Table 1:
Relative bias and coverage probabilities averaged over the simulated ERF estimates when measurement error is present.
| n | m | ω 2 | τ 2 | GPS | Outcome | EPE | Relative Bias |
95% Coverage Probability |
||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| No Correction | Regression Calibration | BART Multiple Imputation | GLM Multiple Imputation | No Correction | Regression Calibration | BART Multiple Imputation | GLM Multiple Imputation | |||||||
|
| ||||||||||||||
| 400 | 2,000 | 1 | 1 | 0.35 (−0.09) | 0.28 (−0.09) | 0.21 (−0.06) | −0.03 (−0.02) | 0.44 (0.52) | 0.53 (0.54) | 0.75 (0.78) | 0.83 (0.90) | |||
| 400 | 2,000 | 1 | 2 | 0.42 (−0.11) | 0.30 (−0.09) | 0.28 (−0.06) | −0.05 (−0.02) | 0.39 (0.40) | 0.52 (0.46) | 0.74 (0.73) | 0.70 (0.91) | |||
| 400 | 2,000 | 2 | 1 | 0.38 (−0.12) | 0.32 (−0.11) | 0.22 (−0.08) | −0.03 (−0.02) | 0.39 (0.45) | 0.45 (0.48) | 0.71 (0.70) | 0.85 (0.90) | |||
| 400 | 2,000 | 2 | 2 | 0.42 (−0.14) | 0.29 (−0.13) | 0.21 (−0.08) | −0.05 (−0.02) | 0.36 (0.35) | 0.45 (0.44) | 0.71 (0.62) | 0.75 (0.92) | |||
| 400 | 4,000 | 1 | 1 | 0.30 (−0.06) | 0.21 (−0.06) | 0.20 (−0.04) | −0.03 (−0.02) | 0.53 (0.66) | 0.63 (0.69) | 0.78 (0.78) | 0.80 (0.91) | |||
| 400 | 4,000 | 1 | 2 | 0.31 (−0.08) | 0.18 (−0.07) | 0.21 (−0.05) | −0.06 (−0.02) | 0.47 (0.58) | 0.63 (0.65) | 0.78 (0.76) | 0.64 (0.92) | |||
| 400 | 4,000 | 2 | 1 | 0.29 (−0.09) | 0.23 (−0.08) | 0.19 (−0.06) | −0.02 (−0.02) | 0.47 (0.58) | 0.55 (0.64) | 0.74 (0.72) | 0.86 (0.92) | |||
| 400 | 4,000 | 2 | 2 | 0.30 (−0.11) | 0.19 (−0.10) | 0.15 (−0.07) | −0.04 (−0.02) | 0.42 (0.48) | 0.58 (0.52) | 0.76 (0.71) | 0.69 (0.93) | |||
| 800 | 4,000 | 1 | 1 | 0.29 (−0.08) | 0.21 (−0.07) | 0.12 (−0.03) | −0.04 (0.00) | 0.42 (0.45) | 0.53 (0.52) | 0.76 (0.75) | 0.77 (0.96) | |||
| 800 | 4,000 | 1 | 2 | 0.35 (−0.11) | 0.22 (−0.09) | 0.12 (−0.04) | −0.07 (0.00) | 0.37 (0.38) | 0.50 (0.44) | 0.72 (0.74) | 0.62 (0.96) | |||
| 800 | 4,000 | 2 | 1 | 0.36 (−0.10) | 0.29 (−0.09) | 0.15 (−0.04) | −0.04 (0.00) | 0.37 (0.38) | 0.41 (0.39) | 0.73 (0.71) | 0.82 (0.96) | |||
| 800 | 4,000 | 2 | 2 | 0.43 (−0.12) | 0.31 (−0.10) | 0.13 (−0.04) | −0.06 (0.00) | 0.35 (0.30) | 0.43 (0.38) | 0.72 (0.72) | 0.65 (0.94) | |||
| 800 | 8,000 | 1 | 1 | 0.20 (−0.06) | 0.15 (−0.05) | 0.10 (−0.03) | −0.04 (−0.01) | 0.50 (0.67) | 0.65 (0.69) | 0.78 (0.83) | 0.74 (0.94) | |||
| 800 | 8,000 | 1 | 2 | 0.26 (−0.06) | 0.16 (−0.05) | 0.10 (−0.03) | −0.07 (0.00) | 0.44 (0.56) | 0.59 (0.60) | 0.72 (0.76) | 0.59 (0.94) | |||
| 800 | 8,000 | 2 | 1 | 0.22 (−0.08) | 0.17 (−0.07) | 0.10 (−0.03) | −0.04 (−0.01) | 0.45 (0.52) | 0.55 (0.58) | 0.78 (0.78) | 0.78 (0.94) | |||
| 800 | 8,000 | 2 | 2 | 0.28 (−0.08) | 0.19 (−0.07) | 0.11 (−0.03) | −0.07 (−0.01) | 0.42 (0.45) | 0.56 (0.53) | 0.73 (0.74) | 0.62 (0.94) | |||
|
| ||||||||||||||
| 800 | 4,000 | 2 | 1 | ✓ | 0.91 (−0.09) | 0.30 (−0.09) | 0.15 (−0.04) | −0.03 (0.00) | 0.30 (0.47) | 0.42 (0.40) | 0.71 (0.73) | 0.76 (0.93) | ||
| 800 | 4,000 | 2 | 1 | ✓ | 0.18 (−0.20) | 0.11 (−0.20) | −0.01 (−0.14) | 0.04 (0.00) | 0.25 (0.02) | 0.32 (0.03) | 0.55 (0.14) | 0.32 (0.96) | ||
| 800 | 4,000 | 2 | 1 | ✓ | ✓ | 0.60 (−0.22) | 0.11 (−0.20) | −0.01 (−0.14) | 0.05 (0.00) | 0.19 (0.02) | 0.30 (0.02) | 0.55 (0.15) | 0.32 (0.97) | |
| 800 | 4,000 | 2 | 1 | ✓ | 0.52 (−0.06) | 0.46 (−0.05) | 0.29 (−0.01) | −0.03 (0.00) | 0.45 (0.54) | 0.49 (0.58) | 0.75 (0.82) | 0.75 (0.93) | ||
| 800 | 4,000 | 2 | 1 | ✓ | ✓ | 1.28 (0.02) | 0.59 (−0.02) | 0.40 (0.02) | −0.03 (0.00) | 0.34 (0.66) | 0.49 (0.61) | 0.74 (0.82) | 0.74 (0.93) | |
| 800 | 4,000 | 2 | 1 | ✓ | ✓ | 0.15 (−0.21) | 0.08 (−0.20) | −0.05 (−0.15) | −0.09 (−0.05) | 0.18 (0.01) | 0.28 (0.01) | 0.44 (0.08) | 0.37 (0.77) | |
| 800 | 4,000 | 2 | 1 | ✓ | ✓ | ✓ | 0.52 (−0.25) | 0.02 (−0.23) | −0.09 (−0.16) | −0.09 (−0.05) | 0.15 (0.00) | 0.25 (0.00) | 0.41 (0.04) | 0.38 (0.72) |
The values in parentheses represent the statistics evaluated at a = 11. The check-marks indicates whether the corresponding model labeled in the column header is misspecified. The GLM approach refers to the multiple imputation implementation using a log-linear outcome model. The BART approach to multiple imputation uses a BART outcome model.