## 5 Results and Discussion
### Rank Correlations between the Aggregated Rankings  
This subsection provides a comparative analysis of the aggregated rankings obtained using different methods. The analysis examined the effects of the application of different methods (TH, AL, and RN) and different weightings for international matches (\(\mathrm{W}=1\) or \(\mathrm{W}=4\)). In the evaluation, the similarity between the rankings was measured by calculating the Kendall's tau rank correlation. Table 5 summarizes the correlation coefficients of the rankings that contained all teams from the five leagues. Table 5 shows that the correlations were higher between the rankings of the same method, even if the weights differed, than between the rankings generated by different methods using the same weights. By applying the same method and different weights, as previously mentioned, we observed hardly any differences in the case of the AL method (the correlation was 0.987). In the case of the TH method, this measurement was 0.884, while in the case of RN, it equaled 0.880. Therefore, the method itself had a stronger effect on the rankings than the choice of weights. The high values of the rank correlation coefficients could be easily understood by comparing the rankings produced using the same method but different weights (Tables 3 and 4). For example, when we compared the rankings generated by the TH method using different weights, we saw that 8 of the top 10 teams were the same. This observation was true for the weakest 10 teams, too. Concerning the AL and RN methods, the number of equally placed teams was at least 9 among both the best and worst 10 teams. Based on our observations, we could say that modifying the weights (\(\mathrm{W}\)) caused only local changes in the rankings. Comparing the different methods, the largest difference from the other methods was in the case of the RN method: The correlation coefficients with TH were 0.716 and 0.641, and those with AL were 0.638 and 0.637, while in the case of TH and AL, the correlation coefficients were 0.763 and 0.814. Therefore, the TH method resembled the AL method more than the RN method, and the same could be stated for the AL method. Finally, we noted that the TM ranking had moderate correlations with each ranking, but the lowest correlation coefficients (0.592 and 0.552) were seen again with the RN method. Table 6 contains the correlation coefficients between the different rankings of English teams. Additionally, the last two columns of the table contain the Transfermark ranking \begin{table}
\begin{tabular}{l l c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**International W = 1**} & \multicolumn{4}{c}{**International W = 4**} \\ \cline{3-8}  & & **TH** & **AL** & **RN** & **TH** & **AL** & **RN** & **TM** \\ \hline \multirow{3}{*}{Int. W = 1} & TH & 1 & 0.763 & 0.716 & 0.884 & 0.772 & 0.685 & 0.622 \\  & AL & 0.763 & 1 & 0.638 & 0.805 & 0.987 & 0.640 & 0.631 \\  & RN & 0.716 & 0.638 & 1 & 0.637 & 0.628 & 0.880 & 0.592 \\  & TH & 0.884 & 0.805 & 0.637 & 1 & 0.814 & 0.641 & 0.621 \\  & AL & 0.772 & 0.987 & 0.628 & 0.814 & 1 & 0.637 & 0.633 \\  & RN & 0.685 & 0.640 & 0.880 & 0.641 & 0.637 & 1 & 0.552 \\  & TM & 0.622 & 0.631 & 0.592 & 0.621 & 0.633 & 0.552 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Rank correlations between all teams’ aggregated rankings computed by different methods and applying different weights for international matches. and the official ranking. Concerning the correlation coefficients of the TM ranking, we observed values that were moderate (around 0.7) but still higher than in the case of the TM ranking for all teams (Table 5). Concerning the official ranking, we realized that the national rankings had larger correlations with the official rankings than the international ones. The larger the weight of the international matches, the lower the correlation with the official ranking, which was an easily explicable phenomenon.