# Central Composite Design (CCD)

Central Composite Design contains an imbedded factorial or fractional factorial design with center points that are augmented with a group of `star points' that allow the estimation of curvature.

If the distance between the center of the design space and a factorial point is ±1 unit for each variable, then the distance between the center of the design space and a star point is ±α with |α| > 1. The precise value of α depends on certain properties desired for the design and on the number of factors involved. The star points represent new extreme values (low and high) for each variable in the design. Similarly, the number of center point runs that the design is to contain also depends on certain properties required for the design.
Table 1. Values of α Defining the Type of Central Composite Design
CCD Type Terminology Definition
Circumscribed CCC Designs are the original form of the central composite design. The star points are at some distance α 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. These designs have circular, spherical, or hyperspherical symmetry and require five levels for each variable. Augmenting an existing factorial or resolution V fractional factorial design with star points can produce this design.
Inscribed CCI Uses the variable 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 variable level of the CCC design altered to generate the CCI design). This design also requires five levels of each variable.
Note: Used for situations in which the limits specified for variable settings are truly limited.
Face Centered CCF

The star points are at the center of each face of the factorial space, so α = ± 1. This variety requires three levels of each variable. Augmenting an existing factorial or resolution V design with appropriate star points can also produce this design.

Note: 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.

The Box Behnken design and the CCF Central Composite Design can be visualized as near compliments of each other. They both essentially suppress selected runs from a Full Factorial matrix in an attempt to maintain the higher order surface definition. For example, for three three-level variables, the Full Factorial run size is 27. The central composite plan drops all of the middle edge nodes, resulting in only fifteen runs. The Box Behnken design is nearly the opposite in that it uses the twelve middle edge nodes and the center node to fit a 2nd order equation. Central Composite Design plus Box Behnken becomes a Full Factorial with extra samples taken at the center.

## Usability Characteristics

• Generally used for fitting a second-order response surface.
• In HyperStudy, the number of centre runs and axial distance, a, are parameters that you need to enter. HyperStudy also offers some preset values for a.
Preset Name Axial Distance No. of Centre Runs
Rotatable 2 User Defined
Orthogonal 1.41421 User Defined
Rotatable & Orthogonal 2 12
On Face 1 User Defined
User Defined User Defined User Defined
• The total number of runs is a function of the number of input variables and the number of center points as the Total runs = 2^k + 2k + N center points (k = input variables).
• When using Central Composite Design, HyperStudy has a limit of 20 input variables.
• Any data in the inclusion matrix is combined with the run data for post-processing. Any run matrix point which is already part of the inclusion data will not be rerun.

## Settings

In the Specifications step, Settings tab, change method settings.
Parameter Default Range Description
Axial Distance 0 0 Automatically calculated when ityp is 1 ~ 4; it can be modified when ityp = 5.
Inscribe On Off or On Choose whether to force the points within the input variable bounds or not.
Off
Points do not need to fill in the bounds.
On
Points need to fill in the bounds.
Center Runs 1   Automatically determined when ityp = 3; you can modify it if ityp is not equal to 3.
Type Rotatable
• Rotatable
• Orthogonal
• Rotatable & Orthogonal
• On Face
• User Specified
Type of axial scaling.
Use Inclusion Matrix Off Off or On Concatenation without duplication between the inclusion and the generated run matrix.