Central composite design

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:

# A factorial (perhaps fractional) design in the factors studied, each having two levels;
# A set of "centre 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;
# 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:
#The matrix F obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1 and −1.
#The matrix C from the centre points, denoted in coded variables as (0,0,0,...,0), where there are "k" zeros.
#A matrix E from the axial points, with 2"k" 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:

: old E = egin{bmatrix} alpha & 0 & 0 & cdots & cdots & cdots & 0 \ { - alpha } & 0 & 0 & cdots & cdots & cdots & 0 \ 0 & alpha & 0 & cdots & cdots & cdots & 0 \ 0 & { - alpha } & 0 & cdots & cdots & cdots & 0 \ vdots & {} & {} & {} & {} & {} & vdots \ 0 & 0 & 0 & 0 & cdots & cdots & alpha \ 0 & 0 & 0 & 0 & cdots & cdots & { - alpha } \end{bmatrix}.

Then d is the vertical concatenation:: old d = egin{bmatrix} old F \ old C \ old E end{bmatrix}.

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:

:old X = egin{bmatrix} old 1 & old d & old d(1)*old d(2) & old d(1)* old d(3) & cdots & old d(k-1)* old d(k) & old d(1)^2 &old d(2)^2 &cdots & old d(k)^2 end{bmatrix},

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" = 2"k" + "n", the number of additional points, where "n" is the number of central points in the design. Common values are as follows (Myers, 1971):
#Orthogonal design:: alpha = (Q*F/4)^{1/4},!, where Q = (sqrt{F + T} -sqrt{F})^2 ;
#Rotatable design: α = "F"1/4 (the design implemented by MATLAB’s "ccdesign" function).


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

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Design of experiments — In general usage, design of experiments (DOE) or experimental design is the design of any information gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms… …   Wikipedia

  • Composite bow — Reconstruction of a Ming dynasty Kaiyuan bow by Chinese bowyer Gao Xiang. This is a horn, bamboo, sinew composite …   Wikipedia

  • Composite material — A cloth of woven carbon fiber filaments, a common element in composite materials Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent …   Wikipedia

  • Composite Corridor — British Rail Composite Corridor W16071 CK at Llangollen on 28th August 2006 In service 1952– Manufacturer BR Derby (C W), Ashford, Eastleigh, and …   Wikipedia

  • Composite Health Care System — The Composite Health Care System (CHCS) is a VMS based medical informatics system designed by Science Applications International Corporation (SAIC) and used by all United States and OCONUS military health care centers. In 1988, SAIC won a… …   Wikipedia

  • Central Drug Research Institute — The Central Drug Research Institute (CDRI) is one of the first laboratories to be established in India right after its independence. CDRI is among the thirty nine laboratories that are functioning under the aegis of the council of scientific and… …   Wikipedia

  • Optimal design — This article is about the topic in the design of experiments. For the topic in optimal control theory, see shape optimization. Gustav Elfving developed the optimal design of experiments, and so minimized surveyors need for theodolite measurements …   Wikipedia

  • Randomized block design — In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) that are similar to one another. Typically, a blocking factor is a source of variability that is not of primary interest to …   Wikipedia

  • Bayesian experimental design — provides a general probability theoretical framework from which other theories on experimental design can be derived. It is based on Bayesian inference to interpret the observations/data acquired during the experiment. This allows accounting for… …   Wikipedia

  • Completely randomized design — In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take other nuisance variables into account. This article describes completely randomized designs that have one… …   Wikipedia