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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Fractional factorial design | 1/2 | https://en.wikipedia.org/wiki/Fractional_factorial_design | reference | science, encyclopedia | 2026-05-05T09:50:18.197962+00:00 | kb-cron |
In statistics, a fractional factorial design is a way to conduct experiments with fewer experimental runs than a full factorial design. Instead of testing every single combination of factors, it tests only a carefully selected portion. This "fraction" of the full design is chosen to reveal the most important information about the system being studied (sparsity-of-effects principle), while significantly reducing the number of runs required. It is based on the idea that many tests in a full factorial design can be redundant. However, this reduction in runs comes at the cost of potentially more complex analysis, as some effects can become intertwined, making it impossible to isolate their individual influences. Therefore, choosing which combinations to test in a fractional factorial design must be done carefully.
== History == Fractional factorial design was introduced by British statistician David John Finney in 1945, extending previous work by Ronald Fisher on the full factorial experiment at Rothamsted Experimental Station. Developed originally for agricultural applications, it has since been applied to other areas of engineering, science, and business.
== Basic working principle == Similar to a full factorial experiment, a fractional factorial experiment investigates the effects of independent variables, known as factors, on a response variable. Each factor is investigated at different values, known as levels. The response variable is measured using a combination of factors at different levels, and each unique combination is known as a run. To reduce the number of runs in comparison to a full factorial, the experiments are designed to confound different effects and interactions, so that their impacts cannot be distinguished. If higher-order interactions between main effects are negligible, it can be considered a reasonable method to study the main effects. This is the sparsity of effects principle. Confounding is controlled by a systematic selection of runs from a full-factorial table.
== Notation == Fractional designs are expressed using the notation lk − p, where l is the number of levels of each factor, k is the number of factors, and p describes the size of the fraction of the full factorial used. Formally, p is the number of generators; relationships that determine the intentionally confounded effects that reduce the number of runs needed. Each generator halves the number of runs required. A design with p such generators is a 1/(lp)=l−p fraction of the full factorial design. For example, a 25 − 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. With two generators, the number of experiments has been halved twice. In practice, one rarely encounters l > 2 levels in fractional factorial designs as the methodology to generate such designs for more than two levels is much more cumbersome. In cases requiring 3 levels for each factor, potential fractional designs to pursue are Latin squares, mutually orthogonal Latin squares, and Taguchi methods. Response surface methodology can also be a much more experimentally efficient way to determine the relationship between the experimental response and factors at multiple levels, but it requires that the levels are continuous. In determining whether more than two levels are needed, experimenters should consider whether they expect the outcome to be nonlinear with the addition of a third level. Another consideration is the number of factors, which can significantly change the experimental labor demand. 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 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 a two-level experiment with two factors can thus be represented as
−
−
{\displaystyle --}
,
+
−
{\displaystyle +-}
,
−
+
{\displaystyle -+}
, and
+
+
{\displaystyle ++}
. The factorial points can also be abbreviated by (1), a, b, and ab, where the presence of a letter indicates that the specified factor is at its high (or second) level and the absence of a letter indicates that the specified factor is at its low (or first) level (for example, "a" indicates that factor A is on its high setting, while all other factors are at their low (or first) setting). (1) is used to indicate that all factors are at their lowest (or first) values. Factorial points are typically arranged in a table using Yates’ standard order: 1, a, b, ab, c, ac, bc, abc, which is created when the level of the first factor alternates with each run.