## 3 Results and Discussion
### Analysis of Response Output Index Results  
#### 3.1.1 Density  
By analyzing the linear model (linear), two-factor model (2 FI), and quadratic equation model (quadratic), the best fitting equation model was found and the coefficient of determination R\({}^{2}\) was used to evaluate the fit. The impact factor was fitted to the three model equations, and SST (sum of squares total) of each model was calculated. Then, the SST was decomposed:  
\[\text{SST}=\Sigma(Y_{i}-\overline{Y})^{2}=\Sigma(Y_{i}-\overline{Y}_{i})^{2}+ 2\Sigma\Big{(}Y_{i}-\hat{Y}_{i}\Big{)}\Big{(}\hat{Y}_{i}-\overline{Y}\Big{)}+ \Sigma(\hat{Y}_{i}-\overline{Y})^{2}, \tag{13}\]  
\[\Sigma\Big{(}Y_{i}-\hat{Y}_{i}\Big{)}\Big{(}\hat{Y}_{i}-\overline{Y}\Big{)}= \Sigma e_{i}\Big{(}\hat{Y}_{i}-\overline{Y}\Big{)}=0, \tag{14}\]  
Therefore, the total dispersion was finally decomposed into the following Equation:  
\[\text{SST}=\Sigma(Y_{i}-\overline{Y})^{2}=\Sigma(Y_{i}-\overline{Y}_{i})^{2} +\Sigma(\hat{Y}_{i}-\overline{Y})^{2}=SSR+SSE, \tag{15}\]  
In the Equation, SSR is the sum of squares within the group (regression square sum); SSE is the sum of squares between groups (sum of squared residuals). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Test Serial** & **A** & **B** & **C** & \begin{tabular}{c} **Density** \\ **(g/cm\({}^{3}\))** \\ \end{tabular}  & \begin{tabular}{c} **Air Voids** \\ **(\%)** \\ \end{tabular}  & \begin{tabular}{c} **Marshall Stability** \\ **(\%)** \\ \end{tabular}  & \begin{tabular}{c} **Flow Value** \\ **(nm)** \\ \end{tabular}  & \ \begin{tabular}{c} **VMA** \\ **(\%)** \\ \end{tabular}  & \begin{tabular}{c} **VFA** \\ **(\%)** \\ \end{tabular}  \\ \hline
1 & 1.0 & 5.0 & 4.0 & 2.383 & 6.9 & 10.30 & 1.9 & 13.0 & 46.5 \\
2 & 1.0 & 5.0 & 7.0 & 2.365 & 3.9 & 8.57 & 3.6 & 16.1 & 75.6 \\
3  
The solvability coefficient R\({}^{2}\) is as follows:  
\[R^{2}=\frac{SSR}{SST}=1-\frac{SSE}{SST}, \tag{16}\]  
The closer SSR is to SST, the smaller the residual is and the closer R\({}^{2}\) is to 1, indicating a higher degree of fit. In order to exclude the influence of too many variables on the degree of fit, R\({}^{2}\) was adjusted by the following Equation to obtain the predicted \(\overline{R}^{2}\):  
\[\overline{R}^{2}=1-\frac{SSE/(n-k-1)}{SST/(n-1)}, \tag{17}\]  
In the Equation, \(n-k-1\) is the degree of freedom of the sum of squares between groups; \(n-1\) is the degree of freedom of the sum of squares of the total dispersion. Table 10 lists the comparison results after fitting with various equations. After analyzing the fitting degree of each model, it was found that the highest fitting degree was the quadratic equation model. The F test was carried out to assess the variance and misfit values of each regression model. Here, we take the linear model as an example. For the test model:  
\[Y_{I}=\beta_{0}+\beta_{1}X_{1i}+\beta_{2}X_{2i}+\cdots+\beta_{k}X_{ki}+u_{i}, \ i=1,2,\cdots,n \tag{18}\]  
If the parameter is significantly non-zero, the following null hypothesis and alternative hypothesis are made:  
\(H_{0}:\beta_{1}=\beta_{2}=\cdots=\beta_{k}=0\ H_{1}:\beta_{i}\) is not all 0. \[\text{F}=\frac{\overline{R}^{2}/k}{\left(1-\overline{R}^{2}\right)(n-k-1)} \tag{19}\]  
It can be found that when R\({}^{2}\) approaches 0, the F value also approaches 0; when R\({}^{2}\) approaches 1, the F value approaches infinity. Therefore, the test hypothesis H\({}_{0}\) is equivalent to the test R\({}^{2}\) = 0. Tables 11-13 list the results for the analysis of variance for each fitted model. According to the analysis of the above tables, it is found that the quadratic equation had the highest degree of fitting, which was recommended. The variance of each variable fitted by the quadratic equation is analyzed below, and the results are shown in Table 14.