Rotatability

Rotatability is a design property used in statistics, particularly within response surface methodology, that concerns the precision of predicted responses. A rotatable design ensures that the variance of the predicted response at points that are the same distance from the design center is the same regardless of the direction from the center. This means prediction accuracy depends only on how far a point is from the center, not on where it lies in the factor space.

Mathematically, rotatability relates to the second-order (quadratic) model y = β0 + ∑βi xi + ∑βii xi^2 + ∑βij xi xj + ε, with coded factors xi. The predicted value y_hat at a new point x has variance Var(y_hat) = σ^2 x^T (X^T X)^{-1} x. A design is rotatable if this variance can be expressed as a function of r^2 = x^T x only, i.e., Var(y_hat) depends solely on distance from the center. This typically requires a symmetric arrangement of design points around the center.

Central composite designs (CCDs) are the canonical rotatable designs for second-order models. In a CCD with k factors, rotatability is achieved by placing axial (star) points at a distance alpha from the center, with alpha chosen as alpha = 2^(k/4). A CCD combines a factorial or fractional-factorial portion, axial points, and center runs to preserve rotatability. By contrast, Box-Behnken designs are generally not perfectly rotatable, though they are efficient for fitting second-order models and can offer near-rotatable properties in some cases.

Rotatability must be balanced with other design goals, such as orthogonality, practical constraints, and the desire for accuracy in different regions of the factor space. See also response surface methodology, central composite design, and design of experiments.