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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aliasing (factorial experiments) | 1/7 | https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments) | reference | science, encyclopedia | 2026-05-05T09:48:49.670316+00:00 | kb-cron |
In the statistical theory of factorial experiments, aliasing is the property of fractional factorial designs that makes some effects "aliased" with each other – that is, indistinguishable from each other. A primary goal of the theory of such designs is the control of aliasing so that important effects are not aliased with each other. In a "full" factorial experiment, the number of treatment combinations or cells (see below) can be very large. This necessitates limiting observations to a fraction (subset) of the treatment combinations. Aliasing is an automatic and unavoidable result of observing such a fraction. The aliasing properties of a design are often summarized by giving its resolution. This measures the degree to which the design avoids aliasing between main effects and important interactions. Fractional factorial experiments have long been a basic tool in agriculture, food technology, industry, medicine and public health, and the social and behavioral sciences. They are widely used in exploratory research, particularly in screening experiments, which have applications in industry, drug design and genetics. In all such cases, a crucial step in designing such an experiment is deciding on the desired aliasing pattern, or at least the desired resolution. As noted below, the concept of aliasing may have influenced the identification of an analogous phenomenon in signal processing theory.
== Overview == Associated with a factorial experiment is a collection of effects. Each factor determines a main effect, and each set of two or more factors determines an interaction effect (or simply an interaction) between those factors. Each effect is defined by a set of relations between cell means, as described below. In a fractional factorial design, effects are defined by restricting these relations to the cells in the fraction. It is when the restricted relations for two different effects turn out to be the same that the effects are said to be aliased. 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. The definition of a fractional design is sometimes broadened to allow multiple observations of some or all treatment combinations – a multisubset of all treatment combinations. A fraction that is a subset (that is, where treatment combinations are not repeated) is called simple. The theory described below applies to simple fractions.
== Contrasts and effects ==
In any design, full or fractional, the expected value of an observation in a given treatment combination is called a cell mean, usually denoted using the Greek letter μ. (The term cell is borrowed from its use in tables of data.) A contrast in cell means is a linear combination of cell means in which the coefficients sum to 0. In the 2 × 3 experiment illustrated here, the expression
μ
11
−
μ
12
{\displaystyle \mu _{11}-\mu _{12}}
is a contrast that compares the mean responses of the treatment combinations 11 and 12. (The coefficients here are 1 and –1.) The effects in a factorial experiment are expressed in terms of contrasts. In the above example, the contrast
μ
11
+
μ
12
+
μ
13
−
μ
21
−
μ
22
−
μ
23
{\displaystyle \mu _{11}+\mu _{12}+\mu _{13}-\mu _{21}-\mu _{22}-\mu _{23}}
is said to belong to the main effect of factor A as it contrasts the responses to the "1" level of factor
A
{\displaystyle A}
with those for the "2" level. The main effect of A is said to be absent if this expression equals 0. Similarly,
μ
11
+
μ
21
−
μ
12
−
μ
22
{\displaystyle \mu _{11}+\mu _{21}-\mu _{12}-\mu _{22}}
and
μ
11
+
μ
21
−
μ
13
−
μ
23
{\displaystyle \mu _{11}+\mu _{21}-\mu _{13}-\mu _{23}}
are contrasts belonging to the main effect of factor B. On the other hand, the contrasts
μ
11
−
μ
12
−
μ
21
+
μ
22
{\displaystyle \mu _{11}-\mu _{12}-\mu _{21}+\mu _{22}}
and
μ
11
−
μ
13
−
μ
21
+
μ
23
{\displaystyle \mu _{11}-\mu _{13}-\mu _{21}+\mu _{23}}
belong to the interaction of A and B; setting them equal to 0 expresses the lack of interaction. These designations, which extend to arbitrary factorial experiments having three or more factors, depend on the pattern of coefficients, as explained elsewhere. Since it is the coefficients of these contrasts that carry the essential information, they are often displayed as column vectors. For the example above, such a table might look like this:
The columns of such a table are called contrast vectors: their components add up to 0. While there are in general many possible choices of columns to represent a given effect, the number of such columns — the degrees of freedom of the effect — is fixed and is given by a well-known formula. In the 2 × 3 example above, the degrees of freedom for
A
,
B
{\displaystyle A,B}
, and the
A
×
B
{\displaystyle A\times B}
interaction are 1, 2 and 2, respectively. In a fractional factorial experiment, the contrast vectors belonging to a given effect are restricted to the treatment combinations in the fraction. Thus, in the half-fraction {11, 12, 13} in the 2 × 3 example, the three effects may be represented by the column vectors in the following table:
The consequence of this truncation — aliasing — is described below.