## 5 Results and Discussion
### Comparison of the Aggregated Rankings and the UEFA Club Coefficient Rankings  
In this subsection, we compare the evaluation results generated by the TH, AL, and RN methods to the UEFA club coefficient rankings. These rankings and ratings are computed every season for the teams playing in international cups. Teams not included in international cups do not receive such coefficients for that season. The method of the calculation is a point-based algorithm, which is detailed on the website [54]. The five-year club coefficient is the sum of the annual club coefficient in the year of interest and the previous four years. Financial support and the granting of spots in international cups are calculated based on these club coefficients. In the 2020/21 season, 34 teams from the investigated leagues played matches in the international cups. The list of these teams and their UEFA club coefficients for the 2020/21 season are contained in Table 7.  
If we consider Table 4, we can observe that Manchester City and Chelsea were first and second, respectively, in the TH and AL rankings, as is also shown in Table 7; moreover, Real Madrid, Bayern Munich, Manchester United, and Liverpool were among the best teams according to the TH, AL, and UEFA club coefficient models. Surprisingly, Villareal was placed third by the UEFA coefficients, but it was not among the best 20 teams as evaluated by the TH, AL, and RN methods. The reason was that it was the champion of the UEL, and it achieved many wins; therefore, it gained many points. However, these victories were against \"weak\" teams. Villareal hardly played any international matches against the strongest teams. In ESP, it played two matches against Real Madrid, was defeated once, and had one tie. Although Real Madrid was among the best teams according to all evaluations and was the winner of ESP, the position of Villareal was only fifth in ESP. That was why it reached the group stage of the UEL. Therefore, the position of Villareal seemed to have been overestimated in the UEFA club coefficient rankings. Unfortunately, point-based methods do not take into consideration the strength of the opponent, which may cause bias. The inconsistency of aggregation based on point-based methods is detailed in [55]. \begin{table}
\begin{tabular}{l l r r r r r r r r r r r} \hline  & & \multicolumn{4}{c}{**ENG National**} & \multicolumn{4}{c}{**International W = 1**} & \multicolumn{4}{c}{**International W = 4**} \\ \cline{3-13}  & & **TH** & **AL** & **RN** & **TH** & **AL** & **RN** & **TH** & **AL** & **RN** & **TM** & **OR** \\ \hline \multirow{3}{*}{ENG Nat.} & TH & 1 & 0.979 & 0.979 & 0.895 & 0.958 & 0.968 & 0.8 & 0.958 & 0.916 & 0.716 & 0.979 \\  & AL & 0.979 & 1 & 0.979 & 0.895 & 0.958 & 0.968 & 0.8 & 0.958 & 0.916 & 0.716 & 0.958 \\  & RN & 0.979 & 0.979 & 1 & 0.874 & 0.958 & 0.968 & 0.8 & 0.937 & 0.916 & 0.716 & 0.958 \\  & TH & 0.895 & 0.895 & 0.874 & 1 & 0.916 & 0.905 & 0.905 & 0.937 & 0.895 & 0.737 & 0.874 \\  & AL & 0.958 & 0.958 & 0.958 & 0.916 & 1 & 0.989 & 0.821 & 0.979 & 0.958 & 0.695 & 0.937 \\  & RN & 0.968 & 0.968 & 0.968 & 0.905 & 0.989 & 1 & 0.811 & 0.968 & 0.947 & 0.705 & 0.947 \\  & TH & 0.800 & 0.800 & 0.800 & 0.905 & 0.821 & 0.811 & 1 & 0.842 & 0.821 & 0.705 & 0.800 \\  & AL & 0.958 & 0.958 & 0.937 & 0.937 & 0.979 & 0.968 & 0.842 & 1 & 0.937 & 0.716 & 0.937 \\  & RN & 0.916 & 0.916 & 0.916 & 0.895 & 0.958 & 0.947 & 0.821 & 0.937 & 1 & 0.653 & 0.937 \\  & TM & 0.716 & 0.716 & 0.716 & 0.737 & 0.695 & 0.705 & 0.705 & 0.716 & 0.653 & 1 & 0.695 \\  & OR & 0.979 & 0.958 & 0.958 & 0.874 & 0.937 & 0.947 & 0.8 & 0.937 & 0.695 & 1 \\ \hline \end{tabular}
\end{table} 
Table 6: Kendall’s tau rank correlations between the different rankings of English teams. As the calculation of the UEFA club coefficients considered only the international matches and not the national ones, its ranking significantly differed from the rankings of the TH, AL, and RN methods.