Factorial design

A factorial design includes every combination of specified factor levels. Balanced replication permits separate estimation of marginal effects and interaction terms. The factor allocation, not the number of subsamples, determines the appropriate error term for each effect.

A main effect compares the mean response at different levels of one factor after averaging over the other factors according to the design. With sign coding, its two-level difference is twice the singleton factorial contrast coefficient. In a fractional factorial design, that coefficient can include aliased higher-order effects.

In a two-level factorial design, a factor subset defines a sign column. The coefficient of this column is its response contrast divided by the number of runs when the columns are orthogonal. A singleton describes a main effect and a pair describes a two-factor interaction. On the full cube, distinct columns have zero inner product: summing over any coordinate in their symmetric difference cancels the terms.

A fractional factorial design uses a selected subset of the full factor-level combinations. In a regular two-level fraction, encode levels by signs and impose $k$ independent equations ${\prod }_{j\in G_i}{x}_j=s_i$. Independence of the generator incidence vectors over ${\mathbb{F}}_2$ leaves $2^{m-k}$ runs. Reduced run count creates aliasing in a fractional factorial design, which must be evaluated against the effects scientifically required.

The resolution of a regular fraction is the shortest nonidentity word in its defining contrast subgroup. Two effects alias only when their word product belongs to that subgroup. Thus resolution VI separates all main effects and two-factor interactions from one another and from three-factor interactions, though some three-factor interactions alias in complementary pairs.

Factor subsets multiply by symmetric difference; their sign columns multiply pointwise. The subgroup generated by independent defining words has $2^k$ elements. On a selected fractional factorial design, every word in this subgroup is constant, with sign determined by the chosen generator signs. The cosets classify identical or opposite contrast columns.

Two factorial contrasts are aliased if their columns are proportional on the selected fractional factorial design. A word $G$ that has fixed sign $s_G$ gives ${\chi }_{AG}=s_G{\chi }_A$. Conversely, averaging a character over the fraction vanishes unless its word belongs to the defining contrast subgroup, so every alias class is exactly a subgroup coset. In a half fraction each class has two distinct words; their separate coefficients cannot be identified without assumptions or additional runs.