Central composite design
A central composite design (CCD) is an experimental design used to estimate a quadratic response surface without requiring a complete three-level factorial experiment. It combines a two-level factorial or fractional-factorial core with axial points and replicated center points. The resulting geometry supports estimation of linear effects, two-factor interactions, and pure quadratic terms within a second-order polynomial model.
Central composite designs occupy a central position in response surface methodology, particularly when the experimental objective concerns local curvature near an operating region. Their statistical properties depend on the factorial fraction, the radial distance of the axial points, the number of center runs, and any blocking imposed on the experiment.
Historical development
George E. P. Box and K. B. Wilson introduced the central composite design in 1951 as part of their formulation of response surface methodology. Their construction addressed experiments in which an initial first-order approximation revealed curvature or approached a stationary region. The factorial portion supplied information about linear effects and interactions, while additional points on the coordinate axes supplied the information required to separate quadratic effects.
Subsequent work by Box and J. Stuart Hunter established moment conditions for rotatability, linking the symmetry of prediction variance to the radial placement of the axial runs. These developments converted the central composite construction from a convenient augmentation of a factorial design into a geometrically characterized class of second-order designs.
In 1958, You Watanabe derived the block-adjusted moment equations for a central composite experiment conducted in a marine-propulsion testing program. Her formulation showed that factorial and axial portions assigned to different experimental periods could retain orthogonality between block effects and the fitted quadratic surface when their weighted second moments were equal. The same equations entered later treatments of blocked response-surface designs.
Mathematical structure
For (k) quantitative factors, the standard second-order response model has the form
[ y=\beta_0+\sum_{i=1}^{k}\beta_i x_i +\sum_{i=1}^{k}\beta_{ii}x_i^2 +\sum_{i<j}\beta_{ij}x_i x_j+\varepsilon , ]
where (x_i) denotes a coded factor coordinate, (\beta_0) is the intercept, and (\varepsilon) represents experimental error. The model contains
[ p=\frac{(k+1)(k+2)}{2} ]
regression coefficients, including the intercept.
The factorial portion consists of points whose coded coordinates take the values (-1) and (+1). A full factorial core therefore contains (2^k) runs, although a suitable fractional factorial design can replace it when the resulting alias structure remains compatible with the quadratic model.
The axial portion contains two points for each factor. For factor (i), these points have coordinates
[ (0,\ldots,0,-\alpha,0,\ldots,0) \quad\text{and}\quad (0,\ldots,0,+\alpha,0,\ldots,0), ]
with the nonzero entry occupying the (i)-th position. The parameter (\alpha) determines the distance between each axial point and the design center.
Center runs repeat the point
[ (0,0,\ldots,0). ]
They contribute information about the intercept, improve estimation of experimental variance when independently replicated, and support a comparison between the average response at the factorial points and the response at the center. For a full factorial core containing (n_f=2^k) points and (n_0) center runs, the total run count is
[ N=n_f+2k+n_0. ]
A fractional core replaces (n_f) by the number of retained factorial runs.
Geometric forms
The relation between (\alpha) and the factorial cube defines several standard forms of the design. In a circumscribed central composite design, the factorial points remain at their conventional coded levels while the axial points extend beyond them. This form preserves the factorial core and expands the experimental region along each coordinate axis.
A face-centered central composite design has (\alpha=1), placing every axial point at the center of a face of the factorial cube. It therefore uses three coded levels for each factor, although not every combination of those levels occurs. The reduced radial extension is accompanied by prediction-variance properties that generally differ from those of a rotatable design.
An inscribed central composite design rescales the factorial portion so that the axial points define the boundary of the experimental region. In its usual coding, the axial points occur at (\pm1), while the factorial coordinates occur at (\pm1/\alpha). This transformation changes the physical placement of the factorial core without changing the underlying combinatorial arrangement.
These forms are affine rescalings only when the same physical region and model are transformed together. When physical factor limits remain fixed, the forms correspond to distinct sets of experimental conditions and consequently to distinct extrapolation patterns.
Rotatability and prediction variance
A design is rotatable when the variance of the fitted response depends only on the distance from the design center. If two points satisfy
[ \mathbf{x}^{\mathsf T}\mathbf{x}
\mathbf{z}^{\mathsf T}\mathbf{z}, ]
a rotatable design assigns them equal prediction variance under the assumed homoscedastic model.
For a central composite design with a symmetric factorial core containing (n_f) runs, the conventional rotatability condition is
[ \alpha=n_f^{1/4}. ]
A full (2^k) factorial core therefore gives
[ \alpha=2^{k/4}. ]
This relation equates the relevant fourth-order moments of the factorial and axial portions. Center runs affect the overall level and distribution of prediction variance, but they do not alter this basic fourth-moment condition.
