A Central Composite Design (CCD) is a specialized statistical experimental design primarily used in Response Surface Methodology (RSM). It is formulated by combining specific types of experimental points to efficiently explore a multidimensional space and model complex relationships between factors and responses.
Based on the provided information, a Central Composite Design is explicitly formulated by combining:
- Factorial Points: These form the core of the design.
- Axial Points: These points are located along the axes at a specific distance (alpha, α) from the center.
- Center Points: These are replicates run at the center of the design space.
This combination of points allows the CCD to estimate not only linear and interaction effects (from factorial points) but also quadratic effects (from axial and center points), which is crucial for modeling curvature in the response surface and identifying optimal conditions.
Components of a Central Composite Design
Understanding the components is key to grasping the formulation:
- Factorial Points: These represent the corners of the experimental region defined by the factor levels. For k factors, there are 2^k factorial points. They are typically run at the "low" and "high" levels (-1 and +1 in coded units).
- Axial Points (Star Points): These points are situated at a distance 'α' from the center along each factor axis. For k factors, there are 2k axial points. The distance α determines the design's properties (e.g., rotatability, orthogonal blocking).
- Center Points: These points are located at the origin (0,0,...0) in the coded design space. Running multiple replicates at the center allows for estimating experimental error and checking for lack-of-fit.
In summary, the structure is:
- 2^k Factorial Points
- 2k Axial Points
- n_c Center Points (where n_c is the number of center point replicates)
The total number of experimental runs in a CCD is thus N = 2^k + 2k + n_c.
Why This Formulation?
The unique combination of these three types of points is the power behind the CCD:
- Factorial Points: Provide information about main effects and two-factor interactions.
- Axial Points: Extend the design space beyond the factorial points and, combined with factorial points, enable the estimation of quadratic terms.
- Center Points: Allow for assessing pure error and testing the curvature of the response surface.
This specific formulation allows the design to support the fitting of a second-order polynomial model, making it suitable for optimizing processes or product formulations where non-linear relationships exist.
