The 44 CCD runs are shown in Appendix 3 in Supplementary File. Particular information (P value, lack of fit, F-static, and R-squared (multiple and adjusted) was used to specify the response-surface portion of each model (
21). The outputs of the models are presented in Appendix 4 in Supplementary File. Checking the lack of fit value was carried out for all studied models and its insignificancy was observed for all models, but a higher value was seen for the reduced model (
22,
28). According to the comparison of models’ outputs, the reduced model was gorgeous well-fitted model. The chosen/well-fitted model should have the greatest F-value, smallest P value, highest R
2, and smallest lack of fit. So, the insignificant factors (including; x
1, x
2 interaction, x
1, x
3 interaction, x
1, x
4 interaction, x
2, x
3 interaction, x
2, x
4 interaction, x
3, x
4 interaction, x
12, and x
42) are abandoned, and the best model was achieved. An ANOVA was applied to assess model adequacy (
20,
24,
29). As presented in Appendix 5 in Supplementary File, four terms (R
2, P value, lack of fit, and F-value) were applied to appraise the adequacy of the model (
21). The P and F-values are ascribed to the significance of the model’s terms and the most significant influences of them, respectively. The results of these computations (F-value, 85.93 on 6 and 37 DF; P value, 0.03; lack of fit, > 0.05; and R
2, 0.93) indicated that the reduced model fits well for La (III) removal by CS@nZVI. This model also provided good congruence between the actual data (
30) and predicted values with high R
2 (equal to 0.917), as represented in
Figure 5.
In addition, for a powerful model, the related correlation coefficients (R-adjusted and multiple) should be close to each other; otherwise, non-important parameters can be present in the model and damage the validity of the model (
20,
29). In our outputs, given the proximity of these two values to each other (0.933 and 0.922, respectively), we can use the model for further studies like optimization. Regression analysis outputs on the reduced model are given in Appendix 6 in Supplementary File. Accordingly, a list of independent variables with significant effects (x
1, x
2, x
3, x
4, x
22, and x
32) on the removal performance was present in the model. Of note, x
1, and quadratics of x
2, and x
3 have a negative influence on the model’ output since x
2, x
3, and x
4 have positive influence on the model’ output. Also, based on the coefficients analysis, x
4, and x
2 were the most significant parameters influencing the selected responses; as reported previously by Kumar Gupta et al. (
31). The obtained model equation is given in Equation 9.