The center point in U22 showed that a fitted plane could miss the response at the center. However, a single center location does not allow us to estimate A² and B² separately. To identify the direction of curvature, we need measurements at several locations along the factor axes.

The response surface is the relationship between input combinations and the average response.

The second-order model for two continuous factors, x and y, has the following terms:

ŷ = β₀ + β₁x + β₂y + β₁₁x² + β₂₂y² + β₁₂xy

• x, y: The slope in each direction.
• x², y²: The curvature in each axial direction.
• xy: The interaction where the slope of one factor changes depending on the location of the other factor.

Response Surface Methodology (RSM) is a sequential method that arranges experimental points to fit this model and then explores candidate conditions on that surface. The goal is not to make the equation complicated, but rather to distinguish the curvature within the measurement range with data.

CCD combines three types of points.

The two-factor Central Composite Design usually consists of the following points:

1. factorial point: (−1,−1), (−1,+1), (+1,−1), (+1,+1)
2. axial point: (±α,0), (0,±α)
3. center point: Repetition of (0,0)

Axial points measure both sides of one axis while keeping the other factors centered. This provides separate information for examining the effects of x² and y². Independent center-point replicates estimate the response at the center and provide pure-error information.

The axial distance α is not the significance level α

Here, α represents the coded distance from the center to the axial point. In a rotatable central composite design (CCD) of a two-factor factorial block, α=(2²)^(1/4)=√2≈1.414 is commonly used. It shares the same name as the significance level of 0.05 but is a completely different quantity.

Rotatability is a design property that ensures points at the same distance from the center have the same prediction variance, regardless of direction. This does not always mean it is the safest design for actual experiments.

CCC, CCI, and CCF have different ranges under the same name

After defining the coded coordinates, convert them to actual units. For example, if the temperature range is 30–40°C, a coded value of 0 corresponds to 35°C, and the CCC axial point with α=1.414 would be approximately 27.9°C and 42.1°C. If this condition is not safe, physical constraints take precedence over mathematical symmetry.

Box-Behnken avoids cube corners

The non-center points of a three-factor Box-Behnken design are located at edge centers, where two factors are at ±1 and the remaining factor is at 0. It does not include cube corners where all factors are simultaneously at their extremes.

This can be useful when the simultaneous extremes of the three factors are risky, but it does not automatically account for the actual forbidden region. An edge-center condition may also be infeasible, so candidate conditions should be reviewed individually. Do not judge the superiority of CCD and BBD based solely on the number of runs.

Fix these four before selecting a design

• Which quadratic terms and interactions will be estimated?
• What is the studied range in actual units?
• How many replicates of the center point and blocks will be included?
• Are the axial points and edge-centers actually feasible and safe?

After that, check the model term estimability and prediction variance by region in Design Evaluation.

In-Silico Lab: Compare coordinates and risk ranges.

1. Change the CCD α from 1 to 1.682 and observe the maximum coded coordinate.
2. Change the number of center points and see how the total number of runs changes.
3. Select Box-Behnken and view the x-y projection of the 3-factor, 15-run structure.
4. Explain whether axial points outside the actual allowable range are generated in the CCD with a maximum of |coded|>1.

Lab is a coordinate generator. It does not use response values, and each row is a single design run. Identical center point rows should be independent replicates in the actual experiment to provide pure error.

In JMP, read evaluation before generation

The candidate list in Response Surface Design is more than a run-count table. Evaluate the design geometry, estimability, prediction variance, and whether the blocks and center-point replicates match the prespecified modeling objectives. Also check whether the coded and actual-unit columns in the generated table match the planned range.

Example of a result statement

A CCF design was constructed with 4 factorial points, 4 axial points, and 5 independent center-point replicates for a full quadratic model of two continuous factors. All coded coordinates were within the prespecified safe range of -1 to +1, and 6 model terms were estimable. This design is intended for estimating the second-order surface within the currently studied range and does not support extrapolation outside the range.

Concluding the section

• RSM connects the coordinates for estimating curvature with the second-order model.
• CCD combines factorial, axial, and center points.
• The axial point distance α is different from the significance level α.
• Box-Behnken does not use cube corners where all factors are simultaneously at their extremes.
• Practical range, safety, and feasibility are more important than mathematical efficiency.

In the next section, we will look at how to input the responses into this design table and then determine if the second-order model is sufficient.

Official Supplementary Materials

• NIST/SEMATECH · Response Surface Designs
• NIST/SEMATECH · Comparing Response Surface Designs
• JMP Knowledge Portal · Box-Behnken Designs

This article and Lab are educational synthetic designs and are not evidence for actual research, process, quality, or regulatory decisions.