Table 2.
Percentage reduction in empirical SE and bias compared with CCA for MCAR and MAR results of the exposure coefficient in the simulation study
| % Missing | Imputation modela,b | % Reduction in SE compared to CCAc |
% Reduction in bias compared to CCAd |
|
|---|---|---|---|---|
| MCAR data | MAR data | MAR data | ||
| 1 | 1: R2 = 0.36 (No aux info) | 0.00% | −0.01% | 1.46% |
| 2: R2 = 0.40 | 0.16% | 0.24% | 1.91% | |
| 3: R2 = 0.52 | 0.24% | 0.11% | 79.03% | |
| 4: R2 = 0.76 | 0.55% | 0.41% | 79.54% | |
| 5: R2 = 0.92 | 0.52% | 0.58% | 81.42% | |
| 5 | 1: R2 = 0.36 (No aux info) | 0.02% | −0.03% | 0.16% |
| 2: R2 = 0.40 | 0.19% | 0.03% | −1.26% | |
| 3: R2 = 0.52 | 1.04% | 0.93% | 97.92% | |
| 4: R2 = 0.76 | 1.99% | 2.63% | 94.91% | |
| 5: R2 = 0.92 | 1.57% | 3.64% | 93.74% | |
| 10 | 1: R2 = 0.36 (No aux info) | −0.05% | −0.06% | 0.40% |
| 2: R2 = 0.40 | 0.37% | 0.75% | −0.35% | |
| 3: R2 = 0.52 | 0.58% | 1.12% | 97.38% | |
| 4: R2 = 0.76 | 2.59% | 4.61% | 96.73% | |
| 5: R2 = 0.92 | 2.89% | 6.76% | 96.41% | |
| 20 | 1: R2 = 0.36 (No aux info) | 0.03% | −0.05% | −0.19% |
| 2: R2 = 0.40 | 1.08% | 1.03% | −0.65% | |
| 3: R2 = 0.52 | 2.59% | 3.42% | 97.94% | |
| 4: R2 = 0.76 | 8.28% | 7.94% | 97.33% | |
| 5: R2 = 0.92 | 10.53% | 10.26% | 97.29% | |
| 40 | 1: R2 = 0.36 (No aux info) | 0.05% | −0.06% | −0.21% |
| 2: R2 = 0.40 | 2.00% | 1.25% | 0.10% | |
| 3: R2 = 0.52 | 5.37% | 5.06% | 97.84% | |
| 4: R2 = 0.76 | 15.56% | 14.11% | 98.56% | |
| 5: R2 = 0.92 | 21.10% | 22.86% | 98.64% | |
| 60 | 1: R2 = 0.36 (No aux info) | −0.04% | −0.02% | 0.21% |
| 2: R2 = 0.40 | 2.55% | 1.68% | 0.02% | |
| 3: R2 = 0.52 | 5.48% | 6.74% | 99.77% | |
| 4: R2 = 0.76 | 21.02% | 18.45% | 99.43% | |
| 5: R2 = 0.92 | 31.59% | 31.96% | 98.22% | |
| 80 | 1: R2 = 0.36 (No aux info) | −0.03% | −0.14% | 0.00% |
| 2: R2 = 0.40 | 2.16% | 1.57% | 1.34% | |
| 3: R2 = 0.52 | 8.18% | 9.86% | 96.47% | |
| 4: R2 = 0.76 | 27.56% | 28.21% | 99.62% | |
| 5: R2 = 0.92 | 45.88% | 44.66% | 98.77% | |
| 90 | 1: R2 = 0.36 (No aux info) | 0.03% | 0.11% | 0.04% |
| 2: R2 = 0.40 | 1.40% | 2.18% | 0.89% | |
| 3: R2 = 0.52 | 12.44% | 8.86% | 99.97% | |
| 4: R2 = 0.76 | 34.82% | 33.76% | 95.78% | |
| 5: R2 = 0.92 | 53.09% | 52.96% | 98.73% | |
Abbreviations: CCA, complete case analysis; MAR, Missing at random; MCAR, Missing completely at random; SE, Standard error.
R2 refers to the squared coefficient of multiple correlation which is used as a measure of auxiliary information.
Models 1 and 2 do not include all variables in the missingness mechanism and so are biased (as expected) for the MAR data. Models 3–5 do include all variables in the missingness mechanism and so are unbiased (as expected).
Calculated using 100 × (seCCA–seMI)/seCCA, where seCCA and seMI are the empirical standard error of the CCA model and the MI model, respectively.
Calculated using 100 × (abs(biasCCA)-abs(biasMI))/abs(biasCCA), where abs(.) is a function giving the absolute value and biasCCA and biasMI are the bias of the CCA model and the MI model, respectively.