## 3 Results and Discussion
### Reduced Model Equation for Phenol Conversion (\(Y_{1}\)) and Hydrogen Yield (\(Y_{2}\))  
In the reduced regression model, all of the insignificant terms were removed and only the variables with a \(p\)-value of lower than 0.05 were considered. The Pareto chart of the model after reduction at \(p\) = 0.05 with 10 degrees of freedom (DF) for \(Y_{1}\) and nine for \(Y_{2}\) is presented in Figure 7. All the values were greater than 2.07 (\(p\) = 0.05). Moreover, the model is statistically significant as indicated by a \(p\)-value less than 0.05 (95% confidence). The reduced linear regression model for production of hydrogen from phenol was fitted for \(Y_{1}\) and \(Y_{2}\) through the application of multiple regression analysis on the experimental data and subsequent removal of insignificant terms as shown in Equations (8) and (9). From the analysis of the regression coefficient, the value of the model helps in making inferences about whether there is a need for a complex model for better fit. The significances of the models were determined by the experimental \(F\)-value derived from the regression coefficient analysis. On the other hand, the tabulated \(F\)-values are calculated based on the degree of freedom for regression and residual at a specific confidence level. If the \(F\) test for the model is significant at the 5% level (\(p\) < 0.05), then the model is fit and can adequately account for the variation observed. The ratio of the mean square due to regression to that due to error is depicted by the \(F\) value. Generally, a good model should have its calculated \(F\) value several times greater than the tabulated. Tables 5 and 6 depict the reduced model coefficients of the factors and interactions, and the \(p\)-values of the effects in the model are shown for \(Y_{1}\) and \(Y_{2}\), respectively. According to the analysis of the regression coefficient of the reduced regression models, the models are highly significant. This was evident from the Fisher (\(F\)) test. The ratio of the mean square regression to mean square residual was 121.1 for \(Y_{1}\) and 60.13 for \(Y_{2}\) responses. Also, the probability value was very low (\(p\)-value = 0.000). \[Y_{1}=-1.87+0.01A-3.13B+4.99C+0.24D+1.18E+0.01 AE-6.3 BE+6.23 CE-1.94D \tag{8}\]  
\[Y_{2}=15.7+0.004A-3.8B+22.16C+0.62D+1.35E+0.004 AE-2.98 BE-1.38 CE \tag{9}\]  
The coefficient of determination (\(R^{2}\)) and adjusted \(R^{2}\) are among other criteria that are commonly used to show the best fitness of a model. Tables 5 and 6 show the calculated \(R^{2}\) and adjusted \(R^{2}\) values for the reduced models developed with respect to phenol conversion (\(Y_{1}\)) and the hydrogen yield (\(Y_{2}\)) responses, respectively. From these values, it was clear that the regression model was adequate. Figure 7: Pareto chart of the reduced model for (**a**) \(Y_{1}\) and (**b**) \(Y_{2}\). The model reduction has little influence on the predicted values of \(Y_{1}\) and \(Y_{2}\), as indicated by the small change of the \(R^{2}\) values from the analysis of the regression coefficient results (Tables 5 and 6). The validity of the regression model developed was further tested by drawing the parity charts. The parity plot helps in detecting any outliers and provides a prompt indication of the accuracy of the correlation. The observed values and predicted values of the responses are scattered close to the 95% prediction limits with 45\({}^{\circ}\), indicating an almost perfect fit of the developed reduced model. Figure 8 shows the parity plot of the experimental and predicted values for the reduced model. In all the cases, one or two data points out of 32 fall outside the 95% prediction limits (as shown with blue dotted lines).