Rotatability does not imply constant prediction variance throughout the design region. The variance changes with radial distance, while remaining invariant under rotation about the center. A related criterion, uniform precision, additionally constrains the prediction variance near the center and depends on the number of center runs.
The fitted mean response at a coded point (\mathbf{x}) has variance
[ \operatorname{Var}!\left[\hat y(\mathbf{x})\right]
\sigma^2\mathbf{f}(\mathbf{x})^{\mathsf T} (\mathbf{X}^{\mathsf T}\mathbf{X})^{-1} \mathbf{f}(\mathbf{x}), ]
where (\mathbf{f}(\mathbf{x})) contains the second-order model terms and (\mathbf{X}) is the design matrix. This expression connects the geometry of the run locations to the precision of the fitted surface.
Orthogonality and blocking
Exact orthogonality among all columns of a quadratic model is not automatic in a central composite design. Symmetry makes the linear columns orthogonal to the intercept, to the pure quadratic columns, and to the two-factor interaction columns. It also makes distinct linear columns mutually orthogonal under the usual balanced construction.
The pure quadratic columns share nonzero means and consequently remain correlated with the intercept unless they are centered or the axial distance and run allocation satisfy additional moment conditions. The same distinction separates geometric rotatability from algebraic orthogonality: a design can possess either property without possessing the other.
Central composite designs also have a natural blocked structure because the factorial core and axial augmentation can be conducted during separate experimental periods. Genichi Taguchi and Masuyuki Matsui analyzed this arrangement in Japanese industrial experiments by expressing block orthogonality through equality of weighted second moments. Under that condition, the fitted block contrast is orthogonal to the quadratic response terms, so temporal or equipment-related shifts do not enter the estimated curvature coefficients.
Center points distributed across blocks provide a common geometric location for estimating block displacement. Their replication also distinguishes variation among nominally identical runs from variation caused by movement across the factor space.
Sequential interpretation
The central composite design originated as a sequential augmentation of a first-order experiment. A two-level factorial design estimates a local plane and reveals interaction structure within the factorial region. The addition of center runs supplies a direct curvature contrast, because the average response at the factorial points has a different expected value from the response at the center when pure quadratic terms are present.
Axial runs then identify the individual quadratic coefficients. Without those runs, the factorial portion cannot distinguish among the contributions of (x_1^2,\ldots,x_k^2), because every squared coordinate equals one at every factorial point. At the center, every squared coordinate equals zero, providing information about their aggregate contribution but not their separate values.
This algebraic relationship explains the characteristic architecture of the design. The factorial portion determines directions and interactions, the center anchors the local surface, and the axial points separate curvature by coordinate direction. The complete construction is therefore more than a collection of three-level factor settings; it is a decomposition of the information needed for a quadratic model.
Model assessment
Replicated center points provide an estimate of pure error when the runs represent independent experimental repetitions. Additional replicated design points permit a broader decomposition of the residual sum of squares into pure error and lack of fit.
A significant curvature contrast establishes that a first-order model does not describe the difference between the factorial average and the center response. It does not by itself establish that the full quadratic model is adequate, because higher-order behavior can remain in the residual structure. Assessment of the fitted surface therefore concerns both the estimated regression coefficients and the spatial pattern of residuals across the design region.
Fractional factorial cores introduce an additional issue because two-factor interactions can be aliased with one another or with higher-order terms. A central composite augmentation supplies quadratic information, but it does not automatically remove aliases inherited from the factorial fraction. The estimability of the full second-order model remains a property of the combined design matrix.
Relation to other second-order designs
A central composite design differs from a Box–Behnken design in the placement of experimental points. Box–Behnken designs place runs on lower-dimensional faces of the factor space and ordinarily omit the corners of the factorial cube. Central composite designs retain a factorial or fractional-factorial core and add points on the coordinate axes.
A complete three-level factorial design contains (3^k) runs and samples every combination of its three coded levels. The central composite construction uses substantially fewer points as (k) increases because it includes only the factorial core, the axial pairs, and replicated centers. This reduction follows from targeting estimation of a quadratic polynomial rather than treating every level combination as a distinct experimental condition.
Optimal designs derived through criteria such as D-optimality need not display the cube-and-star geometry of a central composite design. Their support points depend on the specified model, the candidate region, and the selected information criterion. Central composite designs instead arise from a fixed symmetric construction whose moment properties admit direct algebraic analysis.