Table 6.
Impact of Percentage Missing Under Conditions 1, 2, and 3.
| a. Condition 1a | ||||||||
|---|---|---|---|---|---|---|---|---|
| Scenario | Variable* | Method | Mean β | Mean SE | Mean Bias | MSE | RelMSE | Coverage |
| A1: 20% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.002 | 0.211 | 0.002 | 0.044 | 0.543 | 95.0 (93.6,96.4) | |
| CC | 1.000 | 0.284 | -0.000 | 0.081 | 1.000 | 95.8 (94.6,97.0) | ||
| MI | 1.062 | 0.230 | 0.062 | 0.052 | 0.644 | 95.9 (94.7,97.1) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.549 | 0.378 | 0.049 | 0.166 | 0.068 | 95.4 (94.1,96.7) | |
| CC | 2.713 | 4.822 | 0.213 | 2.436 | 1.000 | 97.0 (95.9,98.1) | ||
| MI | 2.543 | 0.426 | 0.043 | 0.178 | 0.072 | 96.8 (95.7,97.9) | ||
| Interaction (True effect = 1.5) | Full | 1.547 | 0.434 | 0.047 | 0.204 | 0.082 | 95.7 (94.4,97.0) | |
| CC | 1.713 | 4.894 | 0.213 | 2.503 | 1.000 | 96.7 (95.6,97.8) | ||
| MI | 1.482 | 0.483 | -0.018 | 0.205 | 0.082 | 96.1 (94.9,97.3) | ||
| A2: 30% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.003 | 0.211 | 0.003 | 0.044 | 0.358 | 94.6 (93.2,96.0) | |
| CC | 0.992 | 0.345 | -0.008 | 0.122 | 1.000 | 95.0 (93.6,96.4) | ||
| MI | 1.095 | 0.245 | 0.095 | 0.065 | 0.531 | 94.9 (93.5,96.3) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.558 | 0.379 | 0.058 | 0.146 | 0.014 | 97.2 (96.2,98.2) | |
| CC | 3.288 | 25.207 | 0.788 | 10.689 | 1.000 | 98.5 (97.7,99.3) | ||
| MI | 2.557 | 0.462 | 0.057 | 0.178 | 0.017 | 98.3 (97.5,99.1) | ||
| Interaction (True effect = 1.5) | Full | 1.555 | 0.435 | 0.055 | 0.195 | 0.018 | 96.1 (94.9,97.3) | |
| CC | 2.295 | 25.290 | 0.795 | 10.904 | 0.018 | 96.1 (94.9,97.3) | ||
| MI | 1.462 | 0.515 | -0.038 | 0.219 | 0.020 | 97.8 (96.9,98.7) | ||
| A3: 40% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.020 | 0.211 | 0.020 | 0.045 | 0.258 | 95.6 (94.3,96.8) | |
| CC | 0.991 | 0.411 | -0.009 | 0.175 | 1.000 | 95.5 (94.2,96.8) | ||
| MI | 1.137 | 0.263 | 0.137 | 0.083 | 0.474 | 94.1 (92.6,95.6) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.540 | 0.377 | 0.040 | 0.140 | 0.006 | 96.6 (95.5,97.7) | |
| CC | 4.121 | 64.794 | 1.621 | 23.920 | 1.000 | 96.5 (95.4,97.6) | ||
| MI | 2.536 | 0.489 | 0.036 | 0.177 | 0.007 | 97.5 (96.5,98.5) | ||
| Interaction (True effect = 1.5) |
Full | 1.520 | 0.433 | 0.020 | 0.191 | 0.008 | 96.0 (94.8,97.2) | |
| CC | 3.131 | 64.890 | 1.631 | 24.283 | 1.000 | 97.0 (95.9,98.1) | ||
| MI | 1.399 | 0.538 | -0.101 | 0.220 | 0.009 | 97.2 (96.2,98.2) | ||
| b. Condition 2b | ||||||||
| Scenario | Variable | Method | Mean β | Mean SE | Mean Bias | MSE | RelMSE | Coverage |
