## 3 Results and Analysis
### Data Analysis Results
#### 3.2.4 Bivariate MLR Modelling  
Bivariate MLR modelling involved using two variables (among the four) at a time as the independent (predictor) variables while keeping CS as the dependent variable (to be predicted). According to Section 2.2.3, six models were established.  
Suppose that the general form of all these models can be written as: \(CS=\beta_{0}+\beta_{1}(Variable_{1})+\beta_{2}(Variable_{2})\); the six models are summarised in Table 11, where \(Variable_{1}\), \(Variable_{2}\in\{SR\_BJPA,T,WB\_Ratio,\#Days\}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Which Variable Is Used as** & **Model Significance (_p_)** &
\begin{tabular}{c} **Power to Explain Data** \\ **Variances (\(R^{2}\))** \\ \end{tabular} \\ \hline SR-BJPA & 2.2 \(\times\) 10\({}^{-16}\) & 0.3791 \\ T & 0.003916 & 0.03063 \\ W/B Ratio & 2.234 \(\times\) 10\({}^{-9}\) & 0.1219 \\ \#days & 2.2 \(\times\) 10\({}^{-16}\) & 0.3513 (=0.3986 if optimised) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Significance and data–model fitness of the four univariate SLR models.  
Reassembled from Table 11, the six models can be represented as:  
\[CS=46.1198+(-79.1608)SR\_BJPA+0.1218T \tag{5}\]  
\[CS=82.754+(-79.158)SR\_BJPA+(-61.091)WB\_Ratio \tag{6}\]  
\[CS=42.9999+(-80.51617)SR\_BJPA+0.18270\#Days \tag{7}\]  
\[CS=66.8486+0.12179T+(-61.09144)WB\_Ratio \tag{8}\]  
\[CS=26.35699+0.12179T+0.19630\#Days \tag{9}\]  
\[CS=62.9924+(-61.0914)WB\_Ratio+0.1963\#Days \tag{10}\]  
Each plane for an equation in the 3D space is plotted in Figure 6 for visualisation purposes.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Pairs of Dependent** & **Model’s \(\beta_{0}\)** & **Model’s \(\beta_{1}\)** & **Model’s \(\beta_{2}\)** & \(p\) & \(R^{2}\) \\ \hline SR-BIPA \(\oplus\) T & 46.1198 & \(-79.1608\) & 0.1218 & \(2.2\times 10^{-16}\) & 0.4162 \\ \hline SR-BIPA \(\oplus\) W/B Ratio & 82.754 & \(-79.158\) & \(-61.091\) & \(2.2\times 10^{-16}\) & 0.5086 \\ \hline SR-BIPA \(\oplus\) \#days & 42.9999 & \(-80.5062\) & 0.1827 & \(2.2\times 10^{-16}\) & 0.6632 \\ \hline T \(\oplus\) W/B Ratio & 66.8486 & 0.1218 & \(-61.0914\) & \(2.15\times 10^{-10}\) & 0.1534 \\ \hline T \(\oplus\) \#days & 26.357 & 0.1218 & 0.1963 & \(2.2\times 10^{-16}\) & 0.3818 \\ \hline W/B Ratio \(\oplus\) \#days & 62.9924 & \(-61.0914\) & 0.1963 & \(2.2\times 10^{-16}\) & 0.4741 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Estimated model parameters, significance, and data–model fitness of the six bivariate MLR models.  
Figure 6: _Cont._  
The results indicate that all bivariate models except the one using T and the W/B ratio together as the predictor variables (Equation (8)) can effectively predict CS. Figure 6d illustrates this, as the data point distribution is too diversified and the tendency is too vague, compared to other figures.  
This agrees with the discussion in Section 3.2.3, which mentioned that the poorest univariate model (without any predictive power) was the model using T to predict CS, with the highest acceptable \(p\)-value and an \(R^{2}\) value that almost approaches 0. This was followed by the model using the W/B ratio to predict CS (the \(p\)-value was low but the \(R^{2}\) value was questionable), which has moderate predictive power. Combining these two factors only increased the \(R^{2}\) value to 0.1534, and even the \(p\)-value decreased further to 2.15 \(\times\) 10\({}^{-10}\); thus, compared to all other models, the predictive power of the resulting model is questionable. All other five bivariate models are effective as revealed by the statistical analysis (all with \(p\)-values of 2.2 \(\times\) 10\({}^{-16}\)) and acceptable \(R^{2}\) values (>0.2) (Table 11).  
In general, in the context of this study, using two variables together is a more effective way to predict the CS than using only one variable. The effective models can be used to predict the CS to guide the admixture works (i.e., adding BJPA, altering the curing temperature, and changing the W/B ratio) while preparing the cement mortar in practise.