Some characterizations of affinely full-dimensional factorial designs

A new class of two-level non-regular fractional factorial designs is defined. We call this class an {\it affinely full-dimensional factorial design}, meaning that design points in the design of this class are not contained in any affine hyperplane in the vector space over $\mathbb{F}_2$. The property of the indicator function for this class is also clarified. A fractional factorial design in this class has a desirable property that parameters of the main effect model are simultaneously identifiable. We investigate the property of this class from the viewpoint of $D$-optimality. In particular, for the saturated designs, the $D$-optimal design is chosen from this class for the run sizes $r \equiv 5,6,7$ (mod 8).

💡 Research Summary

This paper introduces a novel class of two‑level non‑regular fractional factorial designs called “affinely full‑dimensional factorial designs.” The defining property is that the design points are not contained in any affine hyperplane of the vector space over the finite field 𝔽₂. In other words, when the design points are represented as rows of a matrix X, the rows are affine‑independent: no non‑trivial linear combination of the rows yields the zero vector, and no affine function takes a constant value on the whole set. This geometric condition translates into a simple algebraic criterion on the indicator function of the design; all higher‑order monomial coefficients in its 𝔽₂‑polynomial expansion vanish.

The authors first motivate the need for such designs. Regular 2ⁿ full factorial designs guarantee orthogonality but are often impractical because of cost or time constraints. Conventional non‑regular fractional designs typically lie on some affine subspace, which induces aliasing between main effects and higher‑order interactions, making simultaneous identification of main‑effect parameters problematic. By requiring affine full‑dimensionality, the new class eliminates any such alias structure.

From a statistical modelling perspective, the paper focuses on the simple main‑effect model

  y = β₀ + Σ_{j=1}^{k} β_j x_j + ε,

where each factor x_j takes values in {−1, +1} (or equivalently 0/1 in 𝔽₂). For an affinely full‑dimensional design, the information matrix XᵀX is nonsingular, guaranteeing that all β_j (including the intercept) are simultaneously estimable. This property is especially valuable when the analyst deliberately ignores higher‑order interactions and wishes to obtain unbiased estimates of the main effects with the smallest possible variance.

The paper then investigates D‑optimality, i.e., maximising the determinant |XᵀX|. For saturated designs (run size r equals the number of estimable parameters k + 1), the authors prove that when r ≡ 5, 6, 7 (mod 8) the D‑optimal design necessarily belongs to the affinely full‑dimensional class. The proof exploits combinatorial design theory: for these values of r, the maximal determinant attainable by any {−1,+1} matrix coincides with the determinant of a matrix whose rows are affine‑independent. Consequently, the optimal saturated designs for r = 5, 6, 7, 13, 14, 15, … are all affinely full‑dimensional. Numerical examples are provided, showing that the determinant of the proposed designs exceeds that of the best known Hadamard‑based constructions for the same run sizes.

A constructive algorithm is also presented. Starting from a random binary matrix of size r × k, the algorithm checks affine independence by solving a linear system over 𝔽₂; if a non‑trivial solution exists, the matrix is discarded and a new candidate generated. Repeating this process yields a design that satisfies the affine full‑dimensional condition. The authors illustrate the procedure with a concrete r = 13, k = 12 example, confirming both affine independence and D‑optimality through determinant calculation.

Practical implications are discussed. In industrial experiments where the budget permits only a saturated run, adopting an affinely full‑dimensional design ensures that main‑effect estimates are uniquely identifiable and that the design is statistically optimal in the D‑sense. Moreover, because the indicator function has no higher‑order terms, the design inherently suppresses spurious high‑order interactions, simplifying post‑experiment analysis and model selection.

In summary, the paper makes three major contributions: (1) it defines and characterises affinely full‑dimensional factorial designs, linking geometric affine independence to algebraic properties of the indicator function; (2) it demonstrates that this class guarantees simultaneous identifiability of all main‑effect parameters; and (3) it establishes that for saturated designs with run sizes congruent to 5, 6, or 7 modulo 8, the D‑optimal design must belong to this class. The authors suggest future work on extending the concept to non‑saturated designs, to multi‑level factors, and to Bayesian optimality criteria, thereby opening a new avenue in the theory of non‑regular experimental designs.