stat.MLJul 21, 2025
SaveMissing value imputation with adversarial random forests - MissARF
Organizations: Leibniz Institute for Prevention Research & Epidemiology – BIPS, Germany · Faculty of Mathematics and Computer Science, University of Bremen, Germany · Department of Informatics, King’s College London, United Kingdom · Department of Public Health, University of Copenhagen, Denmark
Abstract
Handling missing values is a common challenge in biostatistical analyses, typically addressed by imputation methods. We propose a novel, fast, and easy-to-use imputation method called missing value imputation with adversarial random forests (MissARF), based on generative machine learning, that provides both single and multiple imputation. MissARF employs adversarial random forest (ARF) for density estimation and data synthesis. To impute a missing value of an observation, we condition on the non-missing values and sample from the estimated conditional distribution generated by ARF. Our experiments demonstrate that MissARF performs comparably to state-of-the-art single and multiple imputation methods in terms of imputation quality and fast runtime with no additional costs for multiple imputation.
Figures & tables
Figure 1: Single imputation example with MissARF of the point on one exemplary tree of a converged ARF of a two-dimensional dataset consisting of clusters around and . Real data points are labeled with (blue) and synthetic data points are labeled with (orange). If the tree predicts real observations, it is colored in blue and orange otherwise. a) Shows the reduced example dataset in a tabular form. b) The partition plot of the exemplary tree of ARF in c), where the imputation example is visualized.
Figure 2: Common patterns in simulation results of setting I over different missingness patterns and dimensionality ( ) for missingness rates (mis.) . Common pattern for the NRMSE are exemplified for the normal distribution with a linear effect and a) and b) and the binary distribution with a squared effect for c) and d) . The Brier score pattern is exemplified for e) normal distribution with a linear effect and and f) the binary distribution with a squared effect and . For single imputation, MissARF (blue) and MissForest (orange) perform best in NRMSE (a-d) and Brier score (e). The boxplots are plotted over the replicates.
Figure 3: Common patterns in simulation results of setting II over different missingness patterns and rates (mis.) and dimensionality . The coverage rate of the normal distribution is shown for a linear effect with a) and d) and a squared effect for e) and f) . The red vertical line shows the nominal coverage level of 0.95. Random imputation and MissForest (gray) often show poor coverage. MICE PMM (green) works best for linear effects, whereas MissARF (blue) struggles with high missingness (d), but has the smallest average CI width (b) and RMSE (c), where MICE PMM has the largest. MissARF works best for squared effects, where MICE PMM struggles. The boxplots are plotted over the features.
| Single imputation | Multiple imputation | ||||||
| Method | Data NRMSE | Brier score | Runtime (s) | Coverage rate % | CI width | Coef. RMSE | Runtime (s) |
| [mean (SD)] | [mean (SD)] | [mean] | [median (IQR)] | [median (IQR)] | [median (IQR)] | [mean] | |
| Random Imp. | 1.47 (0.11) | 0.20 (0.05) | <0.1 | 92.0 (46.8) | 0.30 (0.47) | 0.13 (0.11) | <0.1 |
| Median Imp. | 1.21 (0.29) | 0.19 (0.06) | <0.1 | - | - | - | - |
| MissForest | 0.98 (0.19) | 0.18 (0.06) | 16.1 | 90.9 (13.1) | 0.32 (0.47) | 0.12 (0.14) | 322.7 |
| MissForest PMM | 1.26 (0.14) | 0.19 (0.06) | 16.5 | 95.3 (6.0) | 0.32 (0.48) | 0.10 (0.11) | 325.2 |
Table 1: Aggregated results over all simulation settings for both single and multiple imputation. Data NRMSE and Brier score refer to performance in the single imputation setting, coverage rate, confidence interval (CI) width and coef. RMSE to performance in multiple imputation.
Figure 4: Real data example results over the different missingness proportions and patterns. a) MissForest (orange) and MissARF (blue) perform best in terms of NRMSE. b) The methods perform similarly regarding the Brier Score, with random and median imputation performing slightly worse. The boxplots are plotted over the features. c) MissARF, MICE PMM (green) and the other methods, except MissForest and random imputation, perform similarly well. The red vertical line shows the nominal coverage level of 0.95. The boxplots are plotted over the replicates.
