## 3 Results  
### Semiconductor Manufacturing  
An experimental dataset was reported by [PERSON] and [PERSON] [49] and was introduced by [PERSON] et al. [1]. The process involved applying a coating material to a wafer. Several coating thicknesses at different locations on a wafer were measured. The mean \(y_{1}\) and standard deviation \(y_{2}\) of the thickness were calculated. The influencing variables were \(x_{1}\), speed; \(x_{2}\), pressure; and \(x_{3}\), distance. These datasets illustrate the backward elimination technique for an adequate equation and are listed in Appendix A.1. For the \(y_{1}\) mean of thickness [49], the complete regression procedure is  
\[1.\ y_{1}=327.615+177.011x_{1}+109.422x_{2}+131.472x_{3}+32.022{x_ {1}}^{2}-22.378{x_{2}}^{2}\] \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\] \[-29.061{x_{3}}^{3}+66.033{x_{1}}{x_{2}}+75.458{x_{1}}{x_{3}}+43.5 83{x_{2}}{x_{3}} \tag{5}\] \[(0.363)\ (0.008)\ (0.003)\ (0.064)\] \[\mathrm{R}^{2}=0.927,\ \mathrm{R}^{2}{}_{\mathrm{adj}}=0.888, \mathrm{s}=76.111,\ \mathrm{PRESS}=337,737.94\]  
Delete \({x_{2}}^{2}\) and recalculate the equation. \[2.\ y_{1}=312.696+177.011x_{1}+109.422x_{2}+131.475x_{3}+32.022 {x_{1}}^{2}-29.061{x_{3}}^{2}\] \[(<\!0.001)\ (<\!0.001)\ (<\!0.001)\ (0.310)\ (0.356)\] \[+66.033{x_{1}}{x_{2}}+75.458{x_{1}}{x_{3}}+45.583{x_{2}}{x_{3}} \tag{6}\] \[(0.007)\ (0.003)\ (0.006)\]  
Delete \({x_{3}}^{2}\) and recalculate the equation. \[3.\ y_{1}=293.322+177.011x_{1}+109.422x_{2}+131.475x_{3}+32.022 {x_{1}}^{2}-66.033{x_{3}}^{2}\] \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]0 \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]1 \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]2  
Delete \({x_{1}}^{2}\) and recalculate the equation. \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]3 \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]4 \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]5 \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]6 \[(<\!0.001)\ (<\!0.001)\ (<\!0001)\ (0.317)\ (0.481)\]7  
The \(p\)-values of all variables are \(<\!0.05\); this is an adequate equation. The normality and constant variance tests are passed. Influential points include the 9 th (t\({}_{\mathrm{i}}=-3.316\), DFFITS\({}_{\mathrm{i}}=-2.858\)), 19 th (t\({}_{\mathrm{i}}=2.255\), DFFITS\({}_{\mathrm{i}}=2.056\)), and 25 th (t\({}_{\mathrm{i}}=-2.322\)). The adequate equation involves \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{1}x_{3}\), and \(x_{2}x_{3}\). No quadratic terms exist in this equation. The contour plots of the response form a plateau, not a surface curve. The adequate equation for the \(y_{2}\) standard deviation and three variables is calculated as follows:  
\[ \begin{split} y_{2}=34.904+11.522x_{1}+15.317x_{2}+29.183x_{3}+4.189{ x_{1}}^{2}-1.328{x_{2}}^{2}\\ (0.280)\ (0.156)\ (0.012)\ (0.818)\ (0.942)\\ -16.772{x_{3}}^{3}+7.717x_{1}x_{2}+5.117x_{1}x_{3}+14.075x_{2}x_{3}\\ (0.362)\ (0.550)\ (0.691)\ (0.281)\\ R^{2}=0.454,\ \text{R}^{2}_{\text{adj}}=0.615,\ \text{s}=43.817,\ \text{PRESS = 93,044.256}\end{split}  \tag{9}\]  
After the execution of the backward elimination, the adequate equation for \(y_{2}\) is  
\[ \begin{split} y_{2}=47.993+29.183x_{3}\\ (0.007)\\ R^{2}=0.256,\ \text{R}^{2}_{\text{adj}}=0.227,\ \text{s}=42.171, \ \text{PRESS = 52,441.283}\end{split}  \tag{10}\]  
The normality test is passed. The constant variance test is failed (\(p<0.001\)). The results of the evaluation of adequate RSM models for semiconductor manufacturing in five studies are listed in Table 4.  
[PERSON] et al. [23] employed the response surface method to optimize the final polishing of Si wafers. The experimental design was CCD, the response \(y_{SR}\) was surface roughness, and the variables were \(x_{1}\) applied pressure, \(x_{2}\) platen speed, and \(x_{3}\) mixed slurry ratio. The RSM model was not reported in this study. Contour plot and response surface plot curves were produced by full models [23]. The full equation with these datasets is as follows:  
\[