# Design

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.

Equation 2

## 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 2^{2} design. Therefore, the design in *Figure 7* (b) becomes a 3^{2} 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 2^{2} 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 2^{2} 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 2^{2} 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*.

Table 1

*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 (n_{c}=n_{s}=1),

Table 2

*Suggested Central Composite Designs *(Kutner et al., 2005)