Two-level full factorial designs and coded effects

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 ${\widehat{\tau }}_A(y)=\frac{1}{\mathrm{\mid }{G}_{A,+}\mathrm{\mid }}{\sum }_{x\in G_{A,+}}{y}_x-\frac{1}{\mathrm{\mid }{G}_{A,-}\mathrm{\mid }}{\sum }_{x\in G_{A,-}}{y}_x\mathrm{.}$