Table 2.
Simulation results comparing the ATE estimation under the alternative hypothesis.
| n | method | set | mean bias | log OR | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| π0 | π1 | Δπ | log OR | MSE | MSE* | VEB (%) | coverage (%) | power (%) | |||
| 100 | GC | outcome | −0.001 | −0.002 | −0.001 | −0.003 | 0.404 | 0.634 | −7.3 | 93.2 | 24.7 |
| treatment | −0.002 | −0.001 | 0.000 | 0.004 | 0.477 | 0.727 | −9.5 | 92.4 | 19.9 | ||
| common | −0.001 | −0.002 | −0.001 | −0.002 | 0.434 | 0.650 | −6.6 | 93.5 | 22.1 | ||
| entire | −0.002 | −0.001 | 0.001 | 0.003 | 0.450 | 0.714 | −11.4 | 91.8 | 22.6 | ||
| IPTW | outcome | −0.003 | −0.001 | 0.001 | 0.011 | 0.464 | 0.646 | 12.1 | 97.4 | 12.1 | |
| treatment | −0.006 | 0.002 | 0.008 | 0.046 | 0.633 | 0.769 | −7.6 | 93.8 | 16.7 | ||
| common | −0.002 | −0.001 | 0.001 | 0.010 | 0.480 | 0.657 | 6.3 | 96.3 | 13.5 | ||
| entire | −0.006 | 0.003 | 0.009 | 0.053 | 0.647 | 0.773 | −7.2 | 94.7 | 16.4 | ||
| TMLE | outcome | −0.001 | −0.002 | −0.000 | 0.003 | 0.438 | 0.642 | −14.3 | 89.5 | 26.9 | |
| treatment | −0.004 | 0.002 | 0.006 | 0.039 | 0.572 | 0.757 | −24.9 | 84.3 | 27.5 | ||
| common | −0.001 | −0.002 | −0.001 | 0.002 | 0.469 | 0.657 | −10.7 | 90.9 | 21.2 | ||
| entire | −0.005 | 0.003 | 0.007 | 0.043 | 0.544 | 0.748 | −30.7 | 80.9 | 34.3 | ||
| FM | outcome | −0.005 | 0.002 | 0.006 | 0.039 | 0.549 | 0.710 | −24.3 | 87.1 | 28.5 | |
| treatment | −0.009 | 0.005 | 0.014 | 0.082 | 0.677 | 0.832 | −37.7 | 78.0 | 35.1 | ||
| common | −0.005 | 0.001 | 0.006 | 0.038 | 0.563 | 0.713 | −26.3 | 85.8 | 29.1 | ||
| entire | −0.007 | 0.006 | 0.014 | 0.082 | 0.674 | 0.830 | −37.3 | 78.1 | 34.8 | ||
| 300 | GC | outcome | −0.000 | −0.000 | 0.000 | 0.001 | 0.221 | 0.532 | −1.9 | 94.5 | 59.8 |
| treatment | −0.000 | −0.000 | 0.000 | 0.001 | 0.259 | 0.608 | −2.8 | 94.3 | 47.4 | ||
| common | −0.000 | −0.000 | 0.000 | 0.001 | 0.237 | 0.539 | −1.2 | 94.8 | 53.5 | ||
| entire | −0.000 | −0.000 | 0.000 | 0.001 | 0.241 | 0.600 | −3.4 | 94.0 | 53.0 | ||
| IPTW | outcome | −0.001 | −0.000 | 0.001 | 0.006 | 0.239 | 0.533 | 20.2 | 98.0 | 34.7 | |
| treatment | −0.002 | 0.000 | 0.003 | 0.014 | 0.330 | 0.615 | 4.6 | 96.0 | 29.5 | ||
| common | −0.001 | −0.000 | 0.001 | 0.006 | 0.252 | 0.541 | 13.3 | 97.4 | 36.5 | ||
| entire | −0.002 | 0.000 | 0.002 | 0.013 | 0.326 | 0.607 | 7.9 | 96.6 | 28.5 | ||
| TMLE | outcome | −0.000 | −0.001 | −0.000 | 0.000 | 0.233 | 0.532 | −3.0 | 93.9 | 54.2 | |
| treatment | −0.001 | 0.000 | 0.002 | 0.009 | 0.310 | 0.612 | −10.4 | 90.6 | 40.2 | ||
| common | −0.001 | −0.001 | 0.000 | 0.001 | 0.249 | 0.540 | −1.5 | 94.6 | 48.1 | ||
| entire | −0.001 | 0.000 | 0.001 | 0.008 | 0.290 | 0.603 | −13.2 | 89.6 | 46.1 | ||
| FM | outcome | −0.002 | 0.000 | 0.002 | 0.010 | 0.294 | 0.552 | −20.2 | 88.7 | 51.6 | |
| treatment | −0.003 | 0.003 | 0.006 | 0.032 | 0.389 | 0.652 | −39.3 | 77.0 | 53.3 | ||
| common | −0.001 | −0.000 | 0.001 | 0.008 | 0.315 | 0.588 | −25.5 | 86.2 | 51.3 | ||
