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 independent equations . Independence of the generator incidence vectors over leaves 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 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 that has fixed sign gives . 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.
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Fractional factorial design is a type of experimental design used in statistics and research to study the effects of multiple factors on a response variable while using a reduced number of experimental runs. This design is particularly useful when time, resources, or costs are limited, allowing researchers to efficiently assess the influence of several factors without conducting a full factorial experiment, which could involve an unmanageable number of trials.