## 4 Results
### Algorithm Performance Analysis
#### 4.2.2 Confidence Intervals and Inferential Statistics
In Table 11, the minimum values from the thirty executions per set are displayed. It is observed that, in each column related to the set of data for the corresponding trajectories, the minimum value, marked in boldface, is obtained by the proposed R1B-NCH. According to this analysis for study case 1, it is observed that the R1B-NCH algorithm presents a high probability to find the best solution through thirty executions of the algorithm in the mechanism synthesis problem for lower limb rehabilitation. Study case 2:  
Table 12 shows the confidence intervals obtained by each algorithm for study case 2 using both trajectories. Figure 8a shows the graphical representation of the confidence interval, and Figure 8b shows the close-up view of the confidence intervals closer to \begin{table}
\begin{tabular}{l l l} \hline \hline  & \multicolumn{1}{c}{**Trajectory 1**} \\ Hypothesis & & Bonferroni \\ \hline MUMSA-PF vs. R1B-NCH & 1 \\ MUMSA-PF vs. R1B-PF & 1 \\ R1B-NCH vs. R1B-PF & 1 \\ \hline \multicolumn{3}{c}{**Trajectory 2**} \\ Hypothesis & Bonferroni \\ \hline R1B-FR vs. R1B-NCH & 1 \\ R1B-FR vs. R1B-PF & 0.59012 \\ R1B-NCH vs. R1B-PF & 0.90510 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: \(p\)-values for tests in the multiple comparisons among the three outstanding algorithms in study case 1 using both trajectories. All algorithms draw two times in both trajectories. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Trajectory 1**} \\
**Algorithm-CHT** & **Set 1** & **Set 2** & **Set 3** & **Set 4** \\ \hline R1B-NCH & **0.00219** & **0.00305** & **0.00216** & **0.00214** \\ R1B-PF & 0.00234 & 0.00848 & 0.00401 & 0.00215 \\ MUMSA-PF & 0.01340 & 0.01142 & 0.01166 & 0.00976 \\ \hline \hline \multicolumn{3}{c}{**Trajectory 2**} \\
**Algorithm-CHT** & **Set 1** & **Set 2** & **Set 3** & **Set 4** \\ \hline R1B-NCH & **0.00255** & **0.00257** & **0.00266** & **0.00263** \\ R1B-PF & 0.00257 & 0.00307 & 0.00341 & 0.00354 \\ R1B-FR & 0.00270 & 0.00260 & 0.00271 & 0.00353 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Minimum value of the objective function through different sets of thirty algorithms’ executions in study case 1 in both trajectories. zero. According to the results for both trajectories, R1B-FR, R1B-eC, and R1B-NCH have confidence intervals closer to zero, where R1B-NCH presents the confidence interval lower limit value closest to zero and its upper limit does not exceed the R1B-eC confidence interval upper limit. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**Trajectory 1**} \\ \hline
**Limits** & **R1B-FR** & **R1B-SR** & **R1B-eC** & **R1B-PF** & **R1B-NC** \\ Low & 0.6830 & - & 0.6723 & 0.6780 & 0.6627 \\ Up & 0.7262 & - & 0.7049 & 0.9141 & 0.7019 \\ \hline  & **GA-FR** & **GA-SR** & **GA-eC** & **GA-PF** & \\ Low & 6.7614 & 2.4716 & 4.2015 & 2.5364 & \\ Up & 32.6767 & 3.7737 & 6.1055 & 2.8906 & \\ \hline  & **PSO-FR** & **PSO-SR** & **PSO-eC** & **PSO-PF** & \\ Low & - & 1.5653 & 1.9376 & 2.4854 & \\ Up & - & 5.2484 & 2.7343 & 3.3350 & \\ \hline  & **MUMSA-FR** & **MUMSA-SR** & **MUMSA-eC** & **MUMSA-PF** & \\ Low & 1.3340 & 1.2845 & 1.1647 & 2.0357 & \\ Up & 1.9489 & 2.2735 & 2.4295 & 3.7658 & \\ \hline  & \multicolumn{5}{c}{**Trajectory 2**} \\ \hline
**Limits** & **R1B-FR** & **R1B-SR** & **R1B-eC** & **R1B-PF** & **R1B-NC** \\ Low & 0.6683 & - & 0.6618 & 0.6984 & 0.6442 \\ Up & 0.7339 & - & 0.7101 & 0.7803 & 0.6735 \\ \hline  & **GA-FR** & **GA-SR** & **GA-eC** & **GA-PF** & \\ Low & 3.9029 & 2.9697 & 4.4512 & 2.5120 & \\ Up & 8.8576 & 4.5703 & 6.2872 & 3.1597 & \\ \hline  & **PSO-FR** & **PSO-SR** & **PSO-eC** & **PSO-PF** & \\ Low & - & 1.4245 & 2.2053 & 2.4316 & \\ Up & - & 10.0193 & 2.9865 & 3.5342 & \\ \hline  & **MUMSA-FR** & **MUMSA-SR** & **MUMSA-eC** & **MUMSA-PF** & \\ Low & 1.0532 & 1.2205 & 1.1600 & 1.3086 & \\ Up & 2.1492 & 2.1417 & 2.3506 & 5.0576 & \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Confidence interval values of algorithms for the study case 2 in both trajectories.