| A4: 20% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.007 | 0.108 | 0.007 | 0.012 | 0.590 | 95.5 (94.2,96.8) | |
| CC | 1.010 | 0.147 | 0.010 | 0.021 | 1.000 | 95.9 (94.7,97.1) | ||
| MI | 1.059 | 0.115 | 0.059 | 0.017 | 0.820 | 92.5 (90.9,94.1) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.552 | 0.279 | 0.052 | 0.089 | 0.909 | 93.7 (92.2,95.2) | |
| CC | 2.548 | 0.298 | 0.048 | 0.098 | 1.000 | 93.4 (91.9,94.9) | ||
| MI | 2.593 | 0.281 | 0.093 | 0.095 | 0.966 | 94.5 (93.1,95.9) | ||
| Interaction (True effect = 1.5) |
Full | 1.544 | 0.300 | 0.044 | 0.102 | 0.872 | 93.6 (92.1,95.1) | |
| CC | 1.538 | 0.332 | 0.038 | 0.117 | 1.000 | 94.0 (92.5,95.5) | ||
| MI | 1.534 | 0.304 | 0.034 | 0.101 | 0.863 | 94.0 (92.5,95.5) | ||
| A5: 30% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.003 | 0.108 | 0.003 | 0.012 | 0.437 | 94.5 (93.1,95.9) | |
| CC | 1.008 | 0.168 | 0.008 | 0.027 | 1.000 | 94.6 (93.2,96.0) | ||
| MI | 1.079 | 0.118 | 0.079 | 0.020 | 0.741 | 92.4 (90.8,94.0) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.532 | 0.277 | 0.032 | 0.078 | 0.798 | 95.7 (94.4,97.0) | |
| CC | 2.529 | 0.317 | 0.029 | 0.098 | 1.000 | 96.2 (95.0,97.4) | ||
| MI | 2.612 | 0.284 | 0.112 | 0.091 | 0.928 | 95.6 (94.3,96.9) | ||
| Interaction (True effect = 1.5) |
Full | 1.529 | 0.297 | 0.029 | 0.091 | 0.731 | 95.4 (94.1,96.7) | |
| CC | 1.521 | 0.359 | 0.021 | 0.125 | 1.000 | 94.8 (93.4,96.2) | ||
| MI | 1.533 | 0.308 | 0.033 | 0.093 | 0.743 | 95.4 (94.1,96.7) | ||
| A6: 40% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.005 | 0.108 | 0.005 | 0.011 | 0.330 | 95.8 (94.6,97.0) | |
| CC | 1.007 | 0.197 | 0.007 | 0.035 | 1.000 | 95.9 (94.7,97.1) | ||
| MI | 1.109 | 0.123 | 0.109 | 0.026 | 0.749 | 87.5 (85.5,89.5) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.551 | 0.279 | 0.051 | 0.081 | 0.620 | 95.7 (94.4,97.0) | |
| CC | 2.563 | 0.355 | 0.063 | 0.131 | 1.000 | 96.2 (95.0,97.4) | ||
| MI | 2.675 | 0.295 | 0.175 | 0.112 | 0.852 | 94.3 (92.9,95.7) | ||
| Interaction (True effect = 1.5) |
Full | 1.547 | 0.300 | 0.047 | 0.094 | 0.563 | 95.1 (93.8,96.4) | |
| CC | 1.557 | 0.406 | 0.057 | 0.168 | 1.000 | 95.3 (94.0,96.6) | ||
| MI | 1.565 | 0.319 | 0.065 | 0.098 | 0.588 | 96.9 (95.8,98.0) | ||
| c. Condition 3c | ||||||||
| Scenario | Variable | Method | Mean β | Mean SE | Mean Bias | MSE | RelMSE | Coverage |
| A7: 20% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.012 | 0.108 | 0.012 | 0.012 | 0.076 | 94.7 (93.3,96.1) | |
| CC | 0.616 | 0.119 | -0.384 | 0.162 | 1.000 | 11.9 (9.9,13.9) | ||
| MI | 0.999 | 0.109 | -0.001 | 0.012 | 0.077 | 94.3 (92.9,95.7) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.549 | 0.279 | 0.049 | 0.080 | 0.667 | 96.1 (94.9,97.3) | |
