## 3 Results
### Regression Model for T-Funnel  
The best response model for the T-Funnel was the 2 FI (two-factor interaction), which obtained a predicted R\({}^{2}\) of 15.58% and an adjusted R\({}^{2}\) of 69.72%, which are values that are considered significant. The percentage difference between the predicted and adjusted values was 54.14% (a value well above 20%), which is not recommended, as the errors are expected to be significant. The model, despite being considered significant in its best response, had a significant lack of fit, which is not ideal, as there are high chances of errors. An inverse transform was recommended by the Box-Cox plot diagnostic to improve the overall model fit. A lambda (\(\lambda\)) equal to \(-1\) was considered within the 95% confidence interval, with its limits ranging between of \(-1.55\) and \(-0.93\). After applying the inverse transform (\(\lambda\)= \(-1\)), the linear model yielded the most significant fit (Table 6). The F-value of 101.78 indicates that there is only a 0.01% probability of such a result being attributed to random noise. Furthermore, the terms A, B, C, and D were statistically significant, as their \(p\)-values were less than 0.05. The lack of fit was not significant, thus further demonstrating the well-fitting property of the model. The F-value for the lack of fit was 0.77, and there was a 69.49% chance that the lack of fit could be attributed to noise. The predicted R\({}^{2}\) was 91.79%, while the adjusted R\({}^{2}\) was slightly higher at 93.29% (ANOVA analysis). These results show a strong correlation between the predicted and actual values. The low standard deviation of 0.0054 reinforces the model's significance. A precision value of 36.02 further supports the model's efficacy. Table 7 shows the estimated coefficients for the T-Funnel linear model. All of the input variables exhibited a standard error of 0.0011, thereby indicating the precision of the estimates. Moreover, the variance inflation factor (VIF) has been calculated to be 1.00, thus corroborating the orthogonality of the factors studied here. The residual normal probability plot in Figure 5a reveals that the data points were closely aligned with the straight line, despite adopting an S-shaped pattern, which indicates a satisfactory distribution of the residuals. The Residuals versus Run plot demonstrates random dispersion, thus lacking any discernible trend. Notably, the scattered data points fell well within the specified limits of \(\pm\)3.54047. Figure 5c shows lambda (\(\lambda\)) values of \(-1\), which lie within the confidence interval (CI) of \(-1.55\) to \(-0.93\). Figure 5d illustrates the relationship between the predicted and actual points, thereby providing an intuitive visual \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Source** & **Sum of Squares** & **Mean Square** & **F-Value** & \(p\)**-Value** \\ \hline Model & 0.0117 & 0.0029 & 101.78 & \(<\)0.0001 \\ A-w/c & 0.0019 & 0.0019 & 65.19 & \(<\)0.0001 \\ B-Sp/p & 0.0001 & 0.0001 & 4.70 & 0.0398 \\ C-w/p & 0.0037 & 0.0037 & 128.64 & \(<\)0.0001 \\ D-s/m & 0.0060 & 0.0060 & 208.57 & \(<\)0.0001 \\ Residual & 0.0007 & 0.0000 & & \\ Lack of Fit & 0.0005 & 0.0000 & 0.7718 & 0.6949 \\ Pure Error & 0.0002 & 0.0000 & & \\ Cor Total & 0.0124 & - & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Fitting results for T-Funnel linear model (inverse). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Factor** & **Intercept** & **A-w/c** & **B-Sp/p** & **C-w/p** & **D-s/m** \\ \hline Coefficient Estimate & 0.0499 & 0.0088 & 0.0024 & 0.0124 & \(-\)0.0158 \\ Standard Error & 0.0010 & 0.0011 & 0.0011 & 0.0011 & 0.0011 \\
95\% CI Low & 0.0479 & 0.0066 & 0.0001 & 0.0102 & \(-\)0.0181 \\
95\% CI High & 0.0519 & 0.0111 & 0.0046 & 0.0147 & \(-\)0.0136 \\ VIF & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Coefficients in terms of coded factors—D-Flow. insight into the model's good predictive performance and accuracy.