Response surface designs help researchers explore how multiple factors affect an outcome. Central composite and Box-Behnken designs are two popular options. They allow for efficient estimation of linear, quadratic, and interaction effects with fewer experimental runs.

These designs differ in their structure and properties. Central composite designs use factorial, axial, and center points. Box-Behnken designs use a subset of factorial points. Both aim for rotatability and orthogonality to ensure reliable predictions across the design space.

Central Composite Designs (CCD)

Overview and Components

• Central composite design (CCD) is a response surface design that combines a two-level factorial or fractional factorial design with additional axial and center points
• Consists of three types of design points: factorial points, axial points, and center points
  • Factorial points are the points from a two-level factorial or fractional factorial design
  • Axial points, also called star points, are points located at a distance $\alpha$ from the center of the design space along each factor axis
  • Center points are repeated points at the center of the design space used to estimate experimental error and curvature

Variants of CCD

• Face-centered CCD has axial points positioned at the center of each face of the factorial space, with $\alpha = 1$
  • Requires only three levels for each factor
  • Useful when the factors have established upper and lower bounds
• Circumscribed CCD has axial points located at a distance $\alpha > 1$ from the center, forming a circle circumscribed around the factorial space
  • Requires five levels for each factor
  • Provides high-quality predictions over the entire design space
• Inscribed CCD has axial points located inside the factorial space, with $\alpha < 1$
  • Requires five levels for each factor
  • Useful when the corners of the factorial space are not of interest or are infeasible

Box-Behnken Design

Overview and Characteristics

• Box-Behnken design is an alternative to central composite designs for fitting second-order response surfaces
• Requires only three levels for each factor, making it more efficient than CCD when the factors have established ranges
• Does not include factorial points at the vertices of the design space, making it useful when the corners are not of interest or are infeasible

Properties of Box-Behnken Design

• Rotatability is a desirable property where the variance of the predicted response is constant at a fixed distance from the center of the design
  • Box-Behnken designs are rotatable or nearly rotatable, ensuring consistent prediction quality in all directions
• Orthogonality is another desirable property where the effects of different factors can be estimated independently
  • Box-Behnken designs are orthogonal or nearly orthogonal, allowing for the independent estimation of linear, quadratic, and interaction effects