Results for simulation II (|$n$| = 500; 500 Monte Carlo replicates). Point estimates are means across Monte Carlo replicates; standard deviations are Monte Carlo standard deviations.
| Method . | Model correct . | Posterior mean of |$\hat{\psi}_{1opt}$| . | Posterior mean of |$\hat{\psi}_{3opt}$| . | Estimated outcome train Pop. . | Coverage probability |$\psi_{1opt}, \psi_{2opt}$| . | Mean outcome test Pop. . |
|---|---|---|---|---|---|---|
| Freq. | None | 0.590 (0.126) | 0.103 (0.104) | 2.003 (0.355) | — | 0.526 (0.064) |
| Freq. | Treat | 0.479 (0.157) | 0.101 (0.125) | 1.160 (0.155) | — | 0.530 (0.057) |
| Freq. | Outcome | 0.503 (0.048) | 0.102 (0.020) | 1.004 (0.068) | — | 0.581 (0.010) |
| Freq. | Both | 0.499 (0.031) | 0.100 (0.004) | 1.000 (0.065) | — | 0.585 (0.004) |
| Freq. | IPW | 0.464 (0.157) | 0.089 (0.134) | 1.198 (0.184) | — | 0.529 (0.055) |
| Bayes. | None | 0.589 (0.123) | 0.094 (0.106) | 2.200 (0.351) | 0.952 0.996 | 0.549 (0.022) |
| Bayes. | Treat | 0.481 (0.165) | 0.089 (0.124) | 1.254 (0.150) | 0.992 1 | 0.539 (0.025) |
| Bayes. | Outcome | 0.498 (0.050) | 0.101 (0.016) | 1.008 (0.066) | 0.994 1 | 0.587 (0.005) |
| Bayes. | Both | 0.497 (0.029) | 0.100 (0.004) | 1.001 (0.064) | 1 1 | 0.591 (0.003) |
| Bayes. | IPW | 0.468 (0.163) | 0.072 (0.130) | 1.317 (0.198) | 0.992 1 | 0.537 (0.024) |
| Method . | Model correct . | Posterior mean of |$\hat{\psi}_{1opt}$| . | Posterior mean of |$\hat{\psi}_{3opt}$| . | Estimated outcome train Pop. . | Coverage probability |$\psi_{1opt}, \psi_{2opt}$| . | Mean outcome test Pop. . |
|---|---|---|---|---|---|---|
| Freq. | None | 0.590 (0.126) | 0.103 (0.104) | 2.003 (0.355) | — | 0.526 (0.064) |
| Freq. | Treat | 0.479 (0.157) | 0.101 (0.125) | 1.160 (0.155) | — | 0.530 (0.057) |
| Freq. | Outcome | 0.503 (0.048) | 0.102 (0.020) | 1.004 (0.068) | — | 0.581 (0.010) |
| Freq. | Both | 0.499 (0.031) | 0.100 (0.004) | 1.000 (0.065) | — | 0.585 (0.004) |
| Freq. | IPW | 0.464 (0.157) | 0.089 (0.134) | 1.198 (0.184) | — | 0.529 (0.055) |
| Bayes. | None | 0.589 (0.123) | 0.094 (0.106) | 2.200 (0.351) | 0.952 0.996 | 0.549 (0.022) |
| Bayes. | Treat | 0.481 (0.165) | 0.089 (0.124) | 1.254 (0.150) | 0.992 1 | 0.539 (0.025) |
| Bayes. | Outcome | 0.498 (0.050) | 0.101 (0.016) | 1.008 (0.066) | 0.994 1 | 0.587 (0.005) |
| Bayes. | Both | 0.497 (0.029) | 0.100 (0.004) | 1.001 (0.064) | 1 1 | 0.591 (0.003) |
| Bayes. | IPW | 0.468 (0.163) | 0.072 (0.130) | 1.317 (0.198) | 0.992 1 | 0.537 (0.024) |
Results for simulation II (|$n$| = 500; 500 Monte Carlo replicates). Point estimates are means across Monte Carlo replicates; standard deviations are Monte Carlo standard deviations.
