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


BoxWilson Central Composite Designs  
CCD designs start with a factorial or fractional factorial design (with center points) and add "star" points to estimate curvature 
A BoxWilson 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 


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:
If the factorial is a full factorial, then
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 
 
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:


Additional central composite designs  Examples of other central composite designs will be given after BoxBehnken designs are described. 