# Techniques for Robust Imputation in Incomplete Two-Way Tables
## 3 Results and Discussion  
Table 2 shows the results of the first numerical example. When data were incomplete (10% and 20% of missing values), but without contamination, the SVD88 algorithm always obtained the smallest prediction errors. This behavior is as expected, since without outliers the procedures based on least squares generally perform well [12]. This situation changes completely when there is contamination of the data with 5, 10 and 15% of outliers. In these cases, it can be seen that as the number of outliers increases, the SVD88 prediction errors also increase. Thus, SVD88 is highly affected by contamination, producing low quality imputations. It can also be seen that in these contaminated scenarios the four proposed strategies of robust imputation always surpass SVD88 in terms of \(\text{P}_{\text{e}}\). It remains, then, to analyze in detail the performance of the robust imputations. In general, using the mean and median of the prediction errors, M5 RobSVD provides the best quality of imputations when there are outliers because it minimizes the values of such statistics. However, at the highest imputation and contamination percentages, (20% and 15%, respectively) the M5 GOKimputation strategy marginally outperforms M5 RobSVD. Moreover, the remaining two strategies, rSVD84 and GOKimputation, are also effective in the presence of outliers because they had lower \(\text{P}_{\text{e}}\) values than SVD88, but in no case were they able to improve on M5 RobSVD or M5 GOKimputation. These results show that the elimination of the singular values of the imputation equation within a robust procedure favors convergence and improves the quality of the imputations. Table 3 shows the results of the second numerical example based on the \"Ontario\" matrix. It is noteworthy that the four strategies proposed in this article outperformed both the SVD88 and the missForest method, obtaining \(\text{GF}_{1}\) and \(\text{GF}_{2}\) values close to 1 and minimizing \(\text{P}_{\text{e}}\). In this cross-validation exercise on real data, SVD88 presented the worst performance with very poor-quality imputations, which can be concluded from the negative value in the \(\text{GF}_{1}\) statistic. This situation can be explained by the lack of convergence in the imputation of some observations. On considering the GF\({}_{2}\) and GF\({}_{1}\) statistics, the four methods M5 RobSVD, rSVD84, GOKimputation M5 GOKimputation showed values of approximately 0.98 with very small differences only in the third and fourth decimal places. This indicates that the imputations are all high quality and the performance difference among these methods is very small. On the other hand, the prediction error was minimized using rSVD84, so taking into account the values of all three statistics then rSVD84 can be chosen in this case, because it is the fastest computationally robust method. The numerical results presented so far show that our four robust imputation alternatives (M5 RobSVD, rSVD84, GOKimputation and M5 GOKimputation) work very well and in the case of data contamination the SVD88 algorithm should be replaced by one of them. The reader may therefore be left with the question: which method to choose according to the results obtained in this study? answer this question, outliers must first be detected and for this reason the \"cellWise\" package from R ([13]; cellWise available from CRAN; [[https://cran.r-project.org/web/packages/cellWise/index.html](https://cran.r-project.org/web/packages/cellWise/index.html)]([https://cran.r-project.org/web/packages/cellWise/index.html](https://cran.r-project.org/web/packages/cellWise/index.html)), accessed on 25 June 2021) is recommended. Once outliers are detected [32], the dimension of the matrix under analysis must be taken into account. If the matrix is small (for example 18 \(\times\) 9) rSVD84 can be applied, but if the matrix is larger (for example 100 \(\times\) 8), the M5 RobSVD and M5 GOKimputation systems can be considered.