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.
Definitions The factors in the design are allowed to have different numbers of levels, as in a 2 × 3 × 3 {\displaystyle 2\times 3\times 3} factorial experiment (an asymmetric or mixed-level experiment). Fix a fraction of a full factorial design. Let U {\displaystyle U} be a set of contrast vectors representing an effect (in particular, a main effect or interaction) in the full factorial design, and let U ~ {\displaystyle {\widetilde {U}}} consist of the restrictions of those vectors to the fraction. One says that the effect is
preserved in the fraction if U ~ {\displaystyle {\widetilde {U}}} consists of contrast vectors; completely lost in the fraction if U ~ {\displaystyle {\widetilde {U}}} consists of constant vectors, that is, vectors whose components are equal; and partly lost otherwise. Similarly, let U 1 {\displaystyle U_{1}} and U 2 {\displaystyle U_{2}} represent two effects and let U ~ 1 {\displaystyle {\widetilde {U}}_{1}} and U ~ 2 {\displaystyle {\widetilde {U}}_{2}} be their restrictions to the fraction. The two effects are said to be
unaliased in the fraction if each vector in U ~ 1 {\displaystyle {\widetilde {U}}_{1}} is orthogonal (perpendicular) to all the vectors in U ~ 2 {\displaystyle {\widetilde {U}}_{2}} , and vice versa; completely aliased in the fraction if each vector in U ~ 1 {\displaystyle {\widetilde {U}}_{1}} is a linear combination of vectors in U ~ 2 {\displaystyle {\widetilde {U}}_{2}} , and vice versa; and partly aliased otherwise. Finney and Bush introduced the terms "lost" and "preserved" in the sense used here. Despite the relatively long history of this topic, though, its terminology is not entirely standardized. The literature often describes lost effects as "not estimable" in a fraction, although estimation is not the only issue at stake. Rao referred to preserved effects as "measurable from" the fraction.
Resolution The extent of aliasing in a given fractional design is measured by the resolution of the fraction, a concept first defined by Box and Hunter:
A fractional factorial design is said to have resolution R {\displaystyle R} if every p {\displaystyle p} -factor effect is unaliased with every effect having fewer than R − p {\displaystyle R-p} factors. For example, a design has resolution R = 3 {\displaystyle R=3} if main effects are unaliased with each other (taking p = 1 ) {\displaystyle p=1)} , though it allows main effects to be aliased with two-factor interactions. This is typically the lowest resolution desired for a fraction. It is not hard to see that a fraction of resolution R {\displaystyle R} also has resolution R − 1 , R − 2 {\displaystyle R-1,R-2} , etc., so one usually speaks of the maximum resolution of a fraction. The number p {\displaystyle p} in the definition of resolution is usually understood to be a positive integer, but one may consider the effect of the grand mean to be the (unique) effect with no factors (i.e., with p = 0 {\displaystyle p=0} ). This effect sometimes appears in analysis of variance tables. It has one degree of freedom, and is represented by a single vector, a column of 1's. With this understanding, an effect is
preserved in a fraction if it is unaliased with the grand mean, and completely lost in a fraction if it is completely aliased with the grand mean. A fraction then has resolution R = 2 {\displaystyle R=2} if all main effects are preserved in the fraction. If it has resolution R = 3 {\displaystyle R=3} then two-factor interactions are also preserved.
Computation The definitions above require some computations with vectors, illustrated in the examples that follow. For certain fractional designs (the regular ones), a simple algebraic technique can be used that bypasses these procedures and gives a simple way to determine resolution. This is discussed below.
