## 3 Results and Discussion
### Optimize Parameters Using a Chemometric Approaches  
The results from the CCD were fitted to a quadratic polynomial using several regressions to obtain the following quadratic equation from the encoded data. \[ \begin{array}{l}\text{EE}\%\text{ BES}=71.9718+15.9039\text{A}+8.0273\text{B}+1. 8629\text{C}+4.6801\text{D}+4.5483\text{AB}+\\ 2.0862\text{AC}+1.2078\text{AD}+0.79794\text{BC}+3.7513\text{BD}-3.9001\text{CD} -21.147\text{A}^{2}-\\ 16.705\text{B}^{2}-4.8185\text{C}^{2}-1.5980\text{D}^{2}\text{ (R}^{2}=0.9239)\\ \end{array}  \tag{2}\]  
\[ \begin{array}{l}\text{EE}\%\text{ TES}=87.151+19.155\text{A}+9.9871\text{B}+2.1136\text{C}+6.2539\text{D}+5.4244\text{AB}+\\ 2.7738\text{AC}+1.9213\text{AD}+0.7777\text{5 BC}+5.0723\text{BD}-3.9554\text{CD} -25.545\text{A}^{2}-\\ 19.826\text{B}^{2}-2.4962\text{C}^{2}-6.2847\text{D}^{2}\text{ (R}^{2}=0.9108)\\ \end{array}  \tag{3}\]  
Figure 5: The influence of pH sample solution on the peak area of the extracted analytes using HDESMBME. Other extraction conditions as listed in Table 1. Error bars correspond to standard deviation. \begin{table}
\begin{tabular}{c c c c} \hline  & **BES** & **TES** & **PRO** \\ \hline Binding Mode 1 & \(-11.5\) & \(-13.9\) & \(-13.3\) \\ Binding Mode 2 & \(-11.4\) & \(-15.8\) & \(-15.8\) \\ \hline \end{tabular}
\end{table} 
Table 3: PM3 estimates the interaction energies (in kcal.mol\({}^{-1}\)) of the studied steroid hormones with 4:1 HDEs. EE% PRO = 84.79295 + 17.64394A + 9.32583B + 2.40044C + 6.11878D + 5.21419 AB + 2.79081 AC + 1.93831 AD + 0.52206 BC + 4.91131 BD - 3.83681 CD - 21.93324 A\({}^{2}-18.87824\)B\({}^{2}-4.70274\)C\({}^{2}-6.13174\)D\({}^{2}\) (R\({}^{2}\) = 0.9177) where EE% is the extraction efficiency for selected steroid hormones, A is the number of HDE-SBME, B is the extraction time, C is the volume of the elution solvent, and D is the elution time. In Equation (2), the observed positive value, 71.9718, is the intercept or baseline value of the BES EE% response when all factors are at their low levels. The coefficients for factors A, B, C, and D are positive, indicating that increasing their level will positively affect the EE% response. The quadratic terms for A\({}^{2}\), B\({}^{2}\), C\({}^{2}\), and D\({}^{2}\) factors are negative, indicating that their effect on the response becomes weaker as their level increases beyond a certain point. Furthermore, it is noteworthy that the coefficients pertaining to the interaction terms AB, AC, AD, BC, and BD exhibit a positive value, signifying that the cumulative impact of these factors surpasses the summation of their individual effects. Conversely, the combination of CD yields a negative effect, implying that the collective influence of these factors is inferior to the summation of their individual effects. Equation (3) with a baseline value of and Equation (4) with a baseline value of 84.79295 have the same behavior between all tested factors, quadratic terms, and their interaction terms (A, B, C, D, A\({}^{2}\), B\({}^{2}\), C\({}^{2}\), D\({}^{2}\), AB, AC, AD, BC, and BD) with the EE% response for TES and PRO, respectively. The analysis variances (ANOVA) for the selected steroids EE% of the HDE-SBME method are shown in Tables S2-S4. Regarding the estimated \(p\)-value derived from the ANOVA outcomes, it can be inferred that the impact of extraction conditions on the response of all chosen steroids was deemed significant (\(p\) < 0.0001). The regression equations obtained showed that the coefficients of determination (R\({}^{2}\)) were 0.9239, 0.9108, and 0.9177 for BES, TES, and PRO, respectively. These estimate the fraction of overall variation in the data accounted for by the steroid models. Thus, all models can explain 92.39%, 91.08%, and 91.77% of the variation in response for BES, TES, and PRO, respectively. Also, the 'Model F-value' of BES = 25.27, TES = 21.34, and PRO = 23.30 implied that the models were significant.