## 2 Materials and Methods
### Analysis of Prediction Model
This suggests that the variance of residuals is less than or equal to the variance within groups, thereby confirming the suitability of the model. Following the assumptions outlined and validated in Tables 5 and 6, the parameter estimates for the model are provided in Table 7. The significance of individual effects and their interactions is assessed at a significance level of \(\alpha=0.05\). Table 5 illustrates the estimated effect sizes for each component of the model, which quantifies the influence of various factors on the thickness of the formed layer. Furthermore, the interaction between variables such as concentration, cathodic deposition time, degreasing temperature, and immersion time in the degreasing agent is shown to have a substantial impact on the outcome. \begin{table}
\begin{tabular}{c c} \hline \hline
**Source** & **Value** \\ \hline RSquare Adj & 0.707869 \\ Root Mean Square Error & 1.922936 \\ Mean of Response & 17.71049 \\ Observations (or Sum Wgts) & 82 \\ \hline \end{tabular}
\end{table} 
Table 4: Summary of the model’s suitability. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Sum of Squares** & **Mean Square** & **F Ratio** & **Prob > F** \\ \hline Model & 12 & 770.1278 & 64.1773 & 17.3561 & \textless{}0.0001 \\ Error & 69 & 255.1402 & 3.6977 & - & - \\ C. Total & 81 & 1025.268 & - & - & - \\ \hline \end{tabular}
\end{table} 
Table 5: Analysis of variance (ANOVA). Based on the estimated parameters, it is possible to construct the equation in a coded scale:  
\[ \begin{split}\widehat{y}=15.261-2.089\cdot x_{1}+0.904\cdot x_{5}+0. 806\cdot x_{6}-0.582\cdot x_{3}+3.454\cdot x_{2}-0.517\cdot x_{1}\cdot x_{5}-0.680 \cdot x_{1}\cdot x+0.710\cdot x_{3}\cdot x_{2}+\\ 0.516\cdot x_{1}\cdot x_{5}\cdot x_{3}-0.478\cdot x_{4}-0.689\cdot x_{1} \cdot x_{5}\cdot x_{6}\cdot x_{2}+0.946\cdot x_{1}\cdot x_{5}\cdot x_{3}\cdot x_{2 }\end{split}  \tag{1}\]  
To establish the predictive relationship in the natural scale, it is important to note that the factors used during the analysis were encoded in the coded scale through design of experiments normalization:  
\[x_{\text{d}}(\text{i})=\frac{\text{x}_{\text{i}}-\frac{\text{x}_{\text{max}}+ \text{x}_{\text{min}}}{2}}{\frac{\text{x}_{\text{max}}-\text{x}_{\text{min}}} {2}} \tag{2}\]  
where \(x_{\text{d}}(\text{i})\) represents the coded variable according to the design of experiments, and \(x(\text{i})\) denotes the original base variable, with i representing the set of integers corresponding to the number of base factors. \(\text{x}_{\text{max}}\) is the maximum value of the original variable x(i), and \(\text{x}_{\text{min}}\) is the minimum value of the original variable \(x(\text{i})\). Using the coded equation and the statistical equation, it is possible to define the predictive equation in the natural scale that describes the investigated relationship representing the thickness of the formed layer th:  
\[ \begin{split} th=0.2348441563\cdot DEGR_{con}+0.34283975\cdot DEGR_{ term}-0.3041498125\cdot DEGR_{temp}+3.10826525\cdot KTL_{tim}\\ +0.11258322\cdot KTL_{UI}-0.01334178812\cdot DEGR_{con}+0.00041859125 \cdot DEGR_{con}-0.0000029899375\cdot DEGR_{con}-0.00484984375\cdot\\ DEGR_{tim}-0.02196685\cdot DEGR_{con}-DEGR_{tim}+0.0062359375\cdot DEGR_{ con}\cdot DEGR_{temp}+0.0484984375\cdot\\ DEGR_{tim}\cdot DEGR_{temp}-0.07589475\cdot DEGR_{con}\cdot KTL_{tim}-0.00275624\cdot DEGR_{con}\cdot KTL_{tim}-0.0164625\cdot\\ DEGR_{con}\cdot DEGR_{tim}\cdot KTL_{UI}-0.0012471875\cdot DEGR_{ con}\cdot DEGR_{temp}\cdot KTL_{tim}-0.0082796875\cdot DEGR_{tim}\\ \cdot DEGR_{temp}\cdot KTL_{tim}+0.000551248\cdot DEGR_{con}\cdot KTL_{tim} \cdot KTL_{tl}+0.00241171\cdot DEGR_{tim}\cdot KTL_{tim}.\\ KTL_{UI}+0.0002365625\cdot DEGR_{con}\cdot DEGR_{tim}\cdot DEGR_{ temp}\cdot KTL_{tim}-0.000068906\cdot DEGR_{con}\cdot DEGR_{tim}\\ \cdot KTL_{tim}\cdot KTL_{ll}+8.664520102\end{split}  \tag{3}\]