[1] In a "full" factorial experiment, the number of treatment combinations or cells (see below) can be very large.
[note 1] This necessitates limiting observations to a fraction (subset) of the treatment combinations.
This measures the degree to which the design avoids aliasing between main effects and important interactions.
[5] Fractional factorial experiments have long been a basic tool in agriculture,[6] food technology,[7][8] industry,[9][10][11] medicine and public health,[12][13] and the social and behavioral sciences.
[14] They are widely used in exploratory research,[15] particularly in screening experiments, which have applications in industry, drug design and genetics.
As noted below, the concept of aliasing may have influenced the identification of an analogous phenomenon in signal processing theory.
The presence or absence of a given effect in a given data set is tested by statistical methods, most commonly analysis of variance.
While aliasing has significant implications for estimation and hypothesis testing, it is fundamentally a combinatorial and algebraic phenomenon.
Construction and analysis of fractional designs thus rely heavily on algebraic methods.
In any design, full or fractional, the expected value of an observation in a given treatment combination is called a cell mean,[18] usually denoted using the Greek letter μ.
In the 2 × 3 experiment illustrated here, the expression is a contrast that compares the mean responses of the treatment combinations 11 and 12.
[note 2] These designations, which extend to arbitrary factorial experiments having three or more factors, depend on the pattern of coefficients, as explained elsewhere.
[21][22] Since it is the coefficients of these contrasts that carry the essential information, they are often displayed as column vectors.
factorial experiment described above are of a special kind: Each is the solution set of a linear equation using modular arithmetic.
[41][42][43][44][45] This relation is used to determine the aliasing structure of the fraction: If a given effect is represented by the word
Squaring the latter two expressions does the trick[47] and gives the alias relations Twelve other sets of three aliased effects are given by Wu and Hamada.
It is notable that the alias relations of the fraction depend only on the left-hand side of the defining equations, not on their constant terms.
[51] The length of a word in the effects group is defined to be the number of letters in its name, not counting repetition.
[note 8] Theorem — The maximum resolution of a regular fractional design is equal to the minimum length of a defining word.
While some methods have been developed to deal with aliasing in particular nonregular designs, no overall algebraic scheme has emerged.
Two important facts flow immediately from its definition: To state the next result, it is convenient to enumerate the factors of the experiment by 1 through
With additional assumptions, a stronger conclusion is possible: Theorem[60] — If a fraction has maximum strength
In the 12-run Plackett-Burman design described in the previous section, for example, with factors labeled
Only main effects and components of two-factor interactions are listed, including three pairs of aliases.
This poses a challenge of interpretation, since without more information or further assumptions it is impossible to determine which interaction is responsible for significance.
factorial experiment has 81 treatment combinations, but taking one observation on each of these would leave no degrees of freedom for error.
The fractional design also uses 81 observations, but on just 27 treatment combinations, in such a way that one can make inferences on main effects and on (most) two-factor interactions.
The first statistical use of the term "aliasing" in print is the 1945 paper by Finney,[67] which dealt with regular fractions with 2 or 3 levels.
The 1961 paper in which Box and Hunter introduced the concept of "resolution" dealt with regular two-level designs, but their initial definition[5] makes no reference to lengths of defining words and so can be understood rather generally.
Rao actually makes implicit use of resolution in his 1947 paper[68] introducing orthogonal arrays, reflected in an important parameter inequality that he develops.