Can a factorial experiment be extended into a CCD?

Often, if the existing design and factor settings are suitable; augmentation must account for aliasing, feasibility and changes between stages.

Does alpha always equal square root of two?

No. That value applies to the particular two-factor rotatable construction in the example.

Does the design automatically find the best setting?

No. Model fitting, diagnostics, constrained interpretation and confirmation are still required.

AI Glossary / Statistics and probability / Intermediate

Central composite design

Definition

A central composite design (CCD) is an experimental design for fitting a second-order response surface. It combines a factorial or suitable fractional-factorial portion, center runs and axial or star points. These different locations provide information about linear effects, interactions and individual quadratic terms, supporting local investigation of curvature and possible operating optima.

What it means

A two-level factorial can estimate important effects efficiently, but its corner points do not separate the squared effects of different quantitative factors. Center runs can reveal a combined curvature contrast, while axial points change one factor away from the center with the others held there. Together, the components can support a quadratic approximation. The design is a plan for gathering information, not proof that the true response is quadratic or that the fitted optimum will work outside the studied region.

How it works

Choose quantitative factors, ranges and coding. Add the factorial combinations, paired axial points along each factor axis and suitable center replication. Axial distance is denoted alpha. In a common circumscribed design, alpha exceeds one, placing axial settings outside the factorial cube. An inscribed design rescales the construction to respect outer limits, while a face-centered design uses alpha equal to one. Rotatability concerns equal prediction variance at equal distances from the center under the model; it depends on the design details. Plan randomization or blocking, fit the model, examine residuals and lack of fit, then run independent confirmation experiments at promising settings.

Three practical examples

Manufacturing optimization: model how temperature and processing time jointly affect an output, including curvature and interaction.
Formulation development: study a local region of independently adjustable quantitative settings, while respecting physical constraints and recognizing when mixture-design methods are more appropriate.
AI-system experimentation: investigate a small number of meaningful numeric controls using a structured design, with repeated stochastic evaluations and a carefully defined performance response.

Think of it this way

Think of mapping a hill using measurements at corners, along the coordinate directions and in the middle. The extra axis measurements reveal whether each direction curves. The resulting local map can suggest a summit, but a separate visit is still needed to verify the proposed location.

Worked example

For two coded factors, a full factorial portion has four points: (-1,-1), (-1,+1), (+1,-1) and (+1,+1). A rotatable unblocked construction uses alpha = 4 to the one-quarter power = square root of two, approximately 1.414. Add four axial points, (-alpha,0), (+alpha,0), (0,-alpha) and (0,+alpha), and five independent center runs. The total is 4 + 4 + 5 = 13 runs. The quadratic model has six coefficients: intercept, x, z, xz, x squared and z squared. If the design matrix has full rank, the residual degrees of freedom are 13 - 6 = 7. There are nine distinct settings, so center replication contributes five minus one = four pure-error degrees of freedom, leaving three for lack of fit. If temperature is coded as x = (temperature - 50)/10, the factorial temperatures are 40 and 60, but the axial temperatures are approximately 35.86 and 64.14. Those settings must be feasible; a hard allowable range of 40 to 60 would require a different scaling or design. The run count alone does not establish adequate precision.

Illustrative case study

Fictional case: Brook Process Labs augments a completed factorial experiment to study curvature. The original low and high settings were not physical limits, so appropriate axial runs are feasible. Because the augmentation occurs on another day, analysts consider block effects and include reference runs rather than assuming the two stages are directly interchangeable. They fit a quadratic surface and perform confirmation runs near the proposed operating setting. This fictional case illustrates sequential design and does not claim a verified process optimum.

Common misconceptions and mistakes

A CCD does not always require five factor levels: a face-centered form uses three. Not every CCD is rotatable, and rotatability is a precision property rather than a guarantee of model accuracy. An arbitrary low-resolution fraction may not support the desired quadratic model without problematic aliasing. Center replication estimates local repeatability under its conditions, not automatically variation everywhere. Hard boundaries and infeasible combinations can make a standard CCD unsuitable. A stationary point in a fitted quadratic may be a maximum, minimum or saddle point, and it may lie outside the experimental region.

Business use

Businesses can use CCDs when initial screening has narrowed the factor set and curvature matters for a decision. Verify feasibility before scheduling runs, including settings implied by coded axial distances. Track material batches, run order and other changes that could confound the design. Choose response measures that reflect operational goals and consider trade-offs when several outcomes matter. In AI evaluations, preserve dataset splits and account for stochastic variation; repeated runs on an overused benchmark do not replace independent validation. Treat the fitted response surface as a local decision aid with explicit uncertainty.

Check your understanding

Q: Why does the worked example have only four pure-error degrees of freedom?

Answer: A: Only the center is replicated, with five runs at one setting. The eight other settings each appear once, so they provide no within-setting replication degrees of freedom.

Related terms

Factorial design
Centerpoint
response surface methodology
axial point
quadratic model
rotatability
pure error
lack of fit and confirmation run

Learn next

construct the two-factor design matrix and verify its rank. Compare circumscribed and face-centered settings under the same physical limits, then fit a simulated quadratic and examine how noise changes the estimated stationary point.

Sources

Sources: NIST Engineering Statistics Handbook, central composite designs: https://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm ; NIST, response surface design properties: https://www.itl.nist.gov/div898/handbook/pri/section3/pri336.htm ; NIST, blocking and sequential augmentation: https://www.itl.nist.gov/div898/handbook/pri/section3/pri3364.htm ; DoE.wrapper author documentation, CCD construction: https://search.r-project.org/CRAN/refmans/DoE.wrapper/html/CentralCompositeDesigns.html .

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