| CC | 2.320 | 0.289 | -0.180 | 0.120 | 1.000 | 87.1 (85.0,89.2) | ||
| MI | 2.551 | 0.280 | 0.051 | 0.080 | 0.666 | 96.2 (95.0,97.4) | ||
| Interaction (True effect = 1.5) |
Full | 1.537 | 0.300 | 0.037 | 0.091 | 0.635 | 96.1 (94.9,97.3) | |
| CC | 1.704 | 0.313 | 0.204 | 0.143 | 1.000 | 92.6 (91.0,94.2) | ||
| MI | 1.552 | 0.301 | 0.052 | 0.092 | 0.642 | 95.7 (94.4,97.0) | ||
| A8: 30% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.006 | 0.108 | 0.006 | 0.012 | 0.027 | 95.0 (93.6,96.4) | |
| CC | 0.331 | 0.127 | -0.669 | 0.466 | 1.000 | 0.3 (0,0.6) | ||
| MI | 0.983 | 0.110 | -0.017 | 0.013 | 0.028 | 95.0 (93.6,96.4) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.528 | 0.277 | 0.028 | 0.075 | 0.179 | 95.8 (94.6,97.0) | |
| CC | 1.927 | 0.299 | -0.573 | 0.419 | 1.000 | 48.6 (45.5,51.7) | ||
| MI | 2.492 | 0.277 | -0.008 | 0.072 | 0.171 | 95.6 (94.3,96.9) | ||
| Interaction (True effect = 1.5) |
Full | 1.522 | 0.297 | 0.022 | 0.084 | 0.743 | 96.0 (94.8,97.2) | |
| CC | 1.596 | 0.325 | 0.096 | 0.113 | 1.000 | 95.7 (94.4,97.0) | ||
| MI | 1.509 | 0.298 | 0.009 | 0.081 | 0.713 | 96.1 (94.9,97.3) | ||
| A9: 40% missing | ||||||||
| X1 (X2 = 0) (True effect = 1.0) | Full | 1.007 | 0.108 | 0.007 | 0.011 | 0.012 | 95.8 (94.6,97.0) | |
| CC | 0.027 | 0.139 | -0.973 | 0.966 | 1.000 | 0.0 (0.0,0.0) | ||
| MI | 0.968 | 0.110 | -0.032 | 0.013 | 0.013 | 94.1 (92.6,95.6) | ||
| X1 (X2 = 1) (True effect = 2.5) | Full | 2.537 | 0.277 | 0.037 | 0.079 | 0.092 | 95.6 (94.3,96.9) | |
| CC | 1.627 | 0.312 | -0.873 | 0.864 | 1.000 | 23.1 (20.5,25.7) | ||
| MI | 2.461 | 0.276 | -0.039 | 0.073 | 0.085 | 95.4 (94.1,96.7) | ||
| Interaction (True effect = 1.5) |
Full | 1.531 | 0.298 | 0.031 | 0.089 | 0.683 | 96.1 (94.9,97.3) | |
| CC | 1.600 | 0.342 | 0.100 | 0.130 | 1.000 | 95.2 (93.9,96.5) | ||
| MI | 1.493 | 0.297 | -0.007 | 0.081 | 0.627 | 96.9 (95.8,98.0) | ||
Results From Fitting Full, Complete-Case, and Multiple Imputation Models to 1000 Simulated Data Sets With a Sample Size of 1000 Where the Covariate of Interest and as a Result the Interaction Term Were Missing for Some Subjects and the Auxiliary Information Was Strong.
aCondition 1: X1 is 7.4 times more likely to be missing if X1 = 1, where X1 is binary
bCondition 2: Extreme values of X1 are more likely to be missing (probability of missing is a quadratic function of X1 or the log odds of missing X1 = γ 0+γ 1 X1 + γ 2 X12, where γ 1 = -1 and γ 2 = 2), where X1 is continuous
cCondition 3: A 1-unit increase in X1 corresponds to a 7.4 times decrease in the probability of missing for controls, but a 7.4 times increase for cases, where X1 is continuous