## 4 Discussion
### Correlation Analysis: 4 in. vs. 6 in. Cylinders (Curing Method #C2)
cylinders using Method #C2: (**a**) linear correlation, y = 1067.0 + 0.81x, R-squared = 0.9130; (**b**) linear correlation, y-intercept = 0; (**c**) linear correlation, y-intercept = 0, Day 1; (**d**) linear correlation, y-intercept = 0, summer data; (**e**) linear correlation, y-intercept = 0, fall data; (**f**) linear correlation, y-intercept = 0, winter data; (**g**) linear correlation, y-intercept = 0, spring data; and (**h**) nonlinear, fitted line plot, 4 in. vs. 6 in. cylinders. Figure 7: Residual analyses of the fluences of Method #C2: (**a**) normal probability plot of residuals for linear correlation regression; (**b**) linear correlation regression, residuals vs. fits plot; (**c**) linear correlation regression residual histogram; (**d**) linear correlation regression, residual versus observation order; (**e**) normal probability plot of residuals for nonlinear correlation regression; (**f**) nonlinear, residuals vs. fits plot; (**g**) nonlinear, histogram; and (**h**) nonlinear, residuals versus observation order. Table 8 compares the model performances of the linear and nonlinear equations of the correlation models of the compressive strengths of the 4 in. and 6 in. cylinders cured using Method #C2. The results indicate that the linear model is better in performance. The modified [PERSON]'s correlation method in the table follows the Wilkinson-Rogers notation to describe regression in a simplified manner by making specific coefficient values into zeros. This modified [PERSON]'s correlation method identifies the response variable based on the previous linear and nonlinear correlation analyses. Figure 8a shows the following fitted lines based on the results in Table 6. In general, the compressive strengths of the 4 in. cylinders show slightly higher values than the compressive strengths of the 6 in. cylinders cured using Method #C1. * \(y1=228.0869+0.9325x\), for experiments performed on Days 1, 2, 3, and 7. * \(y2=0.9836x\), for experiments performed on Days 1, 2, 3, and 7 starting from the origin point (0,0). * \(y3=1.0497x\), for estimated strength correlations 24 h (Day 1) after casting concrete cylinders. Figure 8b shows the following fitted lines based on the results in Table 8. In general, the compressive strengths of the 4 in. cylinders show slightly lower values than the compressive strengths of the 6 in. cylinders cured using Method #C2. \begin{table}
\begin{tabular}{c c c} \hline \hline \# & **Explanation** \\ \hline (a) & Linear correlation: y = 1067.0 + 0.81x, R-squared = 0.9130. \\ (b) & When the intercept is set to zero, y = 1.0361x for the data of compressive strengths tested during all curing days including Days 1, 2, 3, and 7. R-squared = 0.9903. \\ (c) & Using Day 1 data and setting the y-intercept to zero. y = 1.1237x, R-squared value = 0.9889. \\ (d) & Using summer data and setting the y-intercept to zero. y = 0.9851x, R-squared value = 0.9959. \\ (e) & Using fall data and setting the y-intercept to zero. y = 1.0407x, R-squared value = 0.9910. \\ (f) & Using winter data and setting the y-intercept to zero. y = 1.1119x, R-squared value = 0.9861. \\ (g) & Using spring data and setting the y-intercept to zero. y = 1.0674x, R-squared value = 0.9925. \\ (h) & For the nonlinear correlation analysis, see Table 8 for details. \\ \hline \end{tabular} Note. The 4 in. cylinders and 6 in. cylinders are cured using only Method #C2.  
\end{table} 
Table 7: Explanations of the correlation analysis in Figure 7.