| Method . | Model correct . | Posterior mean of |$\hat{\psi}_{1opt}$| . | Posterior mean of |$\hat{\psi}_{3opt}$| . | Estimated outcome train Pop. . | Coverage probability |$\psi_{1opt}, \psi_{2opt}$| . | Mean outcome test Pop. . |
|---|---|---|---|---|---|---|
| Freq. | None | 0.590 (0.126) | 0.103 (0.104) | 2.003 (0.355) | — | 0.526 (0.064) |
| Freq. | Treat | 0.479 (0.157) | 0.101 (0.125) | 1.160 (0.155) | — | 0.530 (0.057) |
| Freq. | Outcome | 0.503 (0.048) | 0.102 (0.020) | 1.004 (0.068) | — | 0.581 (0.010) |
| Freq. | Both | 0.499 (0.031) | 0.100 (0.004) | 1.000 (0.065) | — | 0.585 (0.004) |
| Freq. | IPW | 0.464 (0.157) | 0.089 (0.134) | 1.198 (0.184) | — | 0.529 (0.055) |
| Bayes. | None | 0.589 (0.123) | 0.094 (0.106) | 2.200 (0.351) | 0.952 0.996 | 0.549 (0.022) |
| Bayes. | Treat | 0.481 (0.165) | 0.089 (0.124) | 1.254 (0.150) | 0.992 1 | 0.539 (0.025) |
| Bayes. | Outcome | 0.498 (0.050) | 0.101 (0.016) | 1.008 (0.066) | 0.994 1 | 0.587 (0.005) |
| Bayes. | Both | 0.497 (0.029) | 0.100 (0.004) | 1.001 (0.064) | 1 1 | 0.591 (0.003) |
| Bayes. | IPW | 0.468 (0.163) | 0.072 (0.130) | 1.317 (0.198) | 0.992 1 | 0.537 (0.024) |
| Method . | Model correct . | Posterior mean of |$\hat{\psi}_{1opt}$| . | Posterior mean of |$\hat{\psi}_{3opt}$| . | Estimated outcome train Pop. . | Coverage probability |$\psi_{1opt}, \psi_{2opt}$| . | Mean outcome test Pop. . |
|---|---|---|---|---|---|---|
| Freq. | None | 0.590 (0.126) | 0.103 (0.104) | 2.003 (0.355) | — | 0.526 (0.064) |
| Freq. | Treat | 0.479 (0.157) | 0.101 (0.125) | 1.160 (0.155) | — | 0.530 (0.057) |
| Freq. | Outcome | 0.503 (0.048) | 0.102 (0.020) | 1.004 (0.068) | — | 0.581 (0.010) |
| Freq. | Both | 0.499 (0.031) | 0.100 (0.004) | 1.000 (0.065) | — | 0.585 (0.004) |
| Freq. | IPW | 0.464 (0.157) | 0.089 (0.134) | 1.198 (0.184) | — | 0.529 (0.055) |
| Bayes. | None | 0.589 (0.123) | 0.094 (0.106) | 2.200 (0.351) | 0.952 0.996 | 0.549 (0.022) |
| Bayes. | Treat | 0.481 (0.165) | 0.089 (0.124) | 1.254 (0.150) | 0.992 1 | 0.539 (0.025) |
| Bayes. | Outcome | 0.498 (0.050) | 0.101 (0.016) | 1.008 (0.066) | 0.994 1 | 0.587 (0.005) |
| Bayes. | Both | 0.497 (0.029) | 0.100 (0.004) | 1.001 (0.064) | 1 1 | 0.591 (0.003) |
| Bayes. | IPW | 0.468 (0.163) | 0.072 (0.130) | 1.317 (0.198) | 0.992 1 | 0.537 (0.024) |
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