Full factorials

Fix an integer $k\ge 1$ and write $[k]=\{1,\dots ,k\}$. A two-level full factorial design has run set ${\mathcal{X}}_k=\{-1,+1{\}}^{[k]},$ so every one of the $2^k$ sign combinations occurs once. For a run $x\in {\mathcal{X}}_k$ and a subset $A\subseteq [k]$, define its sign column by $x_A={\prod }_{j\in A}{x}_j,$ where the empty product is $x_{\mathrm{\varnothing }}=1$. The design matrix $X$ has one row for each run, one column for each subset of factors, and entry $X_{x,A}=x_A$. Any fixed ordering of the runs and subsets may be used.

The run set and subset index set both have $2^k$ elements: a subset $A$ corresponds to the sign vector whose $j$th coordinate is $+1$ exactly when $j\in A$. Thus $X$ is a square $2^k\times 2^k$ matrix.

For a real response table $y=(y_x{)}_{x\in {\mathcal{X}}_k}$, define its normalized factorial coefficient by ${\widehat{\beta }}_A(y)=2^{-k}{\sum }_{x\in {\mathcal{X}}_k}{x}_A{y}_x\mathrm{.}$ For nonempty $A$, put $G_{A,+}=\{x:x_A=+1\}$ and $G_{A,-}=\{x:x_A=-1\}$. Both sets are nonempty: the all-positive run lies in $G_{A,+}$, and flipping any coordinate in $A$ gives a run in $G_{A,-}$. Define the factorial effect on this response scale as ${\widehat{\tau }}_A(y)=\frac{1}{\mathrm{\mid }{G}_{A,+}\mathrm{\mid }}{\sum }_{x\in G_{A,+}}{y}_x-\frac{1}{\mathrm{\mid }{G}_{A,-}\mathrm{\mid }}{\sum }_{x\in G_{A,-}}{y}_x\mathrm{.}$

For the design in Two-level full factorial designs and coded effects, let $c_A=(x_A(x){)}_{x\in {\mathcal{X}}_k}$ be the column indexed by $A\subseteq [k]$. Then $⟨c_A,c_B⟩=\{\begin{array}{ll}{2}^k,& A=B,\\ 0,& A\mathrm{\ne }B\mathrm{.}\end{array}$ Thus the columns are pairwise orthogonal and $X^{\mathsf{T}}X=2^k{I}_{2^k},2^{-k}{X}^{\mathsf{T}}X=I_{2^k}\mathrm{.}$ This is a deterministic property of the design matrix.

Let $r\ge 1$ be an integer. For every run $x\in {\mathcal{X}}_k$ and replicate $s\in \{1,\dots ,r\}$, suppose $Y_{x,s}={\sum }_{B\subseteq [k]}{\beta }_B{x}_B+{ϵ}_{x,s},$ where the coefficients ${\beta }_B$ are fixed and the errors are mutually independent real random variables on one probability space, with $\mathbb{E}[{ϵ}_{x,s}^2]<\mathrm{\infty }$ and common variance ${\sigma }^2<\mathrm{\infty }$. Define ${\bar{Y}}_x=r^{-1}{\sum }_{s=1}^r{Y}_{x,s}$ and ${\widehat{\beta }}_A=2^{-k}{\sum }_{x\in {\mathcal{X}}_k}{x}_A{\bar{Y}}_x\mathrm{.}$ Then for every $A\subseteq [k]$, $\mathrm{Var}({\widehat{\beta }}_A)=\frac{{\sigma }^2}{r{2}^k}\mathrm{.}$ For nonempty $A$, define ${\widehat{\tau }}_A$ as the difference of the mean run responses ${\bar{Y}}_x$ over $G_{A,+}$ and $G_{A,-}$ from Two-level full factorial designs and coded effects. Then ${\widehat{\tau }}_A=2{\widehat{\beta }}_A,\mathrm{Var}({\widehat{\tau }}_A)=\frac{4{\sigma }^2}{r{2}^k}\mathrm{.}$

The variance calculation assumes equal replication at every run and independence across all run-replicate errors. It requires neither Gaussian errors nor zero interactions. A zero-mean assumption is additionally needed if ${\beta }_B$ is to denote coefficients of the mean response; it is not needed for these variance identities.