## 2 Methods
### Mathematical Model
Quadratic model Regression statistics using the quadratic model are presented in Table 8.  
The following dependence was obtained:  
\[Y=54.67+5.94\;{\rm z_{1}}+8.68\;{\rm z_{2}}+7.125\;{\rm z_{3}}+5.90\;{\rm z_{4}} -4.41\;{\rm z_{1}}^{2}+7.78\;{\rm z_{2}}^{2}++1.15\;{\rm z_{3}}^{2}+3.40\;{\rm z _{4}}^{2}. \tag{11}\]  
Only coefficients of linear terms are significant (according to the Student's test, coefficients at the squares have a level of confidence less than the generally accepted value of 0.95). The coefficient of determination of the model is 0.925, which confirms its high adequacy (according to the [PERSON]'s test, significance is 2.0448 \(\times\) 10\({}^{5}\)). 2. General quadratic model Regression statistics using the general quadratic model are presented in Table 9. \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{Regression statistics} \\ \hline Multiple \(R\) & 0.925099193 \\ \hline \(R\)-square & 0.855808516 \\ \hline Normalized \(R\)-Square & 0.783712774 \\ \hline Standard error & 6.415144034 \\ \hline Observations & 25 \\ \hline Analysis of variance & \\ \hline  & df & MS & F & Significance of F \\ \hline Regression & 8 & 488.517146 & 11.87044467 & \(2.0448\times 10^{5}\) \\ \hline Remainder & 16 & 41.15407298 & \\ \hline Total & 24 & & \\ \hline  & Coefficients & \(t\)-statistics & \(p\)-value \\ \hline \(Y\)-intersection & 54.67322034 & 19.73776195 & \(1.17275\times 10^{12}\) \\ \hline Variable X 1 & 5.937222222 & 3.926568201 & \(0.001204454\) \\ \hline Variable X 2 & 8.681666667 & 5.741600194 & \(3.03206\times 10^{5}\) \\ \hline Variable X 3 & 7.125 & 4.71210229 & \(0.000234952\) \\ \hline Variable X 4 & 5.90333333 & 3.904155863 & \(0.00126267\) \\ \hline Variable X 5 & \(-4.413757062\) & \(-1.097984589\) & \(0.288460441\) \\ \hline Variable X 6 & 7.776242938 & 1.93445058 & \(0.070942481\) \\ \hline Variable X 7 & 1.146242938 & 0.285144167 & \(0.779191732\) \\ \hline Variable X 8 & 3.401242938 & 0.846107359 & \(0.409969175\) \\ \hline \end{tabular}
\end{table} 
Table 8: Using a quadratic model to obtain regression statistics. The following dependence was obtained:  
\[ \begin{array}{l}Y=54.83+8.89\ z_{1}+9.45\ z_{2}+5.83\ z_{3}+5.83\ z_{4}+0.2\ z_{1} ^{2}+5.2\ z_{2}^{2}+\\ +2.7\ z_{3}^{2}+2.7\ z_{4}^{2}-2.5z_{1}z_{2}-1.25\ z_{1}z_{4}+1.86\ z_{2}z_{3}-1.86 \ z_{2}z_{4}-3.12\ z_{3}z_{4}.\end{array}  \tag{12}\]  
Only coefficients of linear terms were significant (according to the Student's test, coefficients at squares and products of factors have a level of confidence less than the generally accepted value of 0.95). The coefficient of determination of the model is 0.965, which confirms its high adequacy (the significance is 0.000481547 according to the [PERSON]'s test). The authors believe that the general quadratic model (the coefficient of determination is 0.965) is the most adequate. The mathematical model allows adjustments to achieve the desired levels of reliability, quality, and durability at any stage of any construction project. The use of the mathematical model, which conveys the essence of the phenomenon considered here, is the optimal solution; it successfully predicts and evaluates the impact of individual factors on the IQI [37; 38; 39; 40; 41; 42]. \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{Regression statistics} \\ \hline Multiple paired & 0.96528285 \\ \hline \(R\)-Square & 0.93177098 \\ \hline Normalized \(R\)-Square & 0.836250352 \\ \hline Standard error & 6.101291117 \\ \hline Observations & 25 \\ \hline \multicolumn{4}{c}{Analysis of variance} \\ \hline  & \(df\) & \(MS\) & \(F\) & Significance of F \\ \hline Regression & 14 & 363.1244619 & 9.754657187 & 0.000481547 \\ \hline Remainder & 10 & 37.2257533 \\ \hline Total & 24 & & \\ \hline  & Coefficients & \  
In our further calculations, the IQI, determined by its parameters, rather than groups of factors, will be referred to as IQI.