Regular two-level fractions, signed defining relations, and resolution

Fix integers $k\ge 1$ and $1\le p\le k$, and put $[k]=\{1,\dots ,k\}$. Let ${\mathbb{F}}_2=\{0,1\}$ with addition and multiplication modulo $2$. For a factor subset $G\subseteq [k]$, its incidence vector $1_G\in {\mathbb{F}}_2^k$ has coordinate $1$ at $j$ exactly when $j\in G$. Choose $p$ subsets $G_1,\dots ,G_p$ whose incidence vectors are linearly independent over ${\mathbb{F}}_2$, and choose signs $s_i\in \{-1,+1\}$. Using the run space and product columns of Two-level full factorial designs and coded effects, define the regular fraction

$\mathcal{F}=\{x\in \{-1,+1{\}}^{[k]}:x_{G_i}=s_i\text{ for }1\le i\le p\},x_G={\prod }_{j\in G}{x}_j\mathrm{.}$

The corresponding design is called a regular $2^{k-p}$ fractional factorial; The defining subgroup gives the exact alias relations ↗ proves that $\mathcal{F}$ has exactly $2^{k-p}$ runs, including the case $p=k$ with one run. The choices $s_i=-1$ are alternate fractions and are part of the definition.

A signed word is a pair $(ϵ,A)$ with $ϵ\in \{-1,+1\}$ and $A\subseteq [k]$. It evaluates at $x$ as $ϵx_A$ and multiplies by

$(ϵ,A)(\delta ,B)=(ϵ\delta ,A\mathrm{△}B),A\mathrm{△}B=(A\setminus B)\cup (B\setminus A)\mathrm{.}$

The identity word is $(+1,\mathrm{\varnothing })$. The signed generator $h_i=(s_i,G_i)$ evaluates to $+1$ on $\mathcal{F}$. The defining subgroup $H$ is generated by $h_1,\dots ,h_p$; the alias theorem proves it is exactly the subgroup of signed words that evaluate to $+1$ on every run in $\mathcal{F}$.

For $A,B\subseteq [k]$, their restricted sign columns are equal-column aliases when $x_A=x_B$ for every $x\in \mathcal{F}$. They are signed aliases with sign $\sigma \in \{-1,+1\}$ when $x_A=\sigma x_B$ on every run; sign $+1$ is the equal-column case and sign $-1$ records opposite columns. The alias theorem gives the exact criterion for each sign.

The word length of $(ϵ,A)$ is $\mathrm{\mid }A\mathrm{\mid }$, the number of factor letters, independent of $ϵ$. The resolution is the minimum $\mathrm{\mid }A\mathrm{\mid }$ among nonidentity words $(ϵ,A)\in H$; it is written as a Roman numeral. Since $p\ge 1$ and the generators are independent, $H$ has a nonidentity word of nonzero length.

• Equal or opposite design columns make the corresponding unrestricted linear-model coefficients nonseparable; see Aliased factorial columns prevent separate coefficient identification. A hierarchy or sparsity premise is an additional modeling assumption, not a consequence of the fraction.
• The resolution records only the shortest defining word. It does not describe the complete alias structure; that is given by the whole subgroup $H$.

Oehlert §§18.2 and 18.4 and NIST/SEMATECH §§5.3.3.4.2–5.3.3.4.4 support the generator, signed-relation, and resolution conventions. The run count and exact subgroup characterization are proved locally in the justified-by theorem.