Fractional factorial aliases and foldover

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.

Let $\mathcal{F}$ be the regular fraction determined by $p$ independent generator equations $x_{G_i}=s_i$ on the full $2^k$ sign design, and let $H$ be its signed defining subgroup. Then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }=2^{k-p}$, $\mathrm{\mid }H\mathrm{\mid }=2^p$, and $H$ consists of all signed words $(ϵ,C)$ for which $ϵx_C=+1$ on every run in $\mathcal{F}$. For all factor subsets $A,B\subseteq [k]$ and signs $\sigma \in \{-1,+1\}$,

$x_A=\sigma x_B\text{ on }\mathcal{F}⟺(\sigma ,A\mathrm{△}B)\in H\mathrm{.}$

In particular, the columns agree exactly when $(+1,A\mathrm{△}B)\in H$; a relation with sign $-1$ means the columns are opposites.

Oehlert §18.2 and NIST/SEMATECH §§5.3.3.4.2–5.3.3.4.4 present signed generators, defining relations, and alias multiplication. The exact row-space characterization and both directions of the column criterion are proved here.

Let $c_A,c_B\in {\mathbb{R}}^{\mathcal{F}}$ be the columns for distinct factorial effects $A,B\subseteq [k]$ in a regular fraction, and suppose $c_A=\sigma c_B$ for a fixed $\sigma \in \{-1,+1\}$. Consider the unrestricted fixed-design linear model $Y=X\beta +ϵ$, where these columns occur in $X$ and the distribution of the error vector $ϵ$ does not depend on $\beta$. A coefficient combination is identifiable if it has the same value for all parameter vectors giving the same distribution of $Y$. The contribution of these two columns depends only on $\sigma {\beta }_A+{\beta }_B$; neither coefficient is separately identifiable while the parameter space remains unrestricted. For equal columns ($\sigma =+1$), the relevant combination is the sum; for opposite columns ($\sigma =-1$), it is the difference. Additional runs can remove the ambiguity only if they break the column relation, and hierarchy or sparsity restrictions are extra modeling assumptions.

Oehlert §18.6 and NIST/SEMATECH §5.3.3.4.3 explain the ambiguity left by aliased effects and the extra role of sparsity assumptions. The parameter nonidentifiability argument is given explicitly above.

Let $\mathcal{F}$ be a regular fraction with signed defining subgroup $H$. For a specified set of factors $J\subseteq [k]$, define ${\tau }_{J}x$ by $({\tau }_{J}x{)}_j=-x_j$ for $j\in J$ and $({\tau }_{J}x{)}_j=x_j$ for $j\mathrm{\notin }J$. The partial foldover on $J$ is the run table formed by stacking the original runs with one transformed copy ${\tau }_J(\mathcal{F})$; the complete mirror foldover is the case $J=[k]$. Its residual signed defining subgroup is exactly

$H_J=\{(ϵ,A)\in H:\mathrm{\mid }A\cap J\mathrm{\mid }\text{ is even}\}\mathrm{.}$

Thus an originally signed-aliased pair $A,B$ remains signed-aliased on the augmented table exactly when $\mathrm{\mid }(A\mathrm{△}B)\cap J\mathrm{\mid }$ is even; if it is odd, the pair is separated. For a complete mirror foldover, a word of order $r=\mathrm{\mid }A\mathrm{△}B\mathrm{\mid }$ changes sign by $(-1{)}^r$. Therefore aliases between effects of opposite order parity separate, while same-parity relations remain. The displayed formula gives the full residual alias set for every partial foldover; a selected-factor follow-up need not remove other aliases.

Oehlert §18.7 develops mirror foldover by reversing every sign. NIST/SEMATECH §§5.3.3.8.1–5.3.3.8.2 covers both mirror and selected-factor follow-ups. The subgroup formula and exact residual condition for arbitrary $J$ are proved here.