## 5 Comparative Experimental Study
### Statistical Analysis of the Overall Performance  
In this work at each study case, thirty executions of the four CHTs are carried out per each of the ten metaheuristics. Each study case groups a total of forty samples (four CHTs per ten metaheuristics). Each sample contains the best thirty objective functions of an algorithm. Descriptive statistics analyzes those samples. The descriptive statistical results are shown in Tables 11-11 of Appendix C, and these consider the following measures: the mean, the standard deviation (std), the median, and the maximum (max) and minimum (min) values of the samples. The graphical visualization of the distribution of the samples is shown in the box plot of Figure 5.  
Based on the results presented in such tables, it is possible to define the CHT that delivers the best performance in search of the most promising solution for each study case. Thus, the best behavior of a metaheuristic algorithm is related to the ability to search for the best solutions and provide more reliable results in different runs. The mean, the standard deviation (std), the median, and the maximum (max) and minimum (min) values of the samples are chosen as the statistical measure to indicate the algorithm capacity to find suitable solutions through runs in the particular samples. The best CHT between two CHTs, called \(A\) and \(B\), included in the algorithms, is obtained by the following comparison procedure:  
* The CHT \(A\) can be compared to the CHT \(B\) if their CHTs are different, and the algorithms that implement such CHTs are the same. On the contrary, they cannot be compared. * The statistical measure from the thirty executions of the CHT \(A\) is better than the CHT \(B\) for the same algorithm if the former presents less value than the latter. When this happens, the CHT \(A\) obtains the point (a win) in favor. Thus, the maximum number of points for comparing each CHT is 150 (three CHT comparisons per five measures per ten algorithms). * The CHT with the highest number of points is the best constraint-handling technique. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & \multicolumn{3}{c}{**Study Case**} \\  & **1** & **2** & **3** \\ \hline NP & 200 & 120 & 120 \\ MNOFE & 5 \(\times\) 10\({}^{6}\) & 2.4 \(\times\) 10\({}^{6}\) & 24 \(\times\) 10\({}^{6}\) \\ \(\varpi_{k}\) & 10,000 & 10,000 & 10,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The Maximum Number of Objective Function Evaluation (MNFE), the Population Number (NP), and the penalty factor chosen for each study case. Figure 5: Box plot of the sample distribution obtained by the overall performance of the CHTs in ten metaheuristics at each study case of mechanism synthesis for lower limb rehabilitation. In the comparison method, the lower values of the statistical measures (the mean, the standard deviation (std), the median, and the maximum (max) and minimum (min) values of the samples) are preferable in each algorithm. All measures are considered in the comparison procedure. The results obtained by the comparison process are presented in Table 7. The best performances of metaheuristic algorithms include the FR for the solution of the three study cases. For study case 1, it wins 115, for study case 2, it wins 104, and for study case 3, it wins 105. The next CHT alternative is the PF because it obtains the second-best performance in two study cases (study cases 1 and 2). The inferential statistical analysis was performed to make general conclusions about the performance of the CHTs incorporated in the metaheuristic algorithms for the solution of the lower limb rehabilitation mechanism synthesis. Nonparametric statistical tests [61] were used because the independence, normality, and homoscedasticity assumptions in the samples were not fulfilled due to the stochastic nature of the algorithms. The inferential statistical analysis consists of performing a multi-comparative post-hoc analysis with Holm post-hoc error correction method [61] in all CHTs incorporated in the algorithms.