# Techniques for Robust Imputation in Incomplete Two-Way Tables
## 3 Results and Discussion
In the latter case, if the imputation and contamination percentages were greater than 20 and 10% respectively, it is suggested to test both algorithms. From a computational point of view, in large matrices (100 \(\times\) 8) M5 GOKimputation presents greater speed if compared to M5 RobSVD. All the proposals in this article are based on a robust SVD, for that reason in the Appendix A function of R is presented to calculate the rSVD84. Finally, it is worth answering one more question that may arise for the applied researcher when analyzing contaminated and unbalanced data: what is the maximum percentage of contamination that can be tolerated if an effective imputation of missing values is to be made? There is a lot of literature on the two problems and very useful \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Percentage of Missing Values and Outliers**} & \multicolumn{6}{c}{**Imputation Method**} \\ \cline{3-8}  & & **SVD88** & **M5 RobSVD** & **rSVD84** & \
\begin{tabular}{c} **GOK** \\ **Imputation** \\ \end{tabular} &
\begin{tabular}{c} **M5 GOK** \\ **Imputation** \\ \end{tabular} \\ \hline
**Matrix** & **\%miss** & **\%out** & **P\({}_{\mathbf{e}}\)** & **P\({}_{\mathbf{e}}\)** & **P\({}_{\mathbf{e}}\)** & **P\({}_{\mathbf{e}}\)** & **P\({}_{\mathbf{e}}\)** \\ \hline
1 & 10 & 0 & 0.9721 & 4.4655 & 4.5945 & 4.5903 & 4.4889 \\
2 & 20 & 0 & 0.7570 & 3.9584 & 4.4752 & 4.4670 & 4.0461 \\
3 & 10 & 5 & 17.5018 & 3.2894 & 3.8948 & 3.8883 & 3.3602 \\
4 & 10 & 10 & 36.1932 & 3.2856 & 3.6275 & 3.6235 & 3.4482 \\
5 & 10 & 15 & 46.3526 & 4.3479 & 4.4465 & 4.4396 & 4.3283 \\
6 & 20 & 5 & 24.0821 & 4.5323 & 5.1784 & 5.1639 & 4.5994 \\
7 & 20 & 10 & 36.8853 & 4.0614 & 4.7675 & 4.7563 & 4.1501 \\
8 & 20 & 15 & 59.2625 & 4.6295 & 4.8285 & 4.8033 & 4.4290 \\ \hline  & Mean & & 27.7508 & **4.0713** & 4.4766 & 4.4665 & 4.1063 \\  & Median & & 30.1377 & **4.2047** & 4.5349 & 4.5286 & 4.2392 \\  & Standard deviation & & 20.9011 & 0.5342 & 0.5036 & 0.4993 & 0.4688 \\  & Interquartile distance & & 25.8827 & 0.6911 & 0.4742 & 0.4663 & 0.5473 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of numerical example 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **GF\({}_{2}\)** & **GF\({}_{1}\)** & **P\({}_{\mathbf{e}}\)** \\ \hline SVD88 & 0.3202 & \(-2.0001\) & 7.4491 \\ M5 RobSVD & 0.9836 & 0.9812 & 0.5891 \\ rSVD84 & 0.9843 & 0.9821 & 0.5756 \\ GOKimputation & 0.9842 & 0.9819 & 0.5788 \\ M5 GOKimputation & 0.9833 & 0.9808 & 0.5958 \\ missForest & 0.8730 & 0.3719 & 3.4083 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of numerical example 2.  
references can be found in the recent study by [32]. However, we suggest following a rationale derived from a simulation study described by [33] and from the results obtained by [4]. In a real situation, algorithms based on SVD can work with up to 60% of missing values [4], but if there is a high proportion of outliers in the remaining data, imputations will necessarily be affected by them and will produce low quality of results [33]. Taking this into account, we recommend only considering imputation on incomplete matrices that still have at least half of the observed data without outliers. For example, one rule of thumb might be to add the percentages of missing data and outliers, and only proceed if that sum is not greater than 40%. This will ensure that at least 60% of \"clean\" data can be effectively used for the purpose of completing the matrix. With less than 60% of \"clean\" data any imputation results should be treated with caution.