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

The paper introduces a new class of two‑level non‑regular fractional factorial designs called “affinely full‑dimensional factorial designs.” A design belongs to this class when, after mapping the levels {−1, 1} to {0, 1}, the set of design points does not lie in any affine hyperplane of the vector space over the field F₂. Equivalently, the binary vectors of the runs span the whole space, so the differences of the run vectors form a basis of F₂^s.

The authors start by formalising the full factorial D = {−1, 1}^s and a fractional design F ⊂ D with r runs. The design matrix M ∈ {−1, 1}^{r×(s+1)} includes a column of ones (the intercept) and s columns for the factor levels. In the main‑effects linear model Y = Mβ + ε (ε ~ N(0,σ²I)), simultaneous estimability of all β parameters is equivalent to M′M being nonsingular. Lemma 2.1 shows that if M′M is singular then F must be a subset of some regular fractional factorial design; conversely, if the design is affinely full‑dimensional then M′M is nonsingular, guaranteeing identifiability of all main‑effect parameters.

The paper also connects this geometric definition with the algebraic indicator‑function representation f(x)=∑{I⊆{1,…,s}} b_I X_I(x), where X_I(x)=∏{i∈I} x_i. Lemma 2.2 states that a design is affinely full‑dimensional precisely when every non‑empty coefficient satisfies |b_I| < b_∅. This provides a practical test based on the coefficients of the indicator polynomial.

The second major contribution concerns D‑optimality. For a saturated design (r = s + 1) the D‑criterion reduces to maximizing |det M|, which is the classic Hadamard maximal determinant problem. Theorem 3.1 establishes a clean equivalence: a saturated design is affinely full‑dimensional if and only if det M is not divisible by 2^r. The proof proceeds by converting M into a binary matrix fM (subtracting the intercept column and dividing the remaining columns by –2). The determinant of fM is odd exactly when det M lacks a factor 2^r, and oddness of det fM is equivalent to fM being nonsingular over F₂, i.e., the design points do not satisfy any non‑trivial affine relation.

Table 1 surveys known maximal determinants for r = 4,…,99, reporting det M / 2^{r‑1} and indicating whether the corresponding design is affinely full‑dimensional. The pattern that emerges is that for run sizes r congruent to 5, 6, 7 (mod 8) the D‑optimal design is affinely full‑dimensional, whereas for r ≡ 0, 1, 2, 3, 4 (mod 8) the D‑optimal design belongs to a regular fractional factorial (i.e., it is a subset design). This observation leads to Conjecture 3.1, which posits exactly this dichotomy for all r. No counter‑example is known at present.

The authors also discuss known determinant bounds. For r ≡ 1 (mod 8) the bound det M ≤ (2^{r‑1})^{½}(r‑1)^{(r‑1)/2} can be attained only when 2^{r‑1} is a perfect square; in those cases the maximal determinant is divisible by 2^r precisely when r ≡ 0 (mod 8). Proposition 3.1 formalises this result, showing that the parity of the exponent of 2 in the maximal determinant determines whether the optimal design is affinely full‑dimensional. A similar analysis is given for r ≡ 2 (mod 4) using the bound det M ≤ 2^{(r‑1)(r‑2)/2}.

In summary, the paper provides a rigorous definition of affinely full‑dimensional factorial designs, demonstrates that they guarantee simultaneous estimability of all main‑effect parameters, links the definition to the indicator‑function coefficients, and establishes a precise algebraic criterion (det M not divisible by 2^r) for saturated designs. By examining known maximal determinant results, the authors argue that for many run sizes the D‑optimal saturated design naturally falls into this new class, especially when r ≡ 5, 6, 7 (mod 8). The work bridges combinatorial matrix theory (Hadamard maximal determinant problem) with experimental design theory, offering a fresh perspective on constructing non‑regular designs that retain desirable statistical properties.