CentralCompositeDesigns: Statistical background of central composite designs
Description
Brief description of the statistical background of
central composite designs
Details
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< code="">,
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.
References
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.