Central Composite vs Box-Behnken: Which Should You Choose?
Compare central composite and Box-Behnken response-surface designs: run counts, factor levels, safety at the corners, and which fits your optimization problem.
By Covariance Labs · · 3 min read
This guide has not yet been reviewed by an independent statistician. If you spot an error, please tell us.
Once screening has told you which factors matter, the next question is usually where is the best setting? Finding a peak or valley requires a model with curvature, which means each factor must be tested at three or more levels. Two-level factorials cannot do it. The two most common designs for this job are the central composite design (CCD) and the Box-Behnken design (BBD).
Both fit a full quadratic (second-order) model, which is the basis of response surface methodology:
How each design is built
Central composite. Take a two-level full factorial (the corners of the factor cube), add axial points along each factor axis, and add replicated centre points. In a face-centred CCD the axial points sit on the faces of the cube at ±1, so every factor has just three levels: low, middle, high. Classic CCDs place the axial points farther out (beyond the original range) to make the design rotatable, which needs five levels per factor.
Box-Behnken. Take pairs of factors, run all four low/high combinations for the pair while holding every other factor at its midpoint, repeat for every pair, and add centre points. BBD runs sit at the midpoints of the edges of the cube. It never includes a run where all factors are at their extremes at the same time.
Run counts
With three centre points:
| Factors | Face-centred CCD | Box-Behnken |
|---|---|---|
| 2 | 11 | not available |
| 3 | 17 | 15 |
| 4 | 27 | 27 |
| 5 | 45 | 43 |
| 6 | 79 | 63 |
The CCD count is 2k + 2k + centre points; the BBD count is 2k(k − 1) + centre points. A full quadratic model with k factors has (k + 1)(k + 2) / 2 terms: 10 for three factors, 15 for four, 21 for five. Both designs leave plenty of runs to estimate error.
Choosing between them
| Consideration | Prefer |
|---|---|
| Extreme corner conditions are unsafe, costly or impossible | Box-Behnken (no corner runs) |
| You already ran a factorial and want to add to it | Central composite (the factorial is its core, so you can add axial points later) |
| Only two factors | Central composite |
| Six or more factors and runs are expensive | Box-Behnken (fewer runs) |
| You want predictions at the extremes of the factor space | Central composite |
| You want to limit each factor to three levels | Face-centred CCD or Box-Behnken |
After you fit the model
Check the lack-of-fit test and the residual plots before trusting the surface, then use the fitted model to find the optimum and confirm it with a real run. Our guide to R², adjusted R² and predicted R² covers how to judge model quality. Covariance Labs fits a full quadratic model for both designs, and its optimum finder (Pro) searches the fitted surface. See the methods page for details, or read the practical guide to DoE for the wider workflow.
Frequently asked questions
- Which needs fewer runs, central composite or Box-Behnken?
- For three factors Box-Behnken needs fewer (12 runs plus centre points against 14 for a face-centred central composite). At four factors they are equal at 24 runs, and from five factors upward Box-Behnken is smaller. Box-Behnken is not available for two factors.
- When should I avoid Box-Behnken?
- Avoid it when you need to estimate the response at the extreme corners of the factor space, or when you want to build the design in stages from an earlier factorial experiment.