Examples
The 2 × 3 experiment The fraction {11, 12, 13} of this experiment was described above along with its restricted vectors. It is repeated here along with the complementary fraction {21, 22, 23}:
In both fractions, the A {\displaystyle A} effect is completely lost (the A {\displaystyle A} column is constant) while the B {\displaystyle B} and interaction effects are preserved (each 3 × 1 column is a contrast vector as its components sum to 0). In addition, the B {\displaystyle B} and interaction effects are completely aliased in each fraction: In the first fraction, the vectors for B {\displaystyle B} are linear combinations of those for A × B {\displaystyle A\times B} , viz.,
[ 1 − 1 0 ] = [ 1 − 1 0 ] + 0 [ 1 0 − 1 ] {\displaystyle {\begin{bmatrix}1\\-1\\0\end{bmatrix}}={\begin{bmatrix}1\\-1\\0\end{bmatrix}}+0{\begin{bmatrix}1\\0\\-1\end{bmatrix}}}
and
[ 0 1 − 1 ] = [ 1 0 − 1 ] − [ 1 − 1 0 ] {\displaystyle {\begin{bmatrix}0\\1\\-1\end{bmatrix}}={\begin{bmatrix}1\\0\\-1\end{bmatrix}}-{\begin{bmatrix}1\\-1\\0\end{bmatrix}}} ; in the reverse direction, the vectors for A × B {\displaystyle A\times B} can be written similarly in terms of those representing B {\displaystyle B} . The argument in the second fraction is analogous. These fractions have maximum resolution 1. The fact that the main effect of A {\displaystyle A} is lost makes both of these fractions undesirable in practice. It turns out that in a 2 × 3 experiment (or in any a × b experiment in which a and b are relatively prime) there is no fraction that preserves both main effects -- that is, no fraction has resolution 2.
The 2 × 2 × 2 (or 2³) experiment This is a "two-level" experiment with factors A , B {\displaystyle A,B} and C {\displaystyle C} . In such experiments the factor levels are often denoted by 0 and 1, for reasons explained below. A treatment combination is then denoted by an ordered triple such as 101 (more formally, (1, 0, 1), denoting the cell in which A {\displaystyle A} and C {\displaystyle C} are at level "1" and B {\displaystyle B} is at level "0"). The following table lists the eight cells of the full 2 × 2 × 2 factorial experiment, along with a contrast vector representing each effect, including a three-factor interaction:
Suppose that only the fraction consisting of the cells 000, 011, 101, and 110 is observed. The original contrast vectors, when restricted to these cells, are now 4 × 1, and can be seen by looking at just those four rows of the table. (Sorting the table on A B C {\displaystyle ABC} will bring these rows together and make the restricted contrast vectors easier to see. Sorting twice puts them at the top.) The following can be observed concerning these restricted vectors:
The A B C {\displaystyle ABC} column consists just of the constant 1 repeated four times. The other columns are contrast vectors, having two 1's and two −1s. The columns for C {\displaystyle C} and A B {\displaystyle AB} are equal. The same holds for A {\displaystyle A} and B C {\displaystyle BC} , and for B {\displaystyle B} and A C {\displaystyle AC} . All other pairs of columns are orthogonal. For example, the column for A {\displaystyle A} is orthogonal to that for B {\displaystyle B} , for C {\displaystyle C} , for A B {\displaystyle AB} , and for A C {\displaystyle AC} , as one can see by computing dot products. Thus
the A B C {\displaystyle ABC} interaction is completely lost in the fraction; the other effects are preserved in the fraction; the effects A {\displaystyle A} and B C {\displaystyle BC} are completely aliased with each other, as are B {\displaystyle B} and A C {\displaystyle AC} , and C {\displaystyle C} and A B {\displaystyle AB} . all other pairs of effects are unaliased. For example, A {\displaystyle A} is unaliased with both B {\displaystyle B} and C {\displaystyle C} and with the A B {\displaystyle AB} and A C {\displaystyle AC} interactions. Now suppose instead that the complementary fraction {001,010,100,111} is observed. The same effects as before are lost or preserved, and the same pairs of effects as before are mutually unaliased. Moreover, A {\displaystyle A} and B C {\displaystyle BC} are still aliased in this fraction since the A {\displaystyle A} and B C {\displaystyle BC} vectors are negatives of each other, and similarly for B {\displaystyle B} and A C {\displaystyle AC} and for C {\displaystyle C} and A B {\displaystyle AB} . Both of these fractions thus have maximum resolution 3.
