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Definitive Screening Designs

Motivation

Classical two-level screening has a blind spot: it cannot see curvature, and its main effects can be biased by two-factor interactions. Running a separate response-surface study afterwards doubles the cost. Definitive Screening Designs (DSDs, Jones & Nachtsheim 2011) collapse both phases into one small, three-level design — the most important screening advance of the last decade.

Theory

For \(m\) factors, a DSD uses \(2m+1\) runs built from a conference matrix \(C\) of order \(m\) (zero diagonal, \(\pm 1\) off-diagonal, \(C^\top C = (m-1)I\)):

\[ \text{DSD} = \begin{bmatrix} C \\ -C \\ \mathbf{0} \end{bmatrix}. \]

The fold \(C / {-}C\) makes the linear effects (odd functions of the coded levels) orthogonal to the quadratic effects and two-factor interactions (even functions). Consequences:

  • main effects are unbiased by 2FI and curvature;
  • every factor is run at three levels \(\{-1, 0, +1\}\), so quadratic effects are estimable;
  • the design projects onto a response surface in the few active factors — no extra runs needed.

doekit builds the conference matrices by the Paley construction (\(m = q+1\), \(q\) prime), covering orders \(4, 6, 8, 12, 14, \dots\); when the smallest available order exceeds \(m\), the surplus columns become "phantom" factors that further reduce estimate bias.

In doekit

import doekit as ed

dsd = ed.definitive_screening({f"x{i+1}": (0, 10) for i in range(6)})  # 13 runs
ed.evaluate(dsd).vif.max()        # ~1.0  (main effects orthogonal)

See also