# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 3 Results
### Multiple Linear Regression (MLR)  
1.1 Multiple Linear Regression Model for the First Group of Predictors (Thickness, Width, Length, Mass of Fresh Paving Block, and Percentage of Water Absorption)  
The multiple linear regression model for the first group of predictors shows us a non-significant \(p\)-value of 0.22736 in the \(T\)-test of the coefficient of the width variable. Therefore, the null hypothesis is not rejected \(\beta_{i\ (width)}=0\), and there is no statistical evidence to affirm that \(\beta_{i\ (width)}\
eq 0\). The discarding of the width variable indicated that it does not significantly influence the prediction of tensile splitting strength. The following model is carried out with the variables (thickness, length, mass of fresh paving block, and the percentage of water absorption), giving rise to significant results (\(p\)-value < 0.05); in all the \(T\)-tests of the coefficients for a confidence level of 95% the null hypothesis is rejected \(\beta_{i}=0\), there being significant statistical evidence to affirm that \(\beta_{i}\
eq 0\), so the \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Variable**} & \multicolumn{1}{c}{**Measures**} & \multicolumn{3}{c}{**Histogram, Density Function,**} \\  & & & **Boxplot (Below), and Dots of Data** \\ \hline \multirow{7}{*}{Mass of fresh paving block, g} & Max: & 2780.30 & g \\  & Min: & 2363.70 & g \\  & Mean: & 2541.55 & g \\  & Median: & 2539.60 & g \\  & Standard deviation: & 77.19 & g \\  & Variance: & 5958.60 & g\({}^{2}\) \\  & Range: & 416.60 & g \\  & Coefficient of variation: & 3.04 & \% \\ \hline \multirow{7}{*}{Density of fresh paving block, kg/m\({}^{3}\)} & Max: & 2385.27 & kg/m\({}^{3}\) \\  & Min: & 1993.58 & kg/m\({}^{3}\) \\  & Mean: & 2181.57 & kg/m\({}^{3}\) \\  & Median: & 2186.00 & kg/m\({}^{3}\) \\  & Standard deviation: & 61.53 & kg/m\({}^{3}\) \\  & Variance: & 3785.66 & (kg/m\({}^{3}\))\({}^{2}\) \\  & Range: & 391.69 & kg/m\({}^{3}\) \\  & Coefficient of variation: & 2.82 & \% \\ \hline \multirow{7}{*}{Percentage of water absorption, g/g\%} & Max: & 13.64 & g/g\% \\  & Min: & 2.12 & g/g\% \\  & Mean: & 5.81 & g/g\% \\  & Median: & 5.46 & g/g\% \\  & Standard deviation: & 2.05 & g/g\% \\  & Variance: & 4.18 & (g/g\%)\({}^{2}\) \\  & Range: & 11.52 & g/g\% \\  & Coefficient of variation: & 35.20 & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. predictor variables do influence the response variable. The resulting adjusted coefficient of determination is 0.7974. In the _F_-test, the _p_-value is much less than 0.05, so for a confidence level of 95%, the null hypothesis is rejected, there being significant evidence to affirm that _at least some of the \(\beta_{i}\
eq 0\)_, obtaining a mean square error of 0.110086. The structural assumptions are verified, and Figure 4 illustrates the adjustment to new values for the prediction of tensile splitting strength vs. the current values of the test dataset; the model can be seen below, where the variable with the most significant inverse influence is the thickness followed by the percentage of water absorption. \[ \begin{array}{ll}\textit{Tensile splitting strength}=&(29.911784)+\\ &(-0.244570)\textit{ Thickness}+\\ &(-0.108435)\textit{ Length}+\\ &(0.004428)\textit{ Mass}\_of\_fresh\_paving\_block}+\\ &(-0.174059)\textit{ Percentage}\_of\_water\_absorption}\\ \end{array}  \tag{32}\]1.2 Multiple Linear Regression Model for the Second Group of Predictors (Density of Fresh Paving Block and Percentage of Water Absorption)  
When carrying out the model with the predictors of the density of the fresh paving block and the percentage of water absorption, a \(p\)-value of less than 0.05 is evidenced for the two variables in the \(T\)-test of the coefficients; for a confidence level of 95%, the null hypothesis is rejected, \(\beta_{i}=0\) exhibiting significant statistical evidence to affirm that \(\beta_{i}\
eq 0\).