## 2 Materials and Methods
### Methodology  
The methodology selected to analyze the influence of the different factors focuses on the Design of Experiments (DoE), also called the Statistical Design of Experiments. This statistical technique is applied for the planning of an experimental study to determine the relationship between the causes or factors that intervene in the problem and the measurable effects in a real process, and it also reveals the interactions between factors, which are often more important than some individual factors alone [17]. Given the industrial origin of the DoE, its main advantage is the possibility of finding significant conclusions with a small number of trials and, therefore, limited cost. Of course, the technique also has its drawbacks: the analysis of the results can be complex in experiments with a large number of factors; for better precision, a greater number of experiments is necessary, which entails greater expense. Furthermore, the method can lead to erroneous conclusions if statistical aspects such as the normality and homoscedasticity of data and independence of events are not checked, as well as if factors external to the experimentation are not controlled [16].  
It is important not to increase the number of tests with the number of levels to be more than necessary. In this study, two levels are handled for each factor, being coded as \((-1)\) or \((+1)\), even if these are numerical or nominal, according to the previously expected negative or positive influence on the result. Thus, in a full factorial design, the total number of tests will be \(N_{\text{tests}}=2^{k}\). The complete, multifactorial DoE has its origin in the theories of design of experiments by [PERSON] (1924-2012) and [PERSON] (1914-2000) and allows us, in a simple way, to determine whether any factor can be discarded when observing that it does not influence the response variable [18].  
The results obtained are analyzed by calculating the effects and contrasts of the main factors and their interactions, and, using the sum of their squares, the statistical analysis is carried out. The effects of each factor or interaction will be equal to the average of the responses of the high level (\(+1\)) minus the average of the responses of the low level (\(-1\)):  
\[Effect=\frac{\sum\,Y_{+}}{n_{+}}-\frac{\sum\,Y_{-}}{n_{-}} \tag{1}\]  
where \(n\) is the number of data collected at each level and \(Y\) are the values of the responses associated with each level. Once the influences of the factors and interactions have been quantified, their reliability is verified with an analysis of variance (ANOVA) and a diagnosis of the \"residual error\" to validate the statistical assumptions.  
The final result of the experiment is reflected in a mathematical model with a regression equation that predicts the response based on the variation of the factors and/or their most influential interactions, within the values of the levels under study, thus being a valuable aid for the interpretation of the experiment. The form of the equation in general terms will be that of the Scheffler equation:  
\[y=\beta_{0}+B_{1}x_{1}+\beta_{2}x_{2}+\ldots+\beta_{k}x_{k}+\varepsilon \tag{2}\]  
with \(y\) being the dependent variable (the response), \(x_{i}\) (\(i=1,\ldots,k\)) are the set of predictor variables, \(\beta_{i}\) (\(i=1,\ldots,k\)) are the regression coefficients, and \(\varepsilon\) is the residual error. The regression coefficients \(\beta\) are obtained with least squares estimation, defined as half of the effect of a particular factor or interaction [18].