Central composite design

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.

The design consists of three distinct sets of experimental runs:

The design matrix for a central composite design experiment involving k factors is derived from a matrix, d, containing the following three different parts corresponding to the three types of experimental runs:

Then d is the vertical concatenation:

The design matrix X used in linear regression is the horizontal concatenation of a column of 1s (intercept), d, and all elementwise products of a pair of columns of d:

where d(i) represents the ith column in d.

There are many different methods to select a useful value of α. Let F be the number of points due to the factorial design and T = 2k + n, the number of additional points, where n is the number of central points in the design. Common values are as follows (Myers, 1971):

Statistical approaches such as RSM can be employed to maximize the production of a special substance by optimization of operational factors. In contrast to conventional methods, the interaction among process variables can be determined by statistical techniques. For instance, in a study, a central composite design was employed to investigate the effect of critical parameters of organosolv pretreatment of rice straw including temperature, time, and ethanol concentration. The residual solid, lignin recovery, and hydrogen yield were selected as the response variables.

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

...
Wikipedia