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

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

1. The matrix obtained from the factorial experiment. This will be denoted by F.
2. 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.
3. 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 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):

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

## Reference

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