Fractional factorial designs

In statistics, fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design. The subset or fraction is chosen so as to exploit the sparsity-of-effects principle to access information about the most important features of the problem studied, while using considerably fewer resources than a full factorial design.

Notation
For example, using the notation of factorial designs, a 25-2 design is a five factor two level fractional factorial design in 8 runs, rather than the 32 runs that would be required in a full factorial experiment. As a result of doing less experiments, the amount of information that is generated from a fractional factorial experiment is less than from a full factorial experiment, but the practical difference may not be great.

General information
A fractional factorial experiment is generated from a factorial experiment. For example, the 2undefined can be generated by using the full factorial experiment involving three factors (say A, B, and C) and then adding two extra factors D and E generated by D=A*B and E=A*C, where the low level of a factor is given by $$-1$$ and the high level of a factor is given by $$+1$$.

An important characteristic of a fractional design is the defining relation, which gives the set of interaction columns equal to a column of plus signs, denoted by 1. For the above example, since D=AB and E=AC, then ABD and ACE are both columns of plus signs, and consequently so is BDCE. In this case the defining relation of the fractional design is 1=ABD=ACE=BCDE. The defining relation allows the alias pattern of the design to be determined.

The resolution of the design is the minimum word length in the defining relation excluding 1, in this case III. The most important fractional designs are those of resolution III, IV, and V. Resolution III designs allow the estimation of the main effects, but they may be biased by two-factor interactions. Resolution IV designs allow the estimation of the main effects unbiased by two-factor interactions, but the two-factor interactions may not be estimable. Resolution V designs allow the estimation of main effects and two-factor interactions. Higher-order interactions are usually assumed to be negligible.

Fractional factorial experiments for three-level factors are possible, but their properties are less satisfactory than for two-level factors.