08(0.01,0.21) | 0.09(0.02,0.22) | 0.10(0.02,0.22) | |
\(R^2\) 1 (0.35) | 0.69(0.33,0.91) | 0.45(0.02,0.93) | 0.53(0.07,0.95) | 0.59(0.10,0.95) | 0.66(0.16,0.95) |
\(R^2\) 1 (0.65) | 0.61(0.07,0.96) | 0.69(0.14,0.98) | 0.71(0.17,0.97) | 0.76(0.25,0.97) | |
\(R^2\) 3 (0.95) | 0.84(0.29,0.99) | 0.85(0.38,1.00) | 0.84(0.37,0.99) | 0.87(0.48,0.99) | |
Setup 3 (Truth) | |||||
\(\alpha _1\) (0) | 0.01(-0.10,0.13) | 0.02(-0.39,0.43) | 0.03(-0.10,0.18) | 0.01(-0.11,0.14) | -0.02(-0.12,0.09) |
\(\alpha _2\) (0) | 0.02(-0.47,0.62) | 0.00(-0.15,0.17) | -0.01(-0.15,0.15) | -0.07(-0.19,0.05) | |
\(\alpha _3\) (0) | 0.05(-0.53,0.81) | 0.00(-0.17,0.20) | -0.01(-0.18,0.18) | -0.09(-0.24,0.05) | |
\(\beta _1\) (-0.25) | -0.56(-0.80,-0.36) | -0.33(-1.90,1.20) | -0.40(-0.80,-0.03) | -0.36(-0.71,-0.03) | -0.25(-0.44,-0.06) |
\(\beta _2\) (-0.6) | -0.65(-2.09,0.51) | -0.59(-0.95,-0.31) | -0.58(-0.91,-0.32) | -0.42(-0.61,-0.24) | |
\(\beta _3\) (-0.6) | -0.68(-1.93,0.26) | -0.60(-0.93,-0.35) | -0.58(-0.87,-0.34) | -0.43(-0.61,-0.26) | |
\(\sigma _{e1}\) (0.15) | 0.11(0.05,0.20) | 0.11(0.04,0.27) | 0.10(0.02,0.23) | 0.10(0.03,0.23) | 0.11(0.03,0.24) |
\(\sigma _{e2}\) (0.06) | 0.09(0.03,0.25) | 0.08(0.01,0.20) | 0.09(0.01,0.21) | 0.10(0.02,0.23) | |
\(\sigma _{e3}\) (0.06) | 0.09(0.03,0.25) | 0.08(0.01,0.20) | 0.08(0.01,0.21) | 0.09(0.02,0.22) | |
\(R^2\) 1 (0.35) | 0.80(0.46,0.95) | 0.51(0.04,0.94) | 0.64(0.11,0.96) | 0.64(0.12,0.96) | 0.69(0.18,0.95) |
\(R^2\) 1 (0.95) | 0.81(0.20,0.99) | 0.87(0.36,1.00) | 0.88(0.41,1.00) | 0.88(0.47,0.99) | |
\(R^2\) 3 (0.95) | 0.86(0.29,0.99) | 0.89(0.42,1.00) | 0.87(0.43,0.99) | 0.89(0.51,0.99) | |
Setup 4 (Truth) | |||||
\(\alpha _1\) (0) | 0.01(-0.11,0.13) | -0.01(-0.45,0.42) | 0.01(-0.13,0.16) | 0.01(-0.12,0.14) | -0.01(-0.12,0.10) |
\(\alpha _2\) (0) | 0.00(-0.64,0.62) | 0.01(-0.16,0.19) | 0.02(-0.13,0.18) | -0.01(-0.13,0.11) | |
\(\alpha _3\) (0) | 0.02(-0.59,0.80) | -0.03(-0.23,0.18) | -0.03(-0.22,0.18) | -0.09(-0.25,0.06) | |
\(\beta _1\) (-0.25) | -0.44(-0.68,-0.23) | -0.22(-1.91,1.50) | -0.31(-0.73,0.10 | -0.31(-0.65,0.02) | -0.23(-0.41,-0.04) |
\(\beta _2\) (-0.25) | -0.26(-1.84,1.35) | -0.31(-0.70,0.06) | -0.33(-0.67,0.00) | -0.23(-0.42,-0.04) | |
\(\beta _3\) (-0.6) | -0.64(-1.93,0.37) | -0.55(-0.89,-0.25) | -0.55(-0.85,-0.28) | -0.41(-0.61,-0.23) | |
\(\sigma _{e1}\) (0.15) | 0.15(0.08,0.24) | 0.12(0.04,0.28) | 0.11(0.03,0.23) | 0.11(0.03,0.24) | 0.12(0.04,0.24) |
\(\sigma _{e2}\) (0.15) | 0.12(0.04,0.27) | 0.11(0.03,0.24) | 0.12(0.03,0.24) | 0.12(0.04,0.25) | |
\(\sigma _{e3}\) (0.06) | 0.09(0.03,0.25) | 0.09(0.02,0.22) | 0.10(0.02,0.23) | 0.11(0.02,0.23) | |
\(R^2\) 1 (0.35) | 0.60(0.24,0.87) | 0.47(0.03,0.93) | 0.50(0.06,0.94) | 0.56(0.08,0.94) | 0.63(0.14,0.94) |
\(R^2\) 1 (0.35) | 0.47(0.03,0.94) | 0.51(0.06,0.93) | 0.55(0.07,0.94) | 0.63(0.12,0.93) | |
\(R^2\) 3 (0.95) | 0.84(0.28,0.99) | 0.82(0.31,0.99) | 0.82(0.34,0.99) | 0.86(0.44,0.99) | |