5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?
5.3.3.6. Response surface designs

Central Composite Designs (CCD)

Box-Wilson Central Composite Designs
CCD designs start with a factorial or fractional factorial design (with center points) and add "star" points to estimate curvature A Box-Wilson Central Composite Design, commonly called 'a central composite design,' contains an imbedded factorial or fractional factorial design with center points that is augmented with a group of 'star points' that allow estimation of curvature. If the distance from the center of the design space to a factorial point is ±1 unit for each factor, the distance from the center of the design space to a star point is |$$\alpha$$| > 1. The precise value of $$\alpha$$ depends on certain properties desired for the design and on the number of factors involved.

Similarly, the number of centerpoint runs the design is to contain also depends on certain properties required for the design.

Diagram of central composite design generation for two factors

FIGURE 3.20: Generation of a Central Composite Design for Two Factors

A CCD design with k factors has 2k star points A central composite design always contains twice as many star points as there are factors in the design. The star points represent new extreme values (low and high) for each factor in the design. Table 3.22 summarizes the properties of the three varieties of central composite designs. Figure 3.21 illustrates the relationships among these varieties.
Description of 3 types of CCD designs, which depend on where the star points are placed
 Central Composite Design Type Terminology Comments Circumscribed CCC CCC designs are the original form of the central composite design. The star points are at some distance from the center based on the properties desired for the design and the number of factors in the design. The star points establish new extremes for the low and high settings for all factors. Figure 5 illustrates a CCC design. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. Augmenting an existing factorial or resolution V fractional factorial design with star points can produce this design. Inscribed CCI For those situations in which the limits specified for factor settings are truly limits, the CCI design uses the factor settings as the star points and creates a factorial or fractional factorial design within those limits (in other words, a CCI design is a scaled down CCC design with each factor level of the CCC design divided by $$\alpha$$ to generate the CCI design). This design also requires 5 levels of each factor. Face Centered CCF In this design the star points are at the center of each face of the factorial space, so $$\alpha$$ = ± 1. This variety requires 3 levels of each factor. Augmenting an existing factorial or resolution V design with appropriate star points can also produce this design.
Pictorial representation of where the star points are placed for the 3 types of CCD designs

FIGURE 3.21: Comparison of the Three Types of Central Composite Designs
Comparison of the 3 central composite designs The diagrams in Figure 3.21 illustrate the three types of central composite designs for two factors. Note that the CCC explores the largest process space and the CCI explores the smallest process space. Both the CCC and CCI are rotatable designs, but the CCF is not. In the CCC design, the design points describe a circle circumscribed about the factorial square. For three factors, the CCC design points describe a sphere around the factorial cube.
Determining $$\alpha$$ in Central Composite Designs
The value of $$\alpha$$ is chosen to maintain rotatability To maintain rotatability, the value of $$\alpha$$ depends on the number of experimental runs in the factorial portion of the central composite design:

$$\alpha = [ \mbox{number of factorial runs} ]^{1/4}$$

If the factorial is a full factorial, then

$$\alpha = \left[ 2^{k} \right] ^{1/4}$$

However, the factorial portion can also be a fractional factorial design of resolution V.

Table 3.23 illustrates some typical values of $$\alpha$$ as a function of the number of factors.

Values of $$\alpha$$ depending on the number of factors in the factorial part of the design
TABLE 3.23: Determining $$\alpha$$ for Rotatability
Number of
Factors
Factorial
Portion
Scaled Value for $$\alpha$$
Relative to ±1

2 22 22/4 = 1.414
3 23 23/4 = 1.682
4 24 24/4 = 2.000
5 25-1 24/4 = 2.000
5 25 25/4 = 2.378
6 26-1 25/4 = 2.378
6 26 26/4 = 2.828
Orthogonal blocking The value of $$\alpha$$ also depends on whether or not the design is orthogonally blocked. That is, the question is whether or not the design is divided into blocks such that the block effects do not affect the estimates of the coefficients in the second order model.
Example of both rotatability and orthogonal blocking for two factors Under some circumstances, the value of $$\alpha$$ allows simultaneous rotatability and orthogonality. One such example for k = 2 is shown below:

 BLOCK X1 X2 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 0 0 1 0 0 2 -1.414 0 2 1.414 0 2 0 -1.414 2 0 1.414 2 0 0 2 0 0

Additional central composite designs Examples of other central composite designs will be given after Box-Behnken designs are described.