Choosing a response surface design for research

It can be daunting to look at the variety of response surface (RSM) designs available and know which one will work the best for your research project. In this article, I will review the most common RSM designs: central composite, Box-Behnken, and optimal design (Figure 1), detailing the similarities and key differences between these options.

Figure 1.

Common response surface designs. (a) Central Composite, (b) Box-Behnken, (c) Optimal.

Overview of RSM designs

Practitioners commonly use response surface designs to optimize a system, especially when the relationship between the factors and the responses is at least partially nonlinear. RSM designs are built to estimate higher-order polynomials, such as quadratic. They expand the capability of two-level factorial designs which simply estimate main effects and two-factor interactions.

Key properties of RSM designs include being able to estimate the chosen polynomial well, with relatively low prediction error across the design space. A good design should include 4–5 more design points than there are coefficients in the model and include some replicated design points to get a pure error estimate. Together, these points are used to form a lack of fit test to check the adequacy of the model. An efficient design has a few extra points for robustness, but not too many to cause unnecessary work.

Central composite

Central composite designs are formed from a two-level factorial base, augmented with center points and axial points. The factorial base is used to estimate the main effects and two-factor interactions, while the other points are used to estimate the quadratic effects. Each factor is set to 5 levels designated in coded units as (-alpha, −1, 0, +1, +alpha). “Alpha” sets the location of the axial points, specifying the distance from the center of the design. This value is modified to meet various statistical properties (mostly beyond the scope of this article). Standard options for the alpha value achieve rotatability of the design or pull the axial points into the faces of the design (making it a 3-level face-centered CCD). If rotatable axial points are used, then the design will have low prediction error within the space defined by the factorial points. A CCD provides its best predictions in a cuboidal design space.

Box-Behnken

Box-Behnken designs are created by placing points on the centers of the edges of a factorial space. A key advantage is that factors are only set to three levels, rather than the five levels needed in a standard central composite design. As a result of the position of the design points, in the 3-factor case the layout is a spherical design space. Many practitioners consider it to be an advantage that the high-high-high and low-low-low combinations of factor settings are not included in the runs. This avoids extreme combinations that may not work well. On the other hand, some practitioners consider this to be a disadvantage because those corners are not being tested. The cuboidal versus spherical design space is the key difference between the central composite and Box-Behnken designs, and it is most important to understand your process space to determine which design will best meet your needs.

From a statistical standpoint, both designs have excellent properties, as long as the center points are replicated as suggested by DOE software (usually 4–5 replicates). The central composite design can estimate the two-factor interaction terms slightly more precisely than the Box-Behnken design (because the design points are at the corners of the cube). The BB design can estimate the quadratic terms slightly more precisely than the CCD (because the design points are at the centers of the edges of the cube). Both designs offer low prediction error across the design space.

Optimal (Custom) design

Algorithmically generated designs are useful when the system has special requirements that do not conform to the limitations of CCD and BB designs. Customization can include:

• Adding multilinear constraints to remove areas of the design space that do not produce measurable output

• Including categoric, or discrete-numeric variables

• Modeling a higher-order polynomial such as cubic

Optimal designs are built by choosing design points that estimate the model terms and also achieve a statistical objective. D-optimal designs estimate the model coefficients as precisely as possible, which is ideal when the DOE objective is to detect effects (like in a factorial design). I-optimal designs reduce the standard error of predictions over the maximum amount of the design space, which is ideal when prediction capability is the DOE goal (like in response surface design). Just like BB and CCD’s, a good optimal design should include replicates although they do not have to be at the center point.

The primary disadvantage of optimal designs is that the number of levels for each factor is variable. The experimenter will likely need to round the recommended design point settings to achieve a practical design. A minor drawback of optimal designs is that they are custom generated each time they are requested and since there are a large number of statistically equivalent point choices, the exact points will differ with each build. Simply remember to save the design so it is not lost! On the other hand, a significant advantage of an optimal design is that when there are a large number of factors, an optimal design may be the most efficient, with the fewest runs.

Conclusion

Table 1 summarizes the three RSM designs covered in this article. All of these response surface design choices are statistically sound. Choose the one that best fits the design space. Remember that best practices for a robust design include replicates so that the lack of fit test can be used as a fit statistic. As a final reminder, after running a DOE and analyzing the data, verify with confirmation runs on the actual process to demonstrate that the results are reproducible.

Table 1.

Common response surface design summary with Stat-Ease® software.

Criteria	Central composite	Box-Behnken	Optimal
Factor levels	5, 3 optional	3	varies
Design space	Axial points located outside factorial box	Factorial cube doesn’t include corners	Defined by factor limits, not limited to cube
Includes Lack of fit	Yes	Yes	With defaults
Includes Replicates	Yes-center point	Yes-center point	With defaults, chosen by optimality
Low Prediction Error	Yes	Yes	With defaults
Choose when:	Region of interest is a cube	Region of interest is a sphere	Region of interest is a custom