Aliasing in regular fractions The two half-fractions of a 2 3 {\displaystyle 2^{3}} factorial experiment described above are of a special kind: Each is the solution set of a linear equation using modular arithmetic. More exactly:
The fraction { 000 , 011 , 101 , 110 } {\displaystyle \{000,011,101,110\}} is the solution set of the equation t 1 + t 2 + t 3 = 0 ( mod 2 ) {\displaystyle t_{1}+t_{2}+t_{3}=0{\pmod {2}}} . For example, 011 {\displaystyle 011} is a solution because 0 + 1 + 1 = 0 ( mod 2 ) {\displaystyle 0+1+1=0{\pmod {2}}} . Similarly, the fraction { 001 , 010 , 100 , 111 } {\displaystyle \{001,010,100,111\}} is the solution set to t 1 + t 2 + t 3 = 1 ( mod 2 ) {\displaystyle t_{1}+t_{2}+t_{3}=1{\pmod {2}}}
Such fractions are said to be regular. This idea applies to fractions of "classical" s k {\displaystyle s^{k}} designs, that is, s k {\displaystyle s^{k}} (or "symmetric") factorial designs in which the number of levels, s {\displaystyle s} , of each of the k {\displaystyle k} factors is a prime or the power of a prime.
A fractional factorial design is regular if it is the solution set of a system of one or more equations of the form
a 1 t 1 + ⋯ + a k t k = b , {\displaystyle a_{1}t_{1}+\cdots +a_{k}t_{k}=b,}
where the equation is modulo s {\displaystyle s} if s {\displaystyle s} is prime, and is in the finite field G F ( s ) {\displaystyle GF(s)} if s {\displaystyle s} is a power of a prime. Such equations are called defining equations of the fraction. When the defining equation or equations are homogeneous, the fraction is said to be principal. One defining equation yields a fraction of size s k − 1 {\displaystyle s^{k-1}} , two independent equations a fraction of size s k − 2 , {\displaystyle s^{k-2},} and so on. Such fractions are generally denoted as s k − r {\displaystyle s^{k-r}} designs. The half-fractions described above are 2 3 − 1 {\displaystyle 2^{3-1}} designs. The notation often includes the resolution as a subscript, in Roman numerals; the above fractions are thus 2 I I I 3 − 1 {\displaystyle 2_{III}^{3-1}} designs. Associated to each expression a 1 t 1 + ⋯ + a k t k {\displaystyle a_{1}t_{1}+\cdots +a_{k}t_{k}} is another, namely A 1 a 1 ⋯ A k a k {\displaystyle A_{1}^{a_{1}}\cdots A_{k}^{a_{k}}} , which rewrites the coefficients as exponents. Such expressions are called "words", a term borrowed from group theory. (In a particular example where k {\displaystyle k} is a specific number, the letters A , B , C … {\displaystyle A,B,C\ldots } are used, rather than A 1 , A 2 , A 3 … {\displaystyle A_{1},A_{2},A_{3}\ldots } .) These words can be multiplied and raised to powers, where the word I = A 1 0 ⋯ A k 0 {\displaystyle I=A_{1}^{0}\cdots A_{k}^{0}} acts as a multiplicative identity, and they thus form an abelian group G {\displaystyle \mathbb {G} } , known as the effects group. When s {\displaystyle s} is prime, one has W s = I {\displaystyle W^{s}=I} for every element (word) W ∈ G {\displaystyle W\in \mathbb {G} } ; something similar holds in the prime-power case. In 2 k {\displaystyle 2^{k}} factorial experiments, each element of G {\displaystyle \mathbb {G} } represents a main effect or interaction. In s k {\displaystyle s^{k}} experiments with s > 2 {\displaystyle s>2} , each one-letter word represents the main effect of that factor, while longer words represent components of interaction. An example below illustrates this with s = 3 {\displaystyle s=3} . To each defining expression (the left-hand side of a defining equation) corresponds a defining word. The defining words generate a subgroup H {\displaystyle \mathbb {H} } of G {\displaystyle \mathbb {G} } that is variously called the alias subgroup, the defining contrast subgroup, or simply the defining subgroup of the fraction. Each element of H {\displaystyle \mathbb {H} } is a defining word since it corresponds to a defining equation, as one can show. The effects represented by the defining words are completely lost in the fraction while all other effects are preserved. If H = { I , W 1 , … , W ℓ } {\displaystyle \mathbb {H} =\{I,W_{1},\ldots ,W_{\ell }\}} , say, then the equation
I = W 1 = ⋯ = W ℓ {\displaystyle I=W_{1}=\cdots =W_{\ell }}
is called the defining relation of the fraction. This relation is used to determine the aliasing structure of the fraction: If a given effect is represented by the word W {\displaystyle W} , then its aliases are computed by multiplying the defining relation by W {\displaystyle W} , viz.,
W = W W 1 = ⋯ = W W ℓ , {\displaystyle W=WW_{1}=\cdots =WW_{\ell },}
where the products W W i {\displaystyle WW_{i}} are then simplified. This relation indicates complete (not partial) aliasing, and W is unaliased with all other effects listed in G {\displaystyle \mathbb {G} } .