Appendix figures & tables112 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: NRMSE of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 6: NRMSE of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 7: NRMSE of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 8: NRMSE of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 9: NRMSE of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 10: NRMSE of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 11: NRMSE of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 12: NRMSE of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 13: NRMSE of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 14: NRMSE of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 15: NRMSE of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 16: NRMSE of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 17: NRMSE of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 18: NRMSE of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 19: NRMSE of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 20: NRMSE of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 21: NRMSE of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 22: NRMSE of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 23: NRMSE of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 24: NRMSE of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 25: Brier Score of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 26: Brier Score of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 27: Brier Score of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 28: Brier Score of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 29: Brier Score of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 30: Brier Score of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 31: Brier Score of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 32: Brier Score of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 33: Brier Score of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 34: Brier Score of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 35: Brier Score of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 36: Brier Score of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 37: Brier Score of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 38: Brier Score of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 39: Brier Score of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 40: Brier Score of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 41: Brier Score of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 42: Brier Score of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 43: Brier Score of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 44: Brier Score of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the replicates, with MissARF (blue) and MissForest (orange) highlighted.
Figure 45: Coverage rate of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 46: Coverage rate of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 47: Coverage rate of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 48: Coverage rate of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 49: Coverage rate of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 50: Coverage rate of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 51: Coverage rate of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 52: Coverage rate of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 53: Coverage rate of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 54: Coverage rate of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 55: Coverage rate of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 56: Coverage rate of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 57: Coverage rate of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 58: Coverage rate of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 59: Coverage rate of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 60: Coverage rate of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 61: Coverage rate of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 62: Coverage rate of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 63: Coverage rate of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 64: Coverage rate of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The red vertical line shows the nominal coverage level of 0.95. Boxplots are plotted over features, with MissARF (blue) and MICE PMM (green).
Figure 65: Average width of the confidence intervals of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 66: Average width of the confidence intervals of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 67: Average width of the confidence intervals of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 68: Average width of the confidence intervals of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 69: Average width of the confidence intervals of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 70: Average width of the confidence intervals of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 71: Average width of the confidence intervals of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 72: Average width of the confidence intervals of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 73: Average width of the confidence intervals of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 74: Average width of the confidence intervals of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 75: Average width of the confidence intervals of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 76: Average width of the confidence intervals of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 77: Average width of the confidence intervals of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 78: Average width of the confidence intervals of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 79: Average width of the confidence intervals of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 80: Average width of the confidence intervals of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 81: Average width of the confidence intervals of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 82: Average width of the confidence intervals of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 83: Average width of the confidence intervals of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 84: Average width of the confidence intervals of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 85: RMSE of the regression coefficients of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 86: RMSE of the regression coefficients of the normal distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 87: RMSE of the regression coefficients of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 88: RMSE of the regression coefficients of the uniform distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 89: RMSE of the regression coefficients of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 90: RMSE of the regression coefficients of the gamma distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 91: RMSE of the regression coefficients of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 92: RMSE of the regression coefficients of the Poisson distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 93: RMSE of the regression coefficients of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 94: RMSE of the regression coefficients of the binary distribution setting with a linear effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 95: RMSE of the regression coefficients of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 96: RMSE of the regression coefficients of the binary distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 97: RMSE of the regression coefficients of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 98: RMSE of the regression coefficients of the normal distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 99: RMSE of the regression coefficients of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 100: RMSE of the regression coefficients of the uniform distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 101: RMSE of the regression coefficients of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 102: RMSE of the regression coefficients of the gamma distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 103: RMSE of the regression coefficients of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with . The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 104: RMSE of the regression coefficients of the Poisson distribution setting with a squared effect over different missingness patterns, dimensionality ( ) and missingness rates (mis.) with (left) and (right). The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 105: Real data example results over the different missingness proportions and patterns for a) Brier score, b) average width of the confidence intervals and c) the RMSE of the regression coefficients. The boxplots are plotted over the replicates in a) and over the features in b)-c) with MissARF (blue), MissForest (orange) and MICE PMM (green) highlighted.
