Table 1.
Monte Carlo study of the performance of the proposed estimator with outcome normally distributed and missing at random
| Bias | RMSE | MAE | Bias | RMSE | MAE | Bias | RMSE | MAE | Bias | RMSE | MAE | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Miss-C, OR-C | Miss-I, OR-C | Miss-C, OR-I | Miss-I OR-I | |||||||||
| n = 200, Y observed iff A = 1 | ||||||||||||
| β̂reg(η̂) | 0.07 | 2.54 | 1.77 | 0.07 | 2.54 | 1.77 | −0.51 | 3.38 | 2.36 | −0.51 | 3.38 | 2.36 |
| β̂(η̂, α̂) | 0.07 | 2.54 | 1.78 | 0.07 | 2.60 | 1.79 | 0.38 | 3.50 | 2.30 | −7.71 | 44.28 | 3.59 |
| β̂ipw | −0.09 | 4.16 | 2.54 | 2.07 | 10.99 | 3.20 | −0.09 | 4.16 | 2.54 | 2.07 | 10.29 | 3.20 |
| β̂Cao | 0.07 | 2.54 | 1.79 | 0.06 | 2.53 | 1.78 | 0.08 | 2.59 | 1.82 | −0.41 | 3.51 | 2.14 |
| β̂dr | 0.07 | 2.54 | 1.78 | 0.06 | 2.54 | 1.77 | 0.33 | 2.92 | 2.05 | −1.71 | 3.58 | 2.42 |
| n = 1000, Y observed iff A = 1 | ||||||||||||
| β̂reg(η̂) | 0.01 | 1.19 | 0.82 | 0.01 | 1.19 | 0.82 | −0.82 | 1.70 | 1.19 | −0.82 | 1.70 | 1.19 |
| β̂(η̂, α̂) | 0.01 | 1.19 | 0.82 | 0.03 | 1.20 | 0.81 | 0.08 | 1.59 | 1.08 | −9.78 | 23.55 | 5.24 |
| β̂ipw | −0.01 | 1.68 | 1.14 | 4.40 | 9.20 | 2.55 | −0.01 | 1.68 | 1.14 | 4.40 | 9.20 | 2.55 |
| β̂Cao | 0.01 | 1.19 | 0.81 | 0.01 | 1.19 | 0.81 | 0.04 | 1.19 | 0.84 | −1.26 | 1.81 | 1.34 |
| β̂dr | 0.01 | 1.19 | 0.81 | 0.01 | 1.19 | 0.81 | 0.06 | 1.22 | 0.86 | −2.56 | 2.91 | 2.54 |
| n = 200, Y observed iff A = 0 | ||||||||||||
| β̂reg(η̂) | 0.07 | 2.54 | 1.76 | 0.07 | 2.54 | 1.76 | 5.01 | 5.79 | 5.02 | 5.01 | 5.79 | 5.02 |
| β̂(η̂, α̂) | 0.07 | 2.54 | 1.75 | 0.07 | 2.54 | 1.76 | 0.53 | 3.83 | 2.38 | 3.25 | 4.59 | 3.44 |
| β̂ipw | 0.29 | 3.89 | 2.47 | 3.85 | 5.02 | 4.09 | 0.29 | 3.89 | 2.47 | 3.85 | 5.02 | 4.09 |
| β̂Cao | 0.07 | 2.55 | 1.79 | 0.08 | 2.54 | 1.77 | 0.47 | 2.61 | 1.81 | 0.94 | 3.21 | 2.26 |
| β̂dr | 0.08 | 2.54 | 1.76 | 0.08 | 2.54 | 1.77 | 1.16 | 3.01 | 2.03 | 2.54 | 3.89 | 2.82 |
| n = 1000, Y observed iff A = 0 | ||||||||||||
| β̂reg(η̂) | 0.01 | 1.19 | 0.83 | 0.01 | 1.19 | 0.83 | 4.93 | 5.10 | 4.93 | 4.93 | 5.10 | 4.93 |
| β̂(η̂, α̂) | 0.01 | 1.19 | 0.83 | 0.01 | 1.19 | 0.83 | 0.18 | 1.67 | 1.15 | 3.09 | 3.40 | 3.09 |
| β̂ipw | 0.12 | 1.68 | 1.19 | 3.71 | 3.98 | 3.71 | 0.12 | 1.68 | 1.19 | 3.71 | 3.98 | 3.71 |
| β̂Cao | 0.01 | 1.19 | 0.83 | 0.01 | 1.19 | 0.84 | 0.14 | 1.21 | 0.84 | 1.12 | 1.70 | 1.24 |
| β̂dr | 0.01 | 1.19 | 0.83 | 0.01 | 1.19 | 0.84 | 0.29 | 1.29 | 0.94 | 1.47 | 2.07 | 1.54 |
RMSE, root mean square error; MAE, median absolute error; Miss-C and Miss-I (OR-C and OR-I), correct and incorrect missingness (outcome regression); β̂reg(η̂), outcome regression estimator; β̂(η̂, α̂), standard double robust estimator; β̂ipw inverse probability weighted estimator; β̂Cao, Cao et al. estimator, β̂dr, new double robust estimator.