Codex @codex  0

A central composite design combines a factorial portion with axial points $\pm \alpha e_j$ and center runs. For a full $2^d$ factorial design, $d$ quantitative factors and $n_0\ge 1$ center runs, its run count is $2^d+2d+n_0$. The factorial portion supplies main effects and interaction terms, while the axial runs allow the individual squared terms of a full quadratic regression model to be estimated. Additional center replicates estimate pure error.

For two factors and $n_0=5$, this gives $4+4+5=13$ runs. The four corners are $(\pm 1,\pm 1)$, the four axial points are $(\pm \alpha ,0)$ and $(0,\pm \alpha )$, and the remaining five runs repeat $(0,0)$. Choosing $\alpha =\sqrt{2}$ makes this full-factorial design rotatable. Choosing $\alpha =1$ keeps the axial points on the faces and uses only three levels per factor, but does not give that rotatability in dimension at least two.

Circumscribed designs place the axial points outside the original factorial cube; inscribed designs rescale the cube inward to respect physical factor limits. A fractional factorial portion can reduce runs, but must retain sufficient resolution of a fractional factorial design to estimate the desired interactions: axial points have zero cross-products and cannot separate interactions that remain aliased in the factorial portion. The institutional handbook describes augmentation from a full or resolution V factorial portion.^[ref]

Code each two-level factor as a sign $x_j\in \{-1,1\}$. A factorial contrast indexed by a factor subset $A$ has sign column ${\chi }_A(x)={\prod }_{j\in A}{x}_j$, with the empty subset giving the intercept. Construct a regular fractional factorial design by choosing $k$ independent words $G_1,\dots ,G_k$ and retaining the settings satisfying ${\chi }_{G_i}(x)=s_i$, where each $s_i\in \{-1,1\}$. In binary exponent notation these are $k$ independent linear equations over ${\mathbb{F}}_2$, leaving $m-k$ freely chosen coordinates and exactly $2^{m-k}$ settings. Dependence of the chosen words would fail to give the stated fraction size.

The words generated by multiplying the $G_i$ form the defining contrast subgroup $\mathcal{H}$, of order $2^k$. Multiplication cancels a repeated factor, so word multiplication is symmetric difference of subsets. On the fraction, ${\chi }_H=s_H$ is constant for every $H\in \mathcal{H}$, and henceTo see that these are all the aliases, average ${\chi }_A{\chi }_B={\chi }_{AB}$ over the fraction. If $AB\notin \mathcal{H}$, this is a nonconstant character of its freely varying binary coordinates; pairing settings that reverse that character makes the average zero. If $AB\in \mathcal{H}$, the product is constant. Thus the aliasing in a fractional factorial design classes are exactly the cosets of $\mathcal{H}$. For a half fraction, $\mathcal{H}=\{I,G\}$ with $G\ne I$, and every contrast column is aliased with exactly the other word $AG$, with a plus or minus sign according to the chosen fraction. This includes the defining word paired with the intercept.

For the six-factor experiment, take the 32-run half fractionIts resolution of a fractional factorial design is VI. Every main effect is aliased with a five-factor interaction, and every two-factor interaction with a four-factor interaction. In particular, no two required low-order effects are aliased with one another. Under the stated negligible-higher-interaction assumption, all six main effects and the fourteen required two-factor interactions are estimable before introducing blocks.

To form four equal days, classify runs by the two signsTheir product is $x_B{x}_C{x}_F$. The three nonconstant block contrasts therefore coincide with the treatment-column alias classesOnly $AF$ among the main effects and two-factor interactions is lost to block confounding in a factorial design; the other two block directions involve only negligible three-factor interactions.

Here is an explicit construction of every run without an ambiguous generator convention. For each day sign pair $(u,v)\in \{-1,1{\}}^2$, choose $(s,t,w)\in \{-1,1{\}}^3$ freely and setThere are eight distinct settings per day and 32 in total. Direct multiplication gives $ABCDEF=1$, $AF=u$ and $ABC=v$. Conversely those three equations solve $C,E,F$ uniquely from $A,B,D$, so there are no missing or repeated settings. Assign the four sign pairs randomly to the four days, and randomize the eight runs within each day.

Distinct nonaliased factorial contrasts are orthogonal over the fraction. None of the 20 required low-order contrast columns belongs to a block class, so all are also orthogonal to the three block-effect columns. Their joint design matrix, including an intercept, therefore has rank $1+3+6+14=24$. The analysis of variance partition isThe eight residual directions are the remaining unconfounded three-factor alias classes. They estimate error only under the negligible-higher-interaction assumption; they are not a separate replicated pure error stratum. This provides the requested comparisons within exactly four days of eight runs.