## 3 Results
### Response Surface Methodology Application  
#### 3.4.1 Analysis of Rate of Penetration (ROP)  
The first step in linear regression is to analyze the statistical performance of the model. Different tools can be used to evaluate the performance and accuracy of the model quantitatively. In this research and for the septic dataset presented throughout this paper, residual value plots (also known as coefficients of determination), r-squared predicted (R\({}^{2}\)), and r-square adjusted (Radi\({}^{2}\)) were used to the goodness of fit of the regression equation [40]. The Radi\({}^{2}\) is employed to assess how well the model fits while considering predictor variables not provided by a large value of R\({}^{2}\), and its value increases by adding significant predictor variables [41; 42]. Figure 5 shows the regression residual plots. Residuals are the differences between the observed values and the predicted values from a regression model. The analysis of residual plots provides an assessment of the model based on the proximity of the residuals to a straight line and the absence of any discernible trend or pattern in these plots [43; 44]. The normal probability plot (Figure 5a) shows that the residual values are clustered around a linear line [40]. This indicates that the residual values are normally distributed and the errors of the model are random [43]. The residual versus fits plot (Figure 5b) shows that the residuals are randomly distributed around 0, while the residual versus order plot (Figure 5c) confirms that the residuals are not trending. This indicates that the models are appropriate and provide accurate predictions [43]. In addition, the model shows a good fit, which can be verified by the R\({}^{2}\) and R\({}_{\text{adj}}\)\({}^{2}\) of 96.53 and 96.38%, respectively. The coefficients of determination show that the model could be significant in predicting the response [40]. Additionally, there is significant proximity  
Figure 5: Residual plots: (**a**) normal probability plot; (**b**) versus fits; (**c**) versus order. between Radj2 and R2 values [43]. It is desirable for adjusted and predicted R2 values to be close to each other for the accuracy of the results [43; 44]. Moreover, a simple hypothesis testing technique is also used to detect heteroscedasticity. Heteroscedasticity is a situation in which the variance of the dependent variables varies across the analyzed data [45]. To detect the existence of heteroscedasticity, the test proposed by [PERSON] (1980) was applied, where the null hypothesis is that the variance in the residuals is constant [46]. The Breusch-Pagan test shows that the null hypothesis of homoscedasticity of the residuals cannot be rejected, i.e., the \(p\)-value of the test is greater than 0.05. In the absence of heteroscedasticity in the model, it is common to go for the ordinary least squares (OLS) method for the regression. Table 2 shows the estimating of the significant effects of these factors on the ROP when considering a level of significance of 5%. Here, effect is used to estimate the impact of an independent variable on the dependent variable; standard deviation is a measure of the dispersion or variation of data around the mean; T-value is used to quantify the relation between the observed effect and its variability, i.e., used to test hypotheses; and the \(p\)-value represents significance and can test the validity of the results. The smaller the \(p\)-value, the more significant the result and the higher the feasibility [47]. The analysis of Table 2 reveals the effects of the factors (WOB, RPM, TOR, and FLOW) on ROP, including the effect of two second-order interactions, namely, WOB vs. WOB and RPM vs. RPM, and the effect of five first-order interactions: WOB vs. TOR, WOB vs. FLOW, RPM vs. TOR, RPM vs. FLOW, and TOR vs. FLOW. For the factors shown in Table 2, the statistical model for predicting the ROP can be expressed by Equation (10).