## 4 Discussion
### Correlation Analysis: 4 in. vs. 6 in. Cylinders (Curing Method #C1)  
Figure 4 shows the fitted line plots, and Table 5 has explanations about the plots. The plots are generated using the compressive strengths of 4 in. cylinders as the x-axis and the compressive strengths of 6 in. cylinders as the y-axis. The nonlinear analysis (Figure 4h) assumes three independent variables (x1, x2, and x3) and one dependent variable. In this case, the degree of freedom of residuals is 186 \(-\) 4 = 182. The degree of freedom of the model is 3. The R-squared value and the Adjusted R-squared value are 0.967, which means that 96.70% of the variation in the y values is accounted for by the x values. The f-statistic of 1799 is greater than the critical value of 2.6790, which means that the null hypothesis that the value of all coefficients = 0 is rejected. Thus, the alternate hypothesis holds, which means that the coefficient related to the predictor variable (the compressive strengths of 4 in. cylinders) is non-zero. The Prob(F-statistic  
Figure 4: Fitted line plots for 4 in. vs. 6 in. cylinders using Method #C1. (**a**) Linear correlation; (**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. \begin{table}
\begin{tabular}{c c} \hline \# & Explanation \\ \hline (a) & The plot shows a linear correlation: y = 228.0869 + 0.9325x, R-squared = 0.9750. \\ \hline  & Further analysis of the linear correlation of (6 in. compressive strength = coefficient \(\times\) 4 in. compressive strength) for all curing days. The y-intercept is zero. Compared with the model equation in Figure 4a, this assumption fits engineering practice because right after the concrete is poured, it does not gain any compressive strength at time zero. The result model equation shown in Figure 4b is y = 0.9836x, for the data of compressive strengths tested during all curing days including Days 1, 2, 3, and 7. The corresponding R-squared is 0.9955, which indicates that the result model equation is accurate in describing the correlation of the observations.  
\end{table} 
Table 5: Explanations for the correlation analysis in Figure 4 (for the correlation analysis of the compressive strengths of 4 in. cylinders and 6 in. cylinders cured using Method#C1). Figure 5b,f shows residual versus fit plots resulting from residual analysis, where residuals are on the y-axis and fitted values (a.k.a., estimated responses) are on the x-axis; the analysis detects non-linearity, unequal error variances, and outliers. The residual = 0 line corresponds to an estimated correlation regression line (e.g., in Figure 4). The residuals in Figure 5b,f are distributed randomly around the corresponding zero lines. Additionally, the residuals roughly form a horizontal box shape around the zero lines in Figure 5b,f, which suggests that the variances of the error terms are equal. A few residuals are quite different from the basic random pattern of residuals, which suggests that there are outliers. Figure 5c,g shows the histograms of the residual frequency. The zero residuals have the most frequencies, and the negative and positive residuals are evenly distributed in these two figures. The histograms show normal distributions. Figure 5d,h shows the standardized residuals in the order of the corresponding observations. When the order of the observations may influence the results, data are collected in a time sequence or some other sequence, or from different geographic areas in a certain order. These plots can help