A Box-Wilson Central Composite Design, commonly called `a central composite design,' contains an imbedded factorial or fractional factorial design with center points that is augmented with a group of `star points' that allow estimation of curvature. Diagram of central composite design generation for two factors. Toyota Central R&D Labs., Inc 15 Central Composite Design (CCD) Mainly used for quadratic polynomial Parametric design 2-level full factorial design nF = 2k Center point n0 1 Two axial point on axis of each design Variable at distance of design origin. NR = 2k Total Number of design n = 2k + 2k + n 0.
Box & Hunter (1957) recommended a set of orthogonally blocked central composite designs (CCD) when the region of interest is spherical. In order to achieve rotatability along with orthogonal blocking, the block size for those designs becomes unequal and it may not be attractive or practical to use such unequally blocked designs in many practical situations. In this paper, a construction method of orthogonally blocked CCD under the assumption of equal block size is proposed and an index of block orthogonality is introduced.
Central Composite Designs
Central composite designs (CCDs), also known as Box-Wilson designs,are appropriate for calibrating full quadratic models. There are threetypes of CCDs—circumscribed, inscribed, and faced—picturedbelow:
Each design consists of a factorial design (the corners of acube) together with center and star pointsthat allow for estimation of second-order effects. For a full quadraticmodel with n factors, CCDs have enough design pointsto estimate the (n+2)(n+1)/2coefficients in a full quadratic model with n factors.
The type of CCD used (the position of the factorial and starpoints) is determined by the number of factors and by the desiredproperties of the design. The following table summarizes some importantproperties. A design is rotatable if the prediction variancedepends only on the distance of the design point from the center ofthe design.
Design | Rotatable | Factor Levels | UsesPoints Outside ±1 | Accuracy of Estimates |
---|---|---|---|---|
Circumscribed (CCC) | Yes | 5 | Yes | Good over entire design space |
Inscribed (CCI) | Yes | 5 | No | Good over central subset of design space |
Faced (CCF) | No | 3 | No | Fair over entire design space; poor for pure quadratic coefficients |
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ccdesign
:The repeated center point runs allow for a more uniform estimateof the prediction variance over the entire design space.