In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a central composite design is an experimental design, useful in response surface methodology

Response surface methodology

In statistics, response surface methodology explores the relationships between several explanatory variables and one or more response variables. The method was introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an...

, for building a second order (quadratic) model for the response variable without needing to use a complete three-level factorial experiment

Factorial experiment

In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be...

.

After the designed experiment is performed, linear regression

Linear regression

In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...

 is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.

Implementation

The design consists of three distinct sets of experimental runs:

1. A factorial (perhaps fractional) design in the factors studied, each having two levels;
2. A set of center points, experimental runs whose values of each factor are the medians of the values used in the factorial portion. This point is often replicated in order to improve the precision of the experiment;
3. A set of axial points, experimental runs identical to the centre points except for one factor, which will take on values both below and above the median of the two factorial levels, and typically both outside their range. All factors are varied in this way.

Design matrix

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:

1. The matrix F obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1 and −1.
2. The matrix C from the center points, denoted in coded variables as (0,0,0,...,0), where there are k zeros.
3. A matrix E from the axial points, with 2k rows. Each factor is sequentially placed at ±α and all other factors are at zero. The value of α is determined by the designer; while arbitrary, some values may give the design desirable properties. This part would look like:

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 i^th column in d.

Choosing α

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):

1. Orthogonal design:: , where ;
2. Rotatable design: α = F^1/4 (the design implemented by MATLAB
   MATLAB
   MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
   
   ’s ccdesign function).