Response Surface Designs

R2026b

In design of experiments (DOE) workflows, two-level fractional factorial and full factorial designs can often be used effectively to screen factors that influence the outcome (response) of a process. However, these designs can be insufficient when the process response involves quadratic (second-order) interactions between factors. In this case, the response surface is curved, and additional factor levels or experimental runs are needed to model the curvature and determine the factor levels that optimize the response. Response surface designs (which include Central Composite Designs and Box–Behnken Designs) can accommodate a full quadratic model, and require a much smaller number of runs than corresponding full factorial designs with the same number of factor levels.

Statistics and Machine Learning Toolbox™ offers several ways to work with response surface designs:

• Create a responseSurfaceDOE object by using the responseSurfaceDOE function. The function provides the following advantages:

  • The responseSurfaceDOE function allows you to specify the design type, design blocks, number of center points, factor names, lower and upper bounds for each factor, and experiment model. For a central composite design, you can also specify the design fraction of the full factorial cube.

  • In addition to returning the design runs, the responseSurfaceDOE function stores your specifications in the responseSurfaceDOE object properties.

  After you create a responseSurfaceDOE object, you can:

  • Fit a linear regression model to the design run responses using the fitlm function.

  • Randomize the run order in the design using the randomizeRunOrder function.

  • Add replicates (duplicates of the original design runs) using the addReplicates function.

• Use the DOE Explorer app to create a response surface design table and fit a linear regression model to the design run responses. Perform factor analysis and generate plots and tables to assess the model fit.

Central Composite Designs

Central composite designs (CCDs), also known as Box–Wilson designs, are appropriate for calibrating full quadratic models. There are three types of CCDs—circumscribed, inscribed, and faced—pictured below for a process involving three factors:

Each design consists of a set of factorial points (runs that correspond to the corners of a cube), and an arrangement of center and star points (runs) that allow for estimation of second-order effects. For a full quadratic model with n factors, CCDs have enough design runs to estimate the (n+2)(n+1)/2 coefficients.

The preferred choice of CCD depends on the number of factors and range of possible levels in the process, and the desired properties of the design. The following table summarizes some important properties. A design is rotatable if the prediction variance depends only on the distance of the design point from the center of the design.

Design	Rotatable	Factor Levels	Uses Level Values Outside ±1	Accuracy of Estimates
Circumscribed	Yes	5	Yes	Good over entire design space
Inscribed	Yes	5	No	Good over central subset of design space
Faced	No	3	No	Fair over entire design space; poor for pure quadratic coefficients

A CCD is characterized by a fractional design index k, where the design is 0.5^k of a full factorial cube. When you create a responseSurfaceDOE object, you can specify this parameter using the FractionalDesignIndex name-value argument. The table below summarizes the default and allowable values of the fractional design index. In the DOE Explorer app, the default fractional design index is always 0.

Number of Factors	Default Value of FractionalDesignIndex and Design Fraction	Allowable Values of FractionalDesignIndex
2 ≤ n ≤ 4	0 (Entire cube)	0
5 ≤ n ≤ 7	1 (1/2 of cube)	0,1
8 ≤ n ≤ 9	2 (1/4 of cube)	0,1,2
n = 10	3 (1/8 of cube)	0,1,2,3
n = 11	4 (1/16 of cube)	0,1,2,3,4
n > 11	0 (Entire cube)	0,1

A CCD can be orthogonal or uniform, depending on the number of center points in the design. An orthogonal design allows the main effects, interaction effects, and quadratic effects in the model to be estimated independently. In a uniform precision design, the variance of the predicted response does not depend on the distance from the center point.

When you select an orthogonal or uniform CCD in DOE Explorer, or specify CenterPoints="orthogonal" or CenterPoints="uniform" when you create a responseSurfaceDOE object, the software selects the number of center points based on the number of runs and fractional design index (see[1]).

For an example of how to create a CCD at the command line, see Create and Visualize Faced Central Composite Design.

Box–Behnken Designs

Like the designs described in Central Composite Designs, Box–Behnken designs are used to calibrate full quadratic models. Box–Behnken designs are rotatable and, for a small number of factors (four or less), require fewer runs than CCDs. By excluding the corners of the design space, you can avoid experimental runs with extreme factor combinations. As in the case with an inscribed CCD, however, extremes are poorly estimated.

The geometry of a three-factor Box–Behnken design is pictured below.

The design points are at the midpoints of edges of the design space and at the center. The design does not include an embedded factorial design or star points.

For an example of how to create a Box–Behnken design at the command line, see Create and Visualize Box–Behnken Design.

References

[1] Box, G. E. P., W. G. Hunter, and J. S. Hunter. "Multi-Factor Experimental Designs for Exploring Response Surfaces." Annals of Mathematical Statistics 28, no. 1 (March 1957): 195–241. https://doi.org/10.1214/aoms/1177707047.