## 4 Strength Regression Analysis  
In order to analyze the relationship between the mechanical properties of RAIW and the five influencing factors, a regression analysis was completed in this study. The hypothesis was that a linear relationship would exist between the mechanical property indexes (the compressive strength, flexural strength, the flexural-compressive ratio) and the five influencing factors (brick granule content, water-cement ratio, aggregate-cement ratio, proportions of mixtures, and the replacement rate). The linear regression model was calculated as follows:  
\[y=a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+a_{4}x_{4}+a_{5}x_{5}+e \tag{1}\]  
where \"\(y\)\" is the compressive strength, flexural strength or flexural-compressive ratio; \"\(a_{i}\)\" (\(i\) = 1, 2, 3, 4, 5) are the regression coefficients; \"\(x_{1}\)\" is the brick powder content; \"\(x_{2}\)\" is the water-cement ratio; \"\(x_{3}\)\" is the aggregate-cement ratio; \"\(x_{4}\)\" is the proportions of the mixtures; \"\(x_{5}\)\" is the replacement of aggregate ratio; and \"\(e\)\" is the test error, respectively. By substituting the data into the regression model in Table 5, the least squares estimation was obtained. The regression equation is given below. Figure 5: Intuitive analysis of flexural strength. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Factor**} & \multicolumn{4}{c}{**Level**} & \multicolumn{6}{c}{**Results**} \\ \cline{2-13}  & **1** & **2** & **3** & **4** & **\(r\)** & **Df** & **F** & **F\({}_{\mathbf{0.1}}\)** & **F\({}_{\mathbf{0.05}}\)** & **F\({}_{\mathbf{0.01}}\)** & _Sig._ \\ \hline A & 0.293 & 0.291 & 0.296 & 0.244 & 0.052 & 3 & 1.346 & 2.490 & 3.290 & 5.420 & - \\ B & 0.318 & 0.252 & 0.279 & 0.276 & 0.066 & 3 & 1.730 & 2.490 & 3.290 & 5.420 & - \\ C & 0.223 & 0.302 & 0.320 & 0.278 & 0.097 & 3 & 4.038 & 2.490 & 3.290 & 5.420 & * \\ D & 0.267 & 0.268 & 0.310 & 0.279 & 0.043 & 3 & 0.962 & 2.490 & 3.290 & 5.420 & - \\ E & 0.261 & 0.255 & 0.316 & 0.292 & 0.061 & 3 & 1.924 & 2.490 & 3.290 & 5.420 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Analysis of the influencing factors on the flexural–compressive ratio. Compressive strength calculation formula:  
\[ \begin{array}{c}y=0.166x_{1}-3.7159x_{2}+78.7234x_{3}-8.3028x_{4}-1.1265x_{5}+0. 188\\ R=0.8765,~{}n=16,~{}F_{I}=15.6068.\end{array}  \tag{2}\]  
Flexural strength calculation formula:  
\[ \begin{array}{c}y=0.149x_{1}-0.6555x_{2}+16.9767x_{3}-1.7216x_{4}-0.1905x_{5}+ 0.041\\ R=0.8893,~{}n=16,~{}F_{I}=17.6770.\end{array}  \tag{3}\]  
Flexural-compressive ratio calculation formula:  
\[ \begin{array}{c}y=0.0322x_{1}+0.1875x_{2}+0.4903x_{3}+0.1296x_{4}+0.0530x_{5 }-0.0039\\ R=0.9470,~{}n=16,~{}F_{I}=39.2829.\end{array}  \tag{4}\]  
In the equations all, \"\(R\)\" is the corresponding coefficient; \"\(n\)\" is the degrees of freedom; and \"\(F_{I}\)\" is the variance in the regression equation. In this study, 96 specimens, of 16 combinations in total, were prepared for the compressive strength and flexural strength tests. Since the number of variables was 6, the number of degrees of freedom was 10, and the confidence level was 1%. The corresponding coefficient of 0.708 can be obtained from the relevant table. As the result of \(R\) = 0.8765 (0.8893/0.9470) > 0.708, the hypothesis of the linear model was found to be reasonable, and the regression equation obtained with this method is meaningful. In addition, from the variance in the regression equation, \(F_{I}\) = 15.6068 (17.6770/39.2829) > \(F_{0.1}\) (10, 5) = 2.52, it can be observed that the regression equation obtained by the linear assumption has great significance. By putting the mix proportions of each group into the regression equation (as shown in Table 9), the theoretical calculation can be obtained.