Response surfaces are usually approximated by a second-order regression model as the higher-order effects are usually unimportant (Box et al., 2005). A second-order regression model (also known as the full quadratic) for k number of factors can be written as in Equation 2. In addition to the most popular method, the central composite design, CCD, Box-Behnken Design will also be demonstrated in the following sections.
Central Composite Design of RSM
Video 3 demonstrates the central composite design.
Video 3. Response Surface Methodology Basic, the Central Composite Design Explained
The most popular method of response surface design is the Central Composite Design, CCD. The CCD is a two-level full factorial or fractional factorial design with added center points and the axial points (also known as star points) as shown in Figure 7. While the center point is added at the center, the axial points are applied in the middle of the levels of a factor for each level of the other factors. Therefore, the coordinators for the axial points are (-1, 0), (1,0), (0, -1), and (0, 1). For this design in Figure 7 (b), the axial (star) points are placed on the face of the square box of the 22 design. Therefore, the design in Figure 7 (b) becomes a 32 factorial for which a full quadratic model (second-order regression) can be fitted for the response surface. With the addition of multiple center points, the lack-of-fit could also be tested. The distance from the center and the axial points are denoted by α (α=1 for this design in Figure 7 (b)).
Figure 7. Central Composite Design, CCD (right) from the 22 Factorial Design (left)
The CCD in Figure 7 (b) is enough to develop a full quadratic model. However, the axial points can be designed more systematically to get even more information from the same number of experiments. For example, the CCD in Figure 8 provides a couple of more advantages over the CCD in Figure 7 (b). As compared to the CCD in Figure 7 (b) with only three levels for each factor allows only up to the full quadratic model, the CCD in Figure 8 consists of five distinctive levels for each factor, which allows for the higher-order model such as the cubic model if there is any lack-of-fit observed in the quadratic model.
Figure 8. Central Composite Design, CCD from the Base 22 Factorial Design with Alpha Value 1.41 for Rotatability
The systematic placement of the axial points along the perimeter of the circle drawn for a 22 factorial design using the corner points provides some additional advantages, including the rotatability function of the design. Rotatability functionality of the design provides good prediction over the interested range of independent variables (x-variables or the predictor variables in RSM and regression). A good prediction model is defined by the ability to produce consistent and stable variance over the entire ranges of the independent/predictor variables. Figure 9 shows a good prediction model with consistent and stable variances over the entire range of the independent variables. Any point from the center has equal variance as this design is rotatable (Figure 9). The rotatability can also be seen in Figure 8.
Figure 9. Central Composite Design, CCD with Consistent and Stable Variance
Video 4 demonstrates the design process for the central composite design using MS Excel.
Video 4. Response Surface Methodology (RSM) Central Composite Design using MS Excel
Video 5 demonstrates the design process for the central composite design using Minitab.
Video 5. Response Surface Design Layout Construction using Minitab
Table 1 provides a two-factor central composite design, CCD with five replications at the center point. The design uses the alpha value of 1.41. The design in Table 1 is simply the arranged coordinates (standard order) of the design shown in Figure 8.
Two-Factor Central Composite Design with 5 Replications at the Center Point
A generalization for the two-factor design is provided in Table 2 for some typical and useful central composite designs. Many of these suggested designs can be found in most statistical software. Generally, for the common central composite designs the following is a guideline for determining the total number of experimental trials (Kutner et al., 2005).
Finally, the alpha value for rotatability is calculated using the following equation.
For example, for two-factor central composite design without any replications for the corner and the axial points (nc=ns=1),
Suggested Central Composite Designs (Kutner et al., 2005)