Box-Behnken Design

Spherical three-level design that avoids extreme corner points while efficiently fitting second-order response surface models.

Design Overview & Structure

What it accomplishes:

Box-Behnken Designs (BBDs) are rotatable or nearly rotatable second-order designs that place experimental points at the midpoints of the edges of the factor space and the center, avoiding potentially risky extreme corner combinations.

Advantages:

• • No extreme corner points (safer)
• • Fewer runs than CCD for many factors
• • Spherical design space
• • Good for 3-7 factors
• • Nearly rotatable properties

Considerations:

• • Cannot be built sequentially
• • No information outside sphere
• • Three-level factors required
• • Not orthogonal

Second-Order Model & Design Point Formula

Second-Order Response Surface Model:

A Box-Behnken Design estimates every term of the full quadratic model because each run places exactly two factors at their ±1 levels while all remaining factors sit at the center (0), systematically covering every factor pair.

Second-order (quadratic) response surface model

= predicted response

= intercept

= linear coefficient for factor i

βii

= quadratic coefficient for factor i

βij

= two-factor interaction coefficient

= random error

Design Size (Number of Runs):

For $k$ factors, a Box-Behnken Design uses balanced incomplete-block pairs of factors set to their edge midpoints plus $n_0$ center-point replicates.

Box-Behnken run count

= number of factors

= number of center-point replicates

For k=3: 12 edge points + 3 center points = 15 runs. For k=4: 24 edge points + 3 center points = 27 runs.

Key Assumptions

• • Response surface is adequately approximated by a second-order polynomial in the sphere of interest
• • Factors are continuous and can be set precisely to three levels (-1, 0, +1)
• • Errors are independent, normally distributed, and have constant variance
• • No information is needed outside the spherical experimental region

Reading the Box-Behnken Design Layout

Example Output & Result Interpretation

Worked Example:

In the 15-run, 3-factor design below (Temperature, Time, Catalyst), the three replicated center points (runs 13-15, all at 0,0,0) yield Yield values of 85.1, 84.7, and 85.4 — a tight cluster that provides the pure-error estimate used to test lack of fit. Fitting the quadratic model to all 15 runs lets the platform estimate all 3 linear, 3 quadratic, and 3 interaction terms simultaneously.

Design Properties:

• • Balanced incomplete block design
• • Good efficiency for quadratic terms
• • Spherical design region
• • Good prediction properties

Response Surface Tools:

• • Contour plots within sphere
• • 3D response surfaces
• • Canonical analysis
• • Desirability optimization

Decision Rules for Model Adequacy:

• Lack-of-fit test non-significant (p ≥ 0.05): Quadratic model adequately describes the response — proceed to optimization

• Lack-of-fit test significant (p < 0.05): Model is missing curvature or higher-order terms — reconsider factor ranges or add runs

• Ridge Analysis: Apply when the predicted optimum falls outside the experimental sphere

• Confirmation Run: Always validate the predicted optimum with a fresh experimental run

sixsigmacenter.com Implementation Guide

Design Construction:

1. Three-Level Setup:

• Low level (-1): Minimum practical value

• Center level (0): Current/nominal setting

• High level (+1): Maximum practical value

2. Edge Point Pattern:

• Each run has exactly two factors at ±1 levels

• Remaining factors at center level (0)

• Systematic coverage of all factor pairs

3. Center Points:

• Multiple replicates at center (0, 0, ..., 0)

• Provide pure error estimate

• Usually 3-6 center points

Platform Implementation:

The sixsigmacenter.com platform automatically generates Box-Behnken designs for 3-7 factors, handles randomization, performs complete second-order analysis, and provides response surface optimization with prediction capabilities.

When to Use Box-Behnken Designs

Ideal Applications:

Safety Concerns:

• • Extreme combinations dangerous/expensive
• • Chemical process optimization
• • Equipment operating limits
• • Quality constraints important

Efficiency Focus:

• • 4-7 factor optimization studies
• • Limited experimental budget
• • Spherical interest region
• • Stand-alone optimization

Factor Count Guidelines:

• 3 factors: 15 runs (12 edge + 3 center)

• 4 factors: 27 runs (24 edge + 3 center)

• 5 factors: 46 runs (40 edge + 6 center)

• 6-7 factors: Competitive with other designs

Example Data Layout for Box-Behnken Design (3 factors)

Standard 15-run BBD for 3 factors: 12 edge points + 3 center points. No extreme corners tested. Sample Size Recommendation: Standard designs available for k=3,4,5,6,7 factors with 3-6 center points.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.