## 4 Discussion
### The FOM Focused on the Difference in Area or Diameter  
It is an alternative suggestion for ultrasound diagnosis of carotid artery stenosis in patients to scan the stenosis diameter. In doing so, the same combination of factors as listed in Table 1 was adopted to measure the difference in the practical survey and theoretical preset in this study (i.e., 10, 8, and 4 mm). As clearly implied in the bottom part of Table 3, group C has an extremely high index of FOM, indicating outstanding performance in analyzing the diameter of stenosis. In contrast, group A (i.e., original group 12 in the L18 orthogonal array, cf. Table 2) does not perform better than the others. This is because the presumption of this study was preset to optimize the imaging quality of the area, not diameter; therefore, the demanded expectation cannot be arbitrarily switched in the domain of robust designation methodology [22; 23]. This is crucial in applying Taguchi analysis to a single expectation demand in reality. Moreover, the conventional setting of ultrasound scan facilities still cannot have the convincing capability to recognize the solid diameter.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \begin{tabular}{c} **Group A** \\ **(Original C12)** \\ \end{tabular} & \begin{tabular}{c} **Group B** \\ **(Conventional)** \\ \end{tabular} & \begin{tabular}{c} **Group C** \\ **(Highest FOM of** \\ **Each Factor)** \\ \end{tabular} & \
\begin{tabular}{c} **Group D** \\ **(Considering** \\ **Cross-Interaction I)** \\ \end{tabular} &
\begin{tabular}{c} **Group E** \\ **(Considering** \\ **Cross-Interaction II)** \\ \end{tabular} \\ \hline \hline \multicolumn{1}{c}{**Factor:**} & & & & & \\ \multicolumn{1}{c}{(A) angle of probe (degree)} & 0 & 0 & 0 & 30 \\ \multicolumn{1}{c}{(B) signal gain (dB/cm)} & 50 & 50 & 55 & 55 & 55 \\ \multicolumn{1}{c}{(C) resolution vs. speed} & S1 & RS & S1 & RS & RS \\ \multicolumn{1}{c}{(D) dynamic range (dB)} & 50 & 50 & 50 & 40 & 50 \\ \multicolumn{1}{c}{(E) XRES (scenario)} & 2 & 2 & 2 & 2 & 2 \\ \multicolumn{1}{c}{(F) zoom} & 0.8 & 1.0 & 0.8 & 0.8 & 0.8 \\ \multicolumn{1}{c}{(G) time gain compensation} & \(-1\) & 0 & \(-1\) & \(-1\) & \(-1\) \\ \multicolumn{1}{c}{**Area:**} & & & & & \\ \multicolumn{1}{c}{Difference in 78.5 mm\({}^{2}\)} & 1.50 & 8.5 & 3.50 & 1.50 & 3.50 \\ \multicolumn{1}{c}{Difference in 50.2 mm\({}^{2}\)} & 1.73 & 2.8 & 1.20 & 0.20 & 1.20 \\ \multicolumn{1}{c}{Difference in 12.6 mm\({}^{2}\)} & 1.07 & 0.6 & 6.40 & 4.40 & 5.40 \\ \multicolumn{1}{c}{Avg.} & 1.43 & 3.97 & 3.70 & 2.03 & 3.37 \\ \multicolumn{1}{c}{Sdev} & 1.18 & 4.08 & 2.61 & 2.15 & 2.10 \\ \multicolumn{1}{c}{\(\beta\)} & 0.86 & 12.07 & 6.00 & 5.64 & 4.21 \\ \multicolumn{1}{c}{\(r^{2}\)} & 0.60 & 0.82 & 0.50 & 0.64 & 0.38 \\ \multicolumn{1}{c}{**FOM**} & **0.413** & **0.004** & **0.012** & **0.031** & **0.013** \\ \multicolumn{1}{c}{**Diameter:**} & & & & & \\ \multicolumn{1}{c}{Difference in 10 mm\({}^{2}\)} & 0.50 & 0.70 & 0.003 & 0.30 & 0.30 \\ \multicolumn{1}{c}{Difference in 8 mm\({}^{2}\)} & 0.20 & 0.02 & 0.002 & 0.00 & 0.38 \\ \multicolumn{1}{c}{Difference in 4 mm\({}^{2}\)} & 0.11 & 0.47 & 0.001 & 0.15 & 0.00 \\ \multicolumn{1}{c}{Avg.} & 0.27 & 0.40 & 0.002 & 0.15 & 0.23 \\ \multicolumn{1}{c}{Sdev} & 0.33 & 0.35 & 0.001 & 0.15 & 0.20 \\ \multicolumn{1}{c}{\(\beta\)} & 0.662 & 0.168 & 0.003 & 0.16 & 0.56 \\ \multicolumn{1}{c}{\(r^{2}\)} & 0.844 & 0.022 & 0.964 & 0.108 & 0.740 \\ \multicolumn{1}{c}{**FOM**} & **14.24** & **0.95** & **1.61E8** & **29.81** & **29.00** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Four groups of factor combinations of ultrasound facility to verify the optimal suggestion according to different scenarios: A. original group 12, the highest FOM among 18 groups; B. conventional suggestion; C. the combination of highest FOM of each factor; D. considering the cross-interaction among factors I; and E. II.  
Low FOMs of the conventional setting show an inappropriate setting in identifying either area or diameter in a routine examination.