## 2 Materials and Methods
### Experimental Design: Analysis of Variance (ANOVA) and Response Surface Methodology (RSM)  
Factorial design and response surface were formulated to investigate the effects of water content (u%) and carbon to nitrogen ratio (C/N) on two objective variables, namely (1) percentage of removed diesel oil and (2) amount of produced fluorescein.  
To evaluate the effects of independent factors, namely u% (x\({}_{1}\)) and C/N ratio (x\({}_{2}\)), each factor was investigated at three levels (low, high, and central). In this way, the factorial design was of type 3\({}^{\mathrm{k}}\), where k = 2 were independent factors and 3 was the number of levels at which two factors were varied. Moreover, for the study of C/N ratio another intermediate point was added, using this variable at four levels. In this way, the data could be interpolated with a cubic polynomial.  
Values for u% and C/N ratio were selected according to the values most used for the optimal biological metabolism. The values of the independent variables investigated in factorial design are shown in Table 2.  
Thus, for the experimental design, 12 microcosms were set up, as shown in Table 3, to define the optimal conditions for the biodegradation process.  
The results of the factorial design were expressed by a regression model and shown in a three-dimensional response surface plot. The surface allowed identification of different areas that depended on the effects of the two independent factors (u% and C/N ratio), and the optimum point could be detected. In this work, two response surfaces were calculated, for (1) the diesel oil removal efficiency and (2) the produced fluorescein.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Independent Variable** & **Low Level** & **High Level** & **Central Point(s)** \\ \hline Water content (u\%), x\({}_{1}\) & 8 & 15 & 12 \\ Carbon to nitrogen ratio (C/N), x\({}_{2}\) & 60 & 300 & 120, 180 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental parameters and levels investigated.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Microcosm No.** & **u\% (By Weight)** & **C/N Ratio** \\ \hline
1 & 8 & 60 \\ \hline
2 & 8 & 120 \\ \hline
3 & 8 & 180 \\ \hline
4 & 8 & 300 \\ \hline
5 & 12 & 60 \\ \hline
6 & 12 & 120 \\ \hline
7 & 12 & 180 \\ \hline
8 & 12 & 300 \\ \hline
9 & 15 & 60 \\ \hline
10 & 15 & 120 \\ \hline
11 & 15 & 180 \\ \hline
12 & 15 & 300 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Microcosms for experimental design.  
In both cases, the data can be represented with a third-order regression polynomial:  
\[\text{y}=\beta_{0}+\beta_{1}\text{x}_{1}+\beta_{2}\text{x}_{2}+\beta_{3}\text{x }_{1}^{2}+\beta_{4}\text{x}_{1}\text{x}_{2}+\beta_{5}\text{x}_{2}^{2}+\beta_{6} \text{x}_{1}\text{x}_{2}^{2}+\beta_{7}\text{x}_{2}^{3}+\epsilon \tag{10}\]  
where \(\text{x}_{1}\) is the coded variable that represents the water content and \(\text{x}_{2}\) represents the carbon to nitrogen ratio; \(\beta_{0}\) is the value of the fixed response at the central point of the design and \(\beta_{1}\) (i = 1 to 7) is the linear, quadratic, cubic, and interaction effects regression terms; and \(\epsilon\) is the random error.  
RSM was used to determine the optimal operative conditions. A maximum point was calculated inside factor space and on the surface.  
The analysis of variance allowed the sequential sum of squares (SS), the Fischer's variance ratio (F), the mean squares (MS), and the probability value (p) to be determined. Because the factorial design had no replicates and the number of independent factors was lower than 4, it was not possible to do the analysis considering \(p<0.05\) with a confident level equal to 95% [22]. The data were considered statistically significant when \(p<1\).