| entire | −0.003 | 0.003 | 0.006 | 0.032 | 0.377 | 0.644 | −37.4 | 77.8 | 52.2 | ||
| 500 | GC | outcome | −0.000 | 0.000 | 0.001 | 0.003 | 0.168 | 0.501 | −0.4 | 94.8 | 81.1 |
| treatment | −0.000 | 0.000 | 0.001 | 0.002 | 0.198 | 0.573 | −1.0 | 94.8 | 69.0 | ||
| common | −0.000 | 0.000 | 0.000 | 0.002 | 0.183 | 0.505 | −0.7 | 94.9 | 75.0 | ||
| entire | −0.000 | 0.000 | 0.001 | 0.004 | 0.183 | 0.569 | −1.0 | 94.8 | 75.3 | ||
| IPTW | outcome | −0.001 | 0.000 | 0.001 | 0.005 | 0.180 | 0.501 | 22.2 | 98.3 | 58.5 | |
| treatment | −0.001 | 0.001 | 0.001 | 0.007 | 0.248 | 0.573 | 8.1 | 96.5 | 42.3 | ||
| common | −0.001 | 0.000 | 0.001 | 0.005 | 0.193 | 0.505 | 13.8 | 97.3 | 58.6 | ||
| entire | −0.001 | 0.000 | 0.001 | 0.006 | 0.239 | 0.569 | 13.1 | 97.2 | 41.3 | ||
| TMLE | outcome | −0.000 | 0.000 | 0.000 | 0.002 | 0.177 | 0.501 | −0.8 | 94.7 | 76.8 | |
| treatment | −0.000 | 0.000 | 0.000 | 0.003 | 0.234 | 0.571 | −5.9 | 92.7 | 56.1 | ||
| common | −0.000 | 0.000 | 0.000 | 0.002 | 0.190 | 0.505 | −0.5 | 94.7 | 69.7 | ||
| entire | −0.000 | 0.000 | 0.000 | 0.003 | 0.218 | 0.566 | −7.5 | 91.8 | 63.1 | ||
| FM | outcome | −0.001 | 0.000 | 0.001 | 0.005 | 0.219 | 0.518 | −17.5 | 89.8 | 70.1 | |
| treatment | −0.002 | 0.002 | 0.003 | 0.018 | 0.302 | 0.598 | −39.8 | 76.2 | 65.5 | ||
| common | −0.001 | −0.000 | 0.001 | 0.005 | 0.266 | 0.555 | −31.8 | 82.3 | 66.4 | ||
| entire | −0.002 | 0.002 | 0.004 | 0.019 | 0.289 | 0.592 | −37.1 | 78.3 | 66.2 | ||
| 2000 | GC | outcome | −0.000 | −0.000 | −0.000 | −0.001 | 0.085 | 0.482 | −0.6 | 94.6 | 100.0 |
| treatment | 0.000 | −0.001 | −0.001 | −0.003 | 0.099 | 0.550 | −0.6 | 94.7 | 99.8 | ||
| common | 0.000 | −0.001 | −0.001 | −0.003 | 0.092 | 0.483 | −0.8 | 94.7 | 99.9 | ||
| entire | −0.000 | −0.000 | −0.000 | −0.001 | 0.091 | 0.550 | −0.6 | 94.7 | 99.9 | ||
| IPTW | outcome | −0.000 | −0.000 | 0.000 | 0.002 | 0.090 | 0.482 | 21.2 | 98.2 | 99.8 | |
| treatment | 0.000 | −0.001 | −0.001 | −0.002 | 0.122 | 0.547 | 9.3 | 96.7 | 95.1 | ||
| common | −0.000 | −0.000 | 0.000 | 0.001 | 0.096 | 0.483 | 13.5 | 97.3 | 99.7 | ||
| entire | 0.000 | −0.000 | −0.001 | −0.002 | 0.117 | 0.546 | 14.3 | 97.5 | 95.6 | ||
| 2000 | TMLE | outcome | −0.000 | −0.000 | −0.000 | −0.001 | 0.088 | 0.482 | −0.6 | 94.8 | 100.0 |
| treatment | 0.000 | −0.001 | −0.001 | −0.003 | 0.116 | 0.545 | −2.2 | 94.4 | 98.7 | ||
| common | 0.000 | −0.000 | −0.001 | −0.002 | 0.095 | 0.483 | −0.3 | 94.8 | 99.9 | ||
| entire | 0.000 | −0.000 | −0.001 | −0.002 | 0.108 | 0.544 | −2.6 | 94.1 | 99.4 | ||
| FM | outcome | −0.000 | −0.000 | −0.000 | 0.000 | 0.129 | 0.497 | −29.9 | 82.9 | 99.0 | |
| treatment | −0.001 | −0.000 | 0.000 | 0.003 | 0.169 | 0.569 | −46.6 | 70.6 | 96.2 | ||
| common | 0.000 | −0.000 | −0.001 | −0.001 | 0.205 | 0.534 | −55.9 | 61.1 | 92.7 | ||
| entire | −0.000 | −0.000 | 0.000 | 0.002 | 0.145 | 0.549 | −37.7 | 77.9 | 98.2 | ||
*MSE in the presence of an unmeasured confounder. Theoretical values: .