Table 3:
Relative empirical variance of outcome model parameters under various imputation strategies and outcome model settings (relative to full data without missingness). Results across 500 simulations are presented. In all settings, X1 was fully-observed and X2 and possibly X3 were imputed.
| Relative variance for effect of X1 | Relative variance for effect of X2 | |||||||
|---|---|---|---|---|---|---|---|---|
| Missingness:† | MCAR | X 1 | Y | X1, Y | MCAR | X 1 | Y | X1, Y |
| Scenario 1: Linear Regression | ||||||||
| Full Data | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| Complete Case | 2.06 | 2.07 | 1.87 | 1.85 | 1.88 | 2.09 | 1.75 | 1.73 |
| MICE with Y* | ||||||||
| ↳ Rubin’s rules | 1.35 | 1.37 | 1.45 | 1.31 | 1.70 | 1.85 | 1.98 | 1.90 |
| ↳ Stacked, 1/M weighted | 1.35 | 1.37 | 1.45 | 1.31 | 1.70 | 1.85 | 1.97 | 1.90 |
| MICE without Y* | ||||||||
| ↳ Rubin’s rules | 0.86 | 0.87 | 0.85 | 0.86 | 0.55 | 0.54 | 0.48 | 0.48 |
| ↳ Stacked, f(Y|X) weighted | 1.34 | 1.37 | 1.45 | 1.31 | 1.69 | 1.83 | 1.95 | 1.89 |
| Bartlett et al. (2014) ⋈ | 1.39 | 1.45 | 1.50 | 1.33 | 1.74 | 1.95 | 2.07 | 1.99 |
| Scenario 2: Logistic Regression | ||||||||
| Full Data | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| Complete Case | 2.52 | 2.29 | 2.02 | 4.08 | 2.36 | 2.46 | 2.15 | 3.66 |
| MICE with Y | ||||||||
| ↳ Rubin’s rules | 1.08 | 1.08 | 1.04 | 1.13 | 1.64 | 1.64 | 1.45 | 2.35 |
| ↳ Stacked, 1/M weighted | 1.08 | 1.07 | 1.04 | 1.12 | 1.63 | 1.63 | 1.45 | 2.33 |
| MICE without Y* | ||||||||
| ↳ Rubin’s rules | 0.93 | 0.95 | 0.92 | 0.94 | 0.54 | 0.45 | 0.60 | 0.43 |
| ↳ Stacked, f(Y|X) weighted | 1.09 | 1.08 | 1.03 | 1.14 | 1.78 | 1.82 | 1.55 | 2.77 |
| Bartlett et al. (2014) | 1.09 | 1.09 | 1.05 | 1.14 | 1.73 | 1.74 | 1.52 | 2.58 |
| Scenario 3: Linear Regression with Interaction | ||||||||
| Full Data | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| Complete Case | 2.14 | 2.13 | 1.78 | 2.37 | 2.11 | 2.04 | 1.83 | 2.50 |
| MICE with Y | ||||||||
| ↳ Rubin’s rules | 2.85 | 2.12 | 1.34 | 5.20 | 3.16 | 3.35 | 1.62 | 4.02 |
| ↳ Stacked, 1/M weighted | 2.85 | 2.12 | 1.34 | 5.21 | 3.16 | 3.35 | 1.62 | 4.05 |
| MICE with Y + interaction* | 2.92 | 2.45 | 1.81 | 4.40 | 4.96 | 4.51 | 2.79 | 5.81 |
| MICE without Y | ||||||||
| ↳ Rubin’s rules | 2.25 | 1.69 | 1.16 | 4.54 | 1.03 | 0.77 | 0.86 | 0.85 |
| ↳ Stacked, f(Y|X) weighted | 1.50 | 1.40 | 1.26 | 2.07 | 1.74 | 1.71 | 1.60 | 2.06 |
| Bartlett et al. (2014) | 1.52 | 1.46 | 1.29 | 2.07 | 1.75 | 1.60 | 1.55 | 1.99 |
| Scenario 4: Cox Proportional Hazards Regression | ||||||||
| Full Data | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| Complete Case | 1.85 | 2.20 | 1.64 | 2.05 | 2.13 | 1.81 | 2.12 | 1.79 |
| MICE with Y | ||||||||
| ↳ Rubin’s rules | 1.06 | 1.13 | 1.02 | 1.17 | 1.62 | 1.57 | 2.02 | 1.95 |
| ↳ Stacked, 1/M weighted | 1.07 | 1.24 | 1.02 | 1.17 | 1.62 | 1.64 | 2.01 | 1.94 |
| MICE without Y | ||||||||
| ↳ Rubin’s rules | 0.97 | 1.01 | 0.95 | 0.99 | 0.42 | 0.42 | 0.45 | 0.44 |
| ↳ Stacked, f(Y|X) weighted | 1.14 | 1.21 | 1.08 | 1.17 | 1.91 | 1.61 | 2.18 | 1.81 |
| Bartlett et al. (2014) | 1.15 | 1.27 | 1.11 | 1.19 | 2.02 | 1.83 | 2.39 | 2.01 |
Missingness is MCAR or MAR dependent on the fully-observed terms listed.
MICE either including or excluding Y from the linear regression imputation models. An interaction between Y and X1 was included in one setting for Scenario 3. MICE with Y for Scenario 4 followed recommendations in White and Royston (2009). Unless otherwise specified, MICE imputations were analyzed using Rubin’s rules.
Xp imputed from distribution proportional to f(Y|X)f(Xp|X−p) using R package smcfcs. Then, apply Rubin’s rules.