Fractional factorial design

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 is chosen so as to exploit the sparsity-of-effects principle to expose information about the most important features of the problem studied, while using a fraction of the effort of a full factorial design in terms of experimental runs and resources.

Fractional designs are expressed using the notation l^k − p, where l is the number of levels of each factor investigated, k is the number of factors investigated, and p describes the size of the fraction of the full factorial used. Formally, p is the number of generators, assignments as to which effects or interactions are confounded, i.e., cannot be estimated independently of each other (see below). A design with p such generators is a 1/(l^p) fraction of the full factorial design.

For example, a 2^5 − 2 design is 1/4 of a two level, five factor factorial design. Rather than the 32 runs that would be required for the full 2⁵ factorial experiment, this experiment requires only eight runs.

In practice, one rarely encounters l > 2 levels in fractional factorial designs, since response surface methodology is a much more experimentally efficient way to determine the relationship between the experimental response and factors at multiple levels. In addition, the methodology to generate such designs for more than two levels is much more cumbersome.

The levels of a factor are commonly coded as +1 for the higher level, and −1 for the lower level. For a three-level factor, the intermediate value is coded as 0.

To save space, the points in a two-level factorial experiment are often abbreviated with strings of plus and minus signs. The strings have as many symbols as factors, and their values dictate the level of each factor: conventionally, ${\displaystyle -}$ for the first (or low) level, and ${\displaystyle +}$ for the second (or high) level. The points in this experiment can thus be represented as ${\displaystyle --}$, ${\displaystyle +-}$, ${\displaystyle -+}$, and ${\displaystyle ++}$.

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