--- title: "Aliasing (factorial experiments)" chunk: 4/7 source: "https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments)" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T09:48:49.670316+00:00" instance: "kb-cron" --- 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} .