## 7 Statistical Test Analysis  
To locate the importance of the differences between the optimization outcomes arrived at by each method, we utilize the [PERSON] statistical test. This test is conducted to find out if there are perceivable differences between the overshoot results of the methods presented in this work [47]. If there are statistically notable differences between the optimization results, the Holm method [47] will be conducted as a posterior statistical procedure that is usually used to compare the best method of performance (i.e., the control method) with the rest of the methods. The level of confidence, pointed out as \(\alpha\), is 5% in all evaluated cases. In more detail, [PERSON]'s test requires the calculation of the average ranked value. Then, a comparison is used to assess the critical values obtained for the level of significance tested (\(\alpha=0.05\)) with [PERSON]'s test to check whether or not the null hypothesis is rejected. In such a context, if the level of significance (i.e., the \(p\)-value) divuyled by [PERSON]'s test is less than or equal to 0.05, the null hypothesis is rejected, denoting that there is a big difference between the performance scores of the algorithms. After that, further steps are needed to determine which algorithm's performance is crucially different from the best algorithm and which algorithms are similar. This is to check whether the outcomes arrived at by the best algorithm differ from the outcomes of the other methods in a statistically significant manner. For this intent, this study looks at a post-hoc statistical analysis method of [PERSON]'s test [48] to see which methods are worse or better than the best algorithm and at what level of significance. [PERSON]'s test is a widely applicable test method based upon the sequential rejection method. It orders all optimization algorithms on the basis of their \(p\)-values and compares them to \(\alpha/k-i\), where \(\alpha\) is the significance level, \(k\) represents the freedom degree and \(i\) denotes the algorithm number. The method starts with the most significant \(p\)-value and consecutively rejects the null hypothesis as long as \(p_{i}<\alpha/k-i\). Once the method is not able to reject the hypothesis, it stops and stops all the remaining hypotheses passable. In applying [PERSON]'s test, the best-performing algorithm obtains the lowest ranking, while the worst performing algorithm obtains the highest rank. The lowest ranked method is used as a control algorithm for post-hoc analysis. The average order of the optimization results arrived at by [PERSON]'s method as per the overshoot metric is tabulated in Table 4. The controlling results based the presented methods in this work are ranked as PSO in the first order, followed, in order, by CSA, GA and ZN. The \(p\)-value, computed using [PERSON]'s statistical test for the overshoot measure in Table 2, is 0.025557; the null hypothesis of the homologous performance level is unacceptable to assert if there are statistically significant differences between the performance of all evaluated methods. The [PERSON]'s method is then appealed as a post-test procedure to determine if there are statistically substantial differences between the best performing algorithm (that is, PSO) and the remaining algorithms (ZN, GA and CSA). [PERSON]'s method dismisses the hypotheses with \(p\)-values of less than 0.025. The results obtained based on the [PERSON]'s procedure are presented in Table 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Ranking** \\ \hline ZN & 4.0 \\ GA & 2.75 \\ CSA & 1.75 \\ PSO & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average order of the optimization methods based on [PERSON]’s test using the overshoot metric measure. The findings of [PERSON]'s method reveal that the optimization results obtained based on PSO in controlling the movement of the quadcopter are statistically better than the optimization results obtained based on the ZN-, GA- and CSA-based approaches.