Brief description of the statistical background of central composite designs
Ulrike Groemping
Central composite designs (ccd's) were invented by Box and Wilson (1951) for response surface experimentation with quantitative factors. They are used for estimation of second order response surface models, i.e. models that allow to estimate linear, quadratic and interaction effects for all factors.
Central composite designs consist of a cube and star points (also called axial points). Both the cube and the star portion of the design should have some center points. The cube is a (fractional) factorial design and should be at least of resolution V. The line between the center points and the star points intersects the faces of the cube in their middle (see the link to the NIST/Sematech e-Handbook for a visualization). There are two star points per factor, i.e. the number of runs for (each block of) the star portion of the design is twice the number of factors plus the number of center points in the star portion.
The tuning parameter alpha determines whether the star points lie on the
faces of the cube (alpha=1, face-centered), inside the cube (alpha<1,
inscribed) or outside the cube (alpha>1, circumscribed).
The latter case is the usual one. The value of
alpha can be chosen such that the design is rotatable (may be useful if the scales
of the factors are comparable) or such that the design is orthogonally blocked
(i.e. the block effects do not affect the effect estimates of interest). The default
is to generate orthogonally blocked designs.
Central composite designs are particularly useful in sequential experimentation, where a (fractional) factorial with center points is followed up by a star portion of the design. While the cube can already estimate the linear and interaction effects, the center points can only estimate the sum of all quadratic effects. If this indicates that quadratic effects are important, a star portion can be added in order to investigate the model more deeply.
Box, G.E.P., Hunter, J.S. and Hunter, W.G. (2005, 2nd ed.). Statistics for Experimenters. Wiley, New York.
Box, G.E.P. and Wilson, K.B. (1951). On the Experimental Attainment of Optimum Conditions. J. Royal Statistical Society, B13, 1-45.
NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm, accessed August 20th, 2009.
Myers, R.H., Montgomery, D.C. and Anderson-Cook, C.M. (2009). Response Surface Methodology. Process and Product Optimization Using Designed Experiments. Wiley, New York.
See Also ccd, ccd.design, ccd.augment