In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whoseexperimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully crossed design. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable.
For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a 2×2 factorial design.
If the number of combinations in a full factorial design is too high to be logistically feasible, a fractional factorial design may be done, in which some of the possible combinations (usually at least half) are omitted.
The notation used to denote factorial experiments conveys a lot of information. When a design is denoted a 23 factorial, this identifies the number of factors (3); how many levels each factor has (2); and how many experimental conditions there are in the design (23=8). Similarly, a 25 design has five factors, each with two levels, and 25=32 experimental conditions; and a 32 design has two factors, each with three levels, and 32=9 experimental conditions. Factorial experiments can involve factors with different numbers of levels. A 243 design has five factors, four with two levels and one with three levels, and has 16 X 3=48 experimental conditions.
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, for the first (or low) level, and for the second (or high) level. The points in this experiment can thus be represented as , , , and .
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.
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