Design of Experiments

Design Efficiency (D, A, G)

Evaluates the quality of an experimental design using three criteria: D-efficiency (determinant), A-efficiency (trace), and G-efficiency (leverage).

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Factorial > Analyze Factorial Design > Design Diagnostics

Description

Design efficiency evaluates how well an experimental design estimates model parameters, relative to a theoretically optimal orthogonal design. A value of 100% means perfect orthogonality. **D-efficiency** is based on the determinant of the information matrix X'X. It minimizes the volume of the confidence ellipsoid of the parameter estimates — i.e., all parameters are estimated as precisely as possible simultaneously. **A-efficiency** is based on the trace of the inverse information matrix. It minimizes the average variance of parameter estimates — i.e., parameters are estimated as accurately as possible on average. **G-efficiency** is based on the maximum leverage value (diagonal of the hat matrix). It minimizes the maximum prediction variance — i.e., no single design point dominates the estimation disproportionately.

Formulas

D_eff

D_{eff} = \frac{( det ( X ^{⊤} X ) ) ^{1/ p}}{n} \times 100%

p = number of model terms (incl. intercept), n = number of runs. Normalizes the p-th root of the determinant by the number of runs.

A_eff

A_{eff} = \frac{p}{n \cdot tr ( ( X ^{⊤} X ) ^{- 1} )} \times 100%

Based on the trace of the inverse information matrix. Minimizes the average variance of parameter estimates.

G_eff

G_{eff} = \frac{p}{n \cdot max _{i} ( h _{ii} )} \times 100%

h_ii = diag(H) are the leverage values of the hat matrix H = X(X'X)⁻¹X'. Minimizes the maximum prediction variance.

H = X (X^{⊤} X)^{- 1} X^{⊤}

The leverage values h_ii on the diagonal measure the influence of each design point on the estimates.

X = 11 ⋮ 1 x_{11} x_{21} x_{n 1} x_{12} x_{22} x_{n 2} \dots \dots \dots x_{11} x_{12} x_{21} x_{22} x_{n 1} x_{n 2} \dots \dots ⋮ \dots

Contains intercept, coded main effects, and 2-factor interactions.

Assumptions

Coded design matrix with values in {−1, 0, +1}
Linear model with main effects and 2-factor interactions
X'X must be invertible (non-singular design)

Limitations

Only considers main effects and 2-factor interactions in the model, no higher orders
Condition number is a simplified approximation (diagonal ratio instead of eigenvalues)

References

Atkinson, A.C. & Donev, A.N., Optimum Experimental Designs, Oxford, 1992 — Chapter 10
Montgomery, D.C., Design and Analysis of Experiments, 10th Ed., Wiley — Chapter 11
Myers, R.H. & Montgomery, D.C., Response Surface Methodology, 4th Ed., Wiley — Chapter 7