## 2 Materials and Methods
### Multiple Linear Regression Model  
Multiple linear regression (MLR) models are widely used to analyze the relationship between a response and several input variables [28]. The full quadratic model form of the multiple regression equation of the dependent variable (or response) (\(\Upsilon\)) on the independent variables (\(X_{1},X_{2},X_{3}\), and \(X_{4}\)) is given by Equation (5):  
\[\begin{split}\Upsilon=\beta_{0}+\beta_{1}\cdot& X_{1}+ \beta_{2}\cdot X_{2}+\beta_{3}\cdot X_{3}+\beta_{4}\cdot X_{4}+\beta_{11}\cdot X _{1}^{2}+\beta_{22}\cdot X_{2}^{2}+\beta_{33}\cdot X_{3}^{2}\\ &+\beta_{44}\cdot X_{4}^{2}+\beta_{12}\cdot X_{1}\cdot X_{2}+\beta _{13}\cdot X_{1}\cdot X_{3}+\beta_{14}\cdot X_{1}\cdot X_{4}+\beta_{23}\\ &\cdot X_{2}\cdot X_{3}+\beta_{24}\cdot X_{2}\cdot X_{4}+\beta_{34 }\cdot X_{3}\cdot X_{4}\end{split} \tag{5}\]  
where \(\beta_{0}\) is the constant coefficient, \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), and \(\beta_{4}\) are the linear coefficients, \(\beta_{11}\), \(\beta_{22}\), \(\beta_{33}\), and \(\beta_{44}\) are the quadratic coefficients, \(\beta_{12}\), \(\beta_{13}\), \(\beta_{14}\), \(\beta_{23}\), \(\beta_{24}\), and \(\beta_{34}\) are the interaction coefficient, and \(X_{1}\), \(X_{2}\), \(X_{3}\), and \(X_{4}\) are the codes of independent parameters.  
In this research, a 95% confidence level with a 5% margin of error has been used, i.e., the significance level is set at 0.05. The significance level, in hypothesis testing, is the criterion used for rejecting or accepting a hypothesis. The hypotheses for testing the significance of the regression coefficient (\(\beta_{i}\)) are shown below [29].  
**Hypothesis 1 (null hypothesis).**\(\beta_{i}=0\) _for \(X_{i}\)._  
**Hypothesis 2 (alternate hypothesis).**\(\beta_{i}\
eq 0\) _for \(X_{i}\)._  
Hypothesis 1 can be rejected if the calculated \(p\)-value for that regression coefficient is less than or equal to the significance level. On the other hand, for a \(p\)-value greater than the significance level, hypothesis 1 is not rejected and this indicates that the regression coefficient is zero and, therefore, the variable (\(X_{i}\)) can be removed from the model [29].  
The linear regression model is commonly used by researchers to obtain the relation between a variable of interest and a set of explicative variables [30]. However, the statistical analyses have uncertainty, called standard error, due to sampling variability [31]. If the residuals have Heteroscedasticity and Autocorrelation Consistent (HAC), then the model will be inconsistent and using it to perform inference will lead to incorrect conclusions [32].  
A simple solution is to adopt \"robust standard errors\", which are used to correct standard errors and avoid the heteroscedasticity effects in linear regression [31; 33]. In the literature, several Heteroscedasticity Consistent Covariance Matrix Estimators (HCCME) have been proposed, including HC0 estimator and its variants, such as HC1, HC2,  
Figure 1: Tukey’s box plot method.  
and HC3 [34; 35]. HC1, HC2, and HC3 estimators were suggested by [PERSON] and [PERSON] (1985). HC1 uses a global degrees-of-freedom adjustment, HC2 is constructed using standardized ordinary least squares residuals, and HC3 approximates a jackknife estimator [32; 36].