# Central composite design

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In statistics, a **central composite design** is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to use a complete three level factorial experiment.

After the designed experiment is performed, linear regression is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.

## Design matrix Edit

The matrix, **d**, for an experiment involving *k* factors consists of the following three different parts:

- The matrix obtained from the factorial experiment. This will be denoted by
**F**. - The centre of the system of interest, denoted in coded variables as (0,0,0,...,0), where there are
*k*zeros. This point is often repeated in order to improve the resolution of the method. This part will be denoted by**C**. - A matrix, with row, where each factor is placed at and all other factors are at zero. The α value is determined by the designer, and it can have just about any value. This part is denoted by
**E**and would look like this:

Thus, the **d** matrix will look like this:

The *X* matrix used in linear regression would be constructed as follows:

where **d**(*i*) represents the column in **d**. The multiplication is to be done memberwise.

### Determining the value of Edit

There are many different methods to determine the value of α. Define , the number of points due to the factorial design and , the number of additional points, where is the number of central points in the design. Common values are as follows (Myers, 1971):

**Orthogonal design:**: , where ;**Rotatable design**: , which is the design implemented by MATLAB’s “ccdesign(k)” function.

## ReferenceEdit

Myers, Raymond H. *Response Surface Methodology*. Boston: Allyn and Bacon, Inc., 1971

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