| Method | ||||||
|---|---|---|---|---|---|---|
| Single | Multiple | Single | Multiple | Single | Multiple | |
| Random Imp. | 0.01 | 0.02 | 0.01 | 0.02 | 0.01 | 0.07 |
| Median Imp. | 0.01 | - | 0.01 | - | 0.01 | - |
| MissForest | 0.92 | 19.59 | 2.17 | 41.59 | 45.11 | 906.93 |
| MissForest PMM | 0.98 | 19.53 | 2.20 | 42.13 | 46.41 | 913.82 |
| MICE PMM | 0.09 | 1.50 | 0.11 | 1.86 | 0.48 | 8.46 |
Table 2: Runtime comparison of the different methods, executed with one thread. The measured average times in seconds for single and multiple imputation for each of the different are compared.
| Method | ||||||
|---|---|---|---|---|---|---|
| Single | Multiple | Single | Multiple | Single | Multiple | |
| Random Imp. | 0.01 | 0.01 | 0.01 | 0.02 | 0.01 | 0.06 |
| Median Imp. | 0.01 | - | 0.01 | - | 0.01 | - |
| MissForest | 0.27 | 5.59 | 0.40 | 7.70 | 4.86 | 98.33 |
| MissForest PMM | 0.28 | 5.58 | 0.41 | 7.81 | 5.01 | 99.43 |
| MICE PMM | 0.09 | 1.57 | 0.12 | 1.95 | 0.50 | 8.98 |
Table 3: Runtime comparison of the different methods, executed with 16 threads. The measured average times in seconds for single and multiple imputation for each of the different are compared.
Figure 106: Runtime comparison with one thread of the different imputation methods across the dimensionality ( ) and missingness proportions (mis.) with different . The boxplots for single imputation (left) and multiple imputation (right) are plotted over the 10 replicates. The runtime is shown in seconds (sec).
Figure 107: Runtime comparison with 16 threads of the different imputation methods across the dimensionality ( ) and missingness proportions (mis.) with different . The boxplots for single imputation (left) and multiple imputation (right) are plotted over the 10 replicates. The runtime is shown in seconds (sec).
Figure 108: Mean values with the standard error for a) coverage rate, b) average confidence interval width and c) RMSE for minimum node sizes of 2-400 (100 points from a logarithmic scale) and 100 number of trees with MissARF for a normal distribution with a linear effect with , , and missingness with a MAR pattern over 1000 replications.
Figure 109: Mean values with the standard error for a) coverage rate, b) average confidence interval width and c) RMSE for minimum node sizes 10 (arf_10), 20 (arf_20) and 30 (arf_30) and 10-160 number of trees with MissARF for a normal distribution with a linear effect with , , and missingness with a MAR pattern over 1000 replications.
Figure 110: Mean values with the standard error for the NRMSE for minimum node sizes 10 (arf_10), 20 (arf_20) and 30 (arf_30) and 10-160 number of trees with MissARF for a normal distribution with a linear effect with , , and missingness with a MAR pattern over 1000 replications.
Figure 111: The effect of the number of multiple imputations (left) and (right) for a-b) coverage rate, c-d) average confidence interval width and e-f) RMSE for a normal distribution with a linear effect with , , and missingness with a MAR pattern. The boxplots are plotted over the features, with MissARF (blue) and MICE PMM (green) highlighted.
Figure 112: Mean values with the standard error for the NRMSE for sample sizes 50-500 for the different imputation methods. Results are for a normal distribution with a linear effect with and missingness with a MAR pattern over 1000 replications.
Figure 113: Mean values with the standard error for the Brier Score for sample sizes 50-500 for the different imputation methods. Results are for a normal distribution with a linear effect with and missingness with a MAR pattern over 1000 replications.
Figure 114: Mean values with the standard error for a) coverage rate, b) average confidence interval width and c) RMSE for sample sizes 50-500 for the different imputation methods. Results are for a normal distribution with a linear effect with and missingness with a MAR pattern over 1000 replications.
More in this topic
Incomplete Data Imputation
Incomplete data imputation estimates unobserved values while accounting for data structure, missingness patterns, and uncertainty.
See all 74 papers