2.4.2. Effect Estimation

The real values *X* have been calculated according to Equation (1).

$$X = \frac{\mathbf{x} - \mathbf{x}\_0}{\Delta \mathbf{x}} \tag{1}$$

Where *X*, is the coded value for the independent variable, *x*, is the natural value, *x0*, is the natural value at the center point and <sup>∆</sup>*x*, is the step change value (the half of the interval (−1 +1)). The mathematical model describing the relation between dependent and independent variables for this process has the quadratic form for the experimental design used:

$$Y\_i = \beta\_0 + \beta\_1 X\_1 + \beta\_2 X\_2 + \beta\_3 X\_3 + \beta\_{11} X\_1^2 + \beta\_{22} X\_2^2 + \beta\_{33} X\_3^2 + \beta\_{12} X\_1 X\_2 + \beta\_{13} X\_1 X\_3 + \beta\_{23} X\_2 X\_3 \tag{2}$$

where *Y*<sup>i</sup> , is the predicted response (in our case, the Biomass production (g/L); *β*0, is offset term; *β*1, *β*2, *β*<sup>3</sup> are the linear effects (showing the predicted response); *β*11, *β*22, *β*<sup>33</sup> are the squared effects *β*12, *β*13, *β*<sup>23</sup> are the interaction terms and *X*1, *X*2, *X*<sup>3</sup> are the independent variables. The calculation of the effect of each variable and the establishment of a correlation between the response *Y*<sup>i</sup> and the variables *X*, were performed using a Minitab 16 software (Minitab, Inc., State College, PA, USA).
