## 5 Results and Discussion
### Comparison of the Aggregated Rankings and the UEFA Club Coefficient Rankings
The Kendall's tau rank correlations between the UEFA coefficient rankings and the TH, AL, RN, and TM rankings applying W = 1, W = 4, and W = 16 weights for international matches are included in Table 8.  
First, we can see that the correlation between the UEFA coefficient rankings and TM was 0.54, which was a moderate correlation. The correlations were moderate in the case of TH, too, with an increasing tendency for W. Certainly, the higher the value of W, the more significant the results of the international matches. In the case of AL, the correlations with the UEFA coefficient rankings were almost the same. The reason was that the elements of the AHP matrix (defined by Equation (8)) were not sensitive to the number of matches. The smallest correlations were between the results of RankNet and the UEFA club coefficient rankings. These correlations were weak. We could establish that the correlations between TM and TH, AL, and RN were stronger than those between the UEFA coefficient rankings and TH, AL, and RN; they were mostly between 0.6 and 0.7. The smallest values appeared again between TM and RN. Summarizing the observations, we could conclude that the best teams identified by TH and AL were the same as those identified by the UEFA club coefficient rankings. The only difference in the top teams (Villareal) could be clearly explained. It seemed to be the consequence of the point-based aggregation of the UCL and UEL, which is inappropriate in some cases. While TH and AL are evaluation methods that take into consideration the strength of the opponent, the UEFA club coefficient rankings do not. On the other hand, the TH and AL methods were based on a much larger set of information than that used for calculating the UEFA coefficient rankings. Not only international matches but also the results of the matches in the national leagues were taken into account when calculating the TH and AL rankings. The results of the national championships provided a more detailed picture of \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Rank** & **Name** & **Score** & **Rank** & **Name** & **Score** \\ \hline
1 & Manchester City & 35 & 18 & Atl. Madrid & 16 \\
2 & Chelsea & 33 & 19 & Tottenham & 15 \\
3 & Villarreal & 30 & 20 & B. Monchengladbach & 15 \\
4 & Bayerm Munich & 27 & 21 & Granada CF & 13 \\
5 & Real Madrid & 26 & 22 & AC Milan & 12 \\
6 & Manchester Utd & 26 & 23 & Hoffenheim & 12 \\
7 & Paris SG & 24 & 24 & Napoli & 10 \\
8 & Liverpool & 24 & 25 & Bayer Leverkusen & 10 \\
9 & AS Roma & 24 & 26 & Leicester & 10 \\
10 & Arsenal & 23 & 27 & Inter & 9 \\
11 & Dortmund & 22 & 28 & Real Sociedad & 8 \\
12 & Juvrentus & 21 & 29 & Lille & 8 \\
13 & Barcelona & 20 & 30 & Marseille & 6 \\
14 & Sevilla & 19 & 31 & Rennes & 5 \\
15 & RB Leipzig & 17 & 32 & Nice & 3 \\
16 & Atlanta & 17 & 33 & Wolfsburg & 2.5 \\
17 & Lazio & 17 & 34 & Reims & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: UEFA club coefficient rankings. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline  & \multicolumn{3}{c}{**International W = 1**} & \multicolumn{3}{c}{**International W = 4**} & \multicolumn{3}{c}{**International W = 16**} & \\ \cline{2-10}  & **TH** & **AL** & **RN** & **TH** & **AL** & **RN** & **TH** & **AL** & **RN** & **TM** \\ \hline UEFA & 0.405 & 0.474 & 0.321 & 0.434 & 0.471 & 0.307 & 0.522 & 0.471 & 0.339 & 0.540 \\ TM & 0.643 & 0.697 & 0.597 & 0.665 & 0.693 & 0.515 & 0.547 & 0.693 & 0.340 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: [PERSON]’s tau rank correlations between the different rankings. the performances of the teams playing in the international cups, and the evaluation results seemed to be more realistic.