# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
### Complete Efficiency Rankings of Vovodeships  
The expected distances (\(Ed\)) and ranks (\(ER\)) allow for the construction of a complete ranking of vovodeships. In this section, we compare such orders with the ones obtained with the most commonly used ranking procedures for DEA, i.e., Cross-efficiency (CE) [30] and Super-efficiency (SE) [31]. We adapted them to a value-based additive efficiency model and ran it on each category of indicators. The rankings generated by all four procedures for the level of inhabitants' health improvement are provided in Table 12. To quantify the agreement level of such rankings for all hierarchy nodes, we used Kendall's \(\tau\) coefficient [28; 32]. Its values are shown in Table 13. Note that \(-1\) means the rankings are inverse, whereas \(1\) denotes a pair of the same rankings. All four procedures provide highly correlated rankings. The two methods proposed in this paper (\(Ed\) and \(ER\)) offer the most similar orders of provinces. The rankings constructed with these two procedures are the same for the comprehensive analysis of the healthcare systems and the customer satisfaction category. For the remaining two categories, Kendall's \(\tau\) coefficient equals \(0.97\). The similarity between the rankings based on \(Ed\), \(ER\), and CE is between \(0.73\) for the comprehensive analysis and \(0.97\) for health improvement and financial management  
Figure 16: Extreme and expected ranks for Polish vovodeships when considering consumer satisfaction level. perspectives. The average measure value is 0.904. When comparing the orders imposed by SE and the two robustness-based methods, Kendall's \(\tau\) is between 0.72 and 0.95 (0.92) for \(Ed\) (\(ER\)), with an average value of 0.879 (0.863). Given that the maximal possible value of Kendall's \(\tau\) is 1, the observed similarity levels are very high. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Comprehensive**} & \multicolumn{4}{c}{**Inhabitants’ Health**} & \multicolumn{4}{c}{**Effective Financial**} & \multicolumn{4}{c}{**Consumer Satisfaction**} \\  & \multicolumn{4}{c}{**Analysis**} & \multicolumn{4}{c}{**Improvement**} & \multicolumn{4}{c}{**Management**} & \\  & \multicolumn{1}{c}{\(Ed\)**} & \multicolumn{1}{c}{\(ER\)**} & \multicolumn{1}{c}{\(CE\)**} & \multicolumn{1}{c}{\(SE\)**} & \multicolumn{1}{c}{\(Ed\)**} & \multicolumn{1}{c}{\(ER\)**} & \multicolumn{1}{c}{\(CE\)**} & \multicolumn{1}{c}{\(SE\)**} & \multicolumn{1}{c}{\(Ed\)**} & \multicolumn{1}{c}{\(ER\)**} & \multicolumn{1}{c}{\(CE\)**} & \multicolumn{1}{c}{\(SE\)**} \\ \hline \(Ed\) & 1.00 & 1.00 & 0.73 & 0.72 & 1.00 & 0.97 & 0.97 & 0.95 & 1.00 & 0.97 & 0.97 & 0.95 & 1.00 & 1.00 & 0.95 & 0.90 \\ \(ER\) & 1.00 & 1.00 & 0.73 & 0.72 & 0.97 & 1.00 & 0.97 & 0.92 & 0.97 & 1.00 & 0.97 & 0.92 & 1.00 & 1.00 & 0.95 & 0.90 \\ \(CE\) & 0.73 & 0.73 & 1.00 & 0.85 & 0.97 & 0.97 & 1.00 & 0.95 & 0.97 & 0.97 & 1.00 & 0.92 & 0.95 & 0.95 & 1.00 & 0.92 \\ \(SE\) & 0.72 & 0.72 & 0.85 & 1.00 & 0.95 & 0.92 & 0.95 & 1.00 & 0.95 & 0.92 & 0.92 & 1.00 & 0.90 & 0.90 & 0.92 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Kendall’s \(\tau\) coefficient for ranking procedures at each hierarchy level. Figure 17: The Hasse diagram of the necessary efficiency preference relation when considering consumer satisfaction level.