Central composite design — fitting curvature without running everything
A two-level factorial can find main effects and interactions, but it sees the world as flat. To capture curvature — to find an optimum rather than a direction — you add star and center points, and that's a central composite design.
A central composite design (CCD) augments a two-level factorial (or fractional factorial) cube with axial ("star") points set a distance α out along each factor axis, plus replicated center points. The factorial corners estimate main effects and interactions; the star and center points add the information needed to fit a full quadratic model — so the design can capture curvature and locate an optimum, which corner points alone cannot. The choice of α controls rotatability; α=1 gives a face-centered design that stays within the cube.
Why a factorial isn't enough
A two-level factorial measures each factor at only its low and high settings. With two points per factor you can fit a line (a main effect) and interactions, but two points can never reveal a curve — you can't tell a peak from a slope. To estimate the squared terms that describe curvature, you need at least three levels of each factor. CCD adds those extra levels efficiently rather than running a full three-level grid.
The three point sets
1 · Factorial points (the cube corners)
A full or fractional two-level factorial. These carry the main effects and two-factor interactions — the linear backbone of the model.
2 · Axial / star points
Two points per factor, set a distance α out along each axis (all other factors at center). These give the third level needed to estimate the quadratic (curvature) terms.
3 · Center points (replicated)
Several runs at the center. They estimate pure error (from replication) and anchor the curvature estimate — and let you test whether curvature is even present.
Choosing α — rotatability
The axial distance α sets the design's geometry. A rotatable CCD (α = (number of factorial runs)^¼) gives equal prediction precision at equal distance from the center — desirable for response surfaces. α = 1 places the star points on the cube faces (a face-centered CCD), useful when factor ranges can't be exceeded. Larger α explores wider but pushes runs to extreme settings.
CCD vs Box-Behnken
Both fit a quadratic. CCD includes the extreme corner-plus-axial combinations and can be built up sequentially from an existing factorial; Box-Behnken uses edge-midpoints, never runs all factors at their extremes simultaneously, and often needs fewer runs — preferable when corner conditions are costly or unsafe.
Common pitfalls
- Skipping center-point replication. Without it you have no pure-error estimate and can't test lack of fit.
- Star points outside feasible ranges. A rotatable α can push a factor beyond what the process allows — switch to face-centered.
- Fitting quadratic when curvature is absent. Check the curvature test; if it's flat, a simpler model is better.
- Ignoring blocking. If the design runs over multiple days/batches, block it so day-to-day shifts don't bias the surface.
Where this gets slow by hand
Laying out a CCD, choosing α, blocking it, then fitting the quadratic, checking lack of fit, and reading the response surface to find the optimum — and iterating when the optimum sits at an edge — is exactly the multi-step modeling loop that eats an engineer's week between experiments.
From design layout to a located optimum
Niobia lays out the central composite design — picking α, adding center replication, and blocking for batch effects — then fits the quadratic model, runs the lack-of-fit and curvature checks, and reads the response surface to locate the optimum with its confidence region. When the optimum sits at a boundary it proposes the augmenting runs to extend the design. The week-long design-fit-interpret loop becomes a worked result you can act on.
Frequently asked
What is a central composite design?
A central composite design (CCD) is a response-surface DOE that combines a two-level factorial (cube) with axial 'star' points along each factor axis and replicated center points. Together these allow a full quadratic model to be fit, capturing curvature and locating an optimum.
What are the star points in a CCD?
The star (axial) points are runs placed a distance alpha out along each factor axis, with all other factors held at their center value. They provide the third factor level needed to estimate the quadratic (curvature) terms of the model.
What is the difference between CCD and Box-Behnken?
Both fit a quadratic response surface. CCD includes extreme corner and axial points and can be built sequentially from a factorial; Box-Behnken uses edge-midpoint runs, never sets all factors to their extremes at once, and often needs fewer runs — preferable when extreme combinations are costly or unsafe.