# Development of a Model Material for Dynamic Geotechnical Model Tests
## 4 Experimental Results and Sensitivity Analysis
### Significance Test of Multiple Linear Regression Equation  
(1) Hypothesis testing  
In the process of multiple linear regression analysis, the basic assumptions of multiple linear regression analysis should be satisfied. The most important is the independence of samples and the absence of collinearity between samples. The two hypotheses are verified by [PERSON]'s principle and the variance inflation coefficient. The closer the Durbin-Watson value is to 2.0, the stronger the independence; the closer the variance inflation factor value is to 1.0, the weaker the collinearity. The process is not shown here, just the results, as shown in the following Table 4.  
Table 4 shows that the Durbin-Watson values of the seven parameters are all close to 2.0, basically meeting the requirements of mutual independence. Meanwhile, the VIF of the seven parameters is 1.0, and there is no autocorrelation among them. In summary, this analysis basically meets the basic assumption of the multiple linear regression equation.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Indicators** & \(\rho\)**/g-cm\({}^{-3}\) & \(\sigma_{t}\)**/MPa** & \(\sigma_{c}\)**/MPa** & \(c\)**/MPa** & \(\varphi\)\(f^{c}\) & **G\({}_{d}\)**/GPa** & **E\({}_{d}\)**/GPa** \\ \hline Durbin–Watson(D-W) & 1.810 & 1.925 & 2.158 & 1.890 & 1.931 & 1.860 & 1.951 \\ Variance inflation factor (VIF) & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Basic hypothesis verification.  
Figure 10: Comparison of fitting and experimental results for dynamic elastic modulus.  
Figure 9: Comparison of fitting and experimental results for dynamic shear modulus.  
(2) Significance testing  
The model obtained through multiple linear regression analysis must pass a certain significance test for verification to determine whether the relationship between the two has good significance. If there is no good significance between the two, then there is no correlation between the dependent variable \(y\) and \(x_{i}\). In the analysis, the whole model was mainly considered to configure the required similar materials, so the significance test of the model was mainly considered here, and the significance of each parameter was no longer considered. In this paper, the significance of the regression equation was tested as follows.  
The significance test of the regression equation generally refers to the F statistic as the constraint condition of the model, and the expression of the \(F\) statistic is  
\[F=\frac{SS_{I}/m}{SS_{r}/(n-m-1)} \tag{16}\]  
In the above formula, \(SS_{I}\) is the sum of squares of regression, \(\text{SS}_{r}\) is the sum of squares of residuals, \(m\) is the number of degrees of freedom, and the \(F\) statistic should obey the distribution of \(F(m,\,n-m-1)\), namely, obey the significance level \(\alpha\), which can be determined by the following expression:  
\[P\{|F|\geq F_{1-\alpha,m,n-m-1}|H_{0}|=\alpha\} \tag{17}\]  
The above formula is the expression of the significance test. If it satisfies \(|F|\geq F_{1-\alpha,m,n-m-1}\), then it indicates that the relationship between \(y\) and \(x_{i}\) has good significance, that is, the model has good significance.  
The statistical quantity \(F\) of each physical quantity and the significance of each variable were calculated by the significance test theory of the regression equation mentioned above, and the calculation results are shown in Table 5.  
Table 5 shows that the significance of all parameters is much less than 0.001, and there is a good correlation between the independent variable and multiple dependent variables, so there is a clear correlation between the dependent variable \(y\) and \(x_{i}\). The obtained physical model can reflect the relationship between various physical parameters and multiple independent variables to a large extent.  
Practical Application of the Proportions of Components in a Model Material Similar to an Actual Material