# Experimental Study of Material Proportioning for Similar Modeling of Brittle Rocks
## 4 Multiple Linear Regression Analysis  
Based on the experimental results, it is evident that each factor influences the physical and mechanical parameters of the analogous material specimens. The adjustments to the proportion of each material component consistently resulted in changes in these parameters. Figure 10: Sensitivity analysis of the brittleness index. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Average Value** & **A** & **B (\%)** & **C (\%)** & **D (\%)** & **E (\%)** \\ \hline Average value 1 & 0.903 & 0.735 & 0.626 & 1.217 & 0.710 \\ Average value 2 & 0.786 & 0.662 & 0.732 & 0.751 & 0.694 \\ Average value 3 & 0.793 & 0.840 & 0.820 & 0.761 & 0.914 \\ Average value 4 & 0.668 & 0.926 & 0.784 & 0.567 & 0.801 \\ Average value 5 & 0.712 & 0.698 & 0.900 & 0.566 & 0.744 \\ Range & 0.235 & 0.264 & 0.274 & 0.651 & 0.220 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Range analysis of the brittleness index. To enhance the efficiency of analogous material proportioning, this study applied a multiple linear regression analysis to the experimental data, aiming to establish correlations between various influencing factors and multiple physical and mechanical parameters. A multiple linear regression analysis was performed on the experimental data, where \(x_{1}\) represents the barite powder-refined iron powder ratio, with \(0.3\leq x_{1}\leq 1.1\); \(x_{2}\) is the quartz sand content (the percentage of the quartz sand mass in the total specimen mass), with \(6\%\leq x_{2}\leq 14\%\); \(x_{3}\) is the gypsum content (the percentage of the gypsum mass in the total specimen mass), with \(2.0\%\leq x_{3}\leq 4.0\%\); \(x_{4}\) is the concentration of the rosin-alcohol solution (the percentage of the rosin mass in the solution mass), with \(5\%\leq x_{4}\leq 25\%\); and \(x_{5}\) is the rosin-alcohol solution dosage (the percentage of the solution mass in the total specimen mass), with \(4.0\%\leq x_{5}\leq 6.0\%\). The resulting multiple linear regression equations for each physical and mechanical parameter are as follows:  
\[\left\{ \begin{array}{l}\rho=2.664-0.099x_{1}-0.912x_{2}-1.250x_{3}-0.205x_{4} -3.650x_{5}\\ \sigma_{\text{c}}=-1.007+0.592x_{1}-3.100x_{2}+5.160x_{3}+5.812x_{4}+15.680x_{ 5}\\ \sigma_{\text{t}}=-0.067+0.040x_{1}-0.350x_{2}+0.360x_{3}+0.492x_{4}+1.680x_{ 5}\\ E=-0.377+0.151x_{1}-1.181x_{2}+6.780x_{3}+2.160x_{4}+4.908x_{5}\\ W_{\text{cf}}=2.176+0.055x_{1}+0.555x_{2}-20.984x_{3}+2.817x_{4}-17.907x_{5}\\ B_{i}=0.763-0.250x_{1}+0.953x_{2}+11.989x_{3}-2.973x_{4}+3.497x_{5}\end{array} \right. \tag{3}\]  
A statistical validation of the multiple linear regression analysis was conducted. As shown in Table 10, the regression equation fit the sample data well overall and explained most of the variation in the dependent variable. However, it also reflected the uncertainty and complexity of parameter selection in the proportioning of the geotechnical materials. The significance tests for the regression equation and its variables indicated that the model is statistically significant overall and that most independent variables have a significant impact. These findings are consistent with the results of the parameter sensitivity analysis conducted in Section 3.2. A comparison between the experimental values of each physical and mechanical parameter of the specimens and the calculated values from the multiple linear regression equations is shown in Figure 11. The multiple linear regression equations provided a good approximation of the actual physical and mechanical parameters of the specimens, demonstrating strong predictive capability for these parameters.