## 3 Research Methodology  
In regression analysis, homoscedasticity refers to the assumption that the variances of errors are constant across levels of an independent variable. If this assumption is violated, it is known as heteroscedasticity, which can affect the validity of a regression model. In this  
Figure 1: Theoretical framework. research, one assumption was the homoscedasticity of the variance of errors (or residuals), which is verified in the Discussion section using regression models and variance analyses, using scatterplots with residuals against dependent variables to check homoscedasticity. An analysis was conducted on the compressive and flexural strengths of 420 samples: 210 from the Turner Lab, 86 from the box culvert project, and 28 from the wing wall project. [PERSON] and [PERSON] [9] showed the details of data collection using sensors for the temperature and moisture of ambient air, specimens, curing boxes, and in situ concrete structure items, along with the concrete strengths using experiments and a universal testing machine. MATLAB (R2022a) software, Python 3.11.5, and Minitab Statistical (R21.3) software were used for data analysis. This research implemented a crossmatching method to create coordinate pairs and help understand the correlation between two variables [9]. Figure 2 shows the procedure for generating regression results for the linear correlation analysis of the compressive strengths of 4 in. cylinders and 6 in. cylinders cured using only Method #C1. The procedure was based on ordinary least squares (OLS), which is a linear regression analysis method used to minimize the sum of squares of the differences between the output (y values) of the target linear function of the independent variable and the values of the variable being observed in the input dataset (x values). The OLS method was selected for its simplicity, interpretability, and good fit to the data, as evidenced by the high R-squared value and low \(p\)-values for the coefficients (see Discussion). The R-squared (R\({}^{2}\)) value is the coefficient of determination (see Equation (1)). The higher the value, the better the goodness of fit of a model. In Figure 3, the R-squared value is 0.975, which means that 97.50% of the variation in the y values is accounted for by the x values. The R-squared indicator measures the strength of the relationship between the prediction of a linear model and its dependent variable to show how well a regression model describes observed data. The main purpose of using an R-squared indicator is to avoid overfitting a model, meaning preventing the model from picking up noise. Nevertheless, the criteria value of R-squared depends on the context. In this research, the R-squared indicator is set to be 0.8 or higher [16]. Figure 3 shows an example of the result from the regression analysis (in Figure 2). There are 186 observations in the paired dataset. The calculation of degrees of freedom of residuals is to calculate the total number of observations (rows) subtracted by the number of  
Figure 2: Pseudocode for how to perform correlation analysis. variables being estimated using the dataset. In this case, the degree of freedom of residuals is \(186-2=184\). The degree of freedom of the model is \(1\). \[R^{2}=1-\frac{\textit{sum squared regression}\ (SSR)}{\textit{total sum of squares}\ (SST)}=1-\frac{\sum(y_{i}-\hat{y_{i}})^{2}}{\sum(y_{i}-\overline{y})^{2}}, \tag{1}\]  
where i is a variable, yi is the observed data, \(\overline{y}\) is the mean of the observed data, and \(\hat{y_{i}}\) is the regression estimate. The Adjusted (Adj.) R-squared in Figure 3 is the modified version of the R-squared value for the number of predictors in the model. The higher the better. The F-statistic is calculated as (MSR/MSE) = Mean sum of squares regression/Mean sum of squares error. Using an F Statistics table (F-Alpha, n.d.), alpha = 0.05, denom. degree of freedom = 184, and numerator degrees of freedom = 1, the f(critical value) = 3.890.