Example 1 In either of the 2 3 − 1 {\displaystyle 2^{3-1}} fractions described above, the defining word is A B C {\displaystyle ABC} , since the exponents on these letters are the coefficients of t 1 + t 2 + t 3 {\displaystyle t_{1}+t_{2}+t_{3}} . The A B C {\displaystyle ABC} effect is completely lost in the fraction, and the defining subgroup H {\displaystyle \mathbb {H} } is simply { I , A B C } {\displaystyle \{I,ABC\}} , since squaring does not generate new elements ( ( A B C ) 2 = A 2 B 2 C 2 = I ) {\displaystyle ((ABC)^{2}=A^{2}B^{2}C^{2}=I)} . The defining relation is thus
I = A B C {\displaystyle I=ABC} , and multiplying both sides by A {\displaystyle A} gives A = A 2 B C {\displaystyle A=A^{2}BC} ; which simplifies to
A = B C , {\displaystyle A=BC,}
the alias relation seen earlier. Similarly, B = A C {\displaystyle B=AC} and C = A B {\displaystyle C=AB} . Note that multiplying both sides of the defining relation by A B , A C {\displaystyle AB,AC} and B C {\displaystyle BC} does not give any new alias relations. For comparison, the 2 3 − 1 {\displaystyle 2^{3-1}} fraction with defining equation t 1 + t 2 = 0 ( mod 2 ) {\displaystyle t_{1}+t_{2}=0{\pmod {2}}} has the defining word A B {\displaystyle AB} (i.e., A 1 B 1 C 0 {\displaystyle A^{1}B^{1}C^{0}} ). The effect A B {\displaystyle AB} is completely lost, and the defining relation is I = A B {\displaystyle I=AB} . Multiplying this by A {\displaystyle A} , by C {\displaystyle C} , and by A C {\displaystyle AC} gives the alias relations A = B {\displaystyle A=B} , C = A B C {\displaystyle C=ABC} , and A C = B C {\displaystyle AC=BC} among the six remaining effects. This fraction only has resolution 2 since all effects (except A B {\displaystyle AB} ) are preserved but two main effects are aliased. Finally, solving the defining equation t 1 + t 2 = 0 ( mod 2 ) {\displaystyle t_{1}+t_{2}=0{\pmod {2}}} yields the fraction {000, 001, 110, 111}. One may verify all of this by sorting the table above on column A B {\displaystyle AB} .
The use of arithmetic modulo 2 explains why the factor levels in such designs are labeled 0 and 1.
Example 2 In a 3-level design, factor levels are denoted 0, 1 and 2, and arithmetic is modulo 3. If there are four factors, say A , B , C {\displaystyle A,B,C} and D {\displaystyle D} , the effects group G {\displaystyle \mathbb {G} } will have the relations
A 3 = B 3 = C 3 = D 3 = I . {\displaystyle A^{3}=B^{3}=C^{3}=D^{3}=I.}
From these it follows, for example, that D 4 = D {\displaystyle D^{4}=D} and D 6 = I {\displaystyle D^{6}=I} .
A defining equation such as t 